This is an archive of email messages concerning the Geometric Langlands Seminar for 2012-13.

As usual, the seminar will meet on Mondays and/or Thursdays in room E206 at 4:30 p.m. First meetings: October 4 (Thursday) and October 8 (Monday). The seminar will begin with a talk by Beilinson followed by a series of talks by Nick Rozenblyum (NWU). The latter will be devoted to a new approach to the foundations of D-module theory developed by Gaitsgory and Rozenblyum. Beilinson's talk is intended to be a kind of introduction to those by Rozenblyum.

Thursday (October 4), 4:30 p.m, room E 206. Alexander Beilinson. Crystals, D-modules, and derived algebraic geometry. Abstract This is an introduction to a series of talks of Nick Rosenblum on his foundational work with Dennis Gaitsgory that establishes the basic D-module functoriality in the context of derived algebraic geometry (hence for arbitrary singular algebraic varieties) over a field of characteristic 0. I will discuss the notion of crystals and de Rham coefficients that goes back to Grothendieck, the derived D-module functoriality for smooth varieties (due to Bernstein and Kashiwara), and some basic ideas of the Gaitsgory-Rosenblum theory. No previous knowledge of the above subjects is needed.

Monday (October 8), 4:30 p.m, room E 206. Nick Rozenblyum (NWU). Duality and D-modules via derived algebraic geometry. Abstract I will describe joint work with D. Gaitsgory formulating the theory of D-modules using derived algebraic geometry. I will begin with an overview of Grothendieck-Serre duality in derived algebraic geometry via the formalism of ind-coherent sheaves. The theory of D-modules will be built as an extension of this theory. A key player in the story is the deRham stack, introduced by Simpson in the context of nonabelian Hodge theory. It is a convenient formulation of Gorthendieck's theory of crystals in characteristic 0. I will explain its construction and basic properties. The category of D-modules is defined as sheaves in the deRham stack. This construction has a number of benefits; for instance, Kashiwara's Lemma and h-descent are easy consequences of the definition. I will also explain how this approach compares to more familiar definitions.

Thursday (October 11), 4:30 p.m, room E 206. Nick Rozenblyum (NWU). Duality and D-modules via derived algebraic geometry. II Abstract > I will describe joint work with D. Gaitsgory formulating the theory of > D-modules using derived algebraic geometry. I will begin with an overview > of Grothendieck-Serre duality in derived algebraic geometry via the > formalism of ind-coherent sheaves. The theory of D-modules will be built > as an extension of this theory. > > A key player in the story is the deRham stack, introduced by Simpson in > the context of nonabelian Hodge theory. It is a convenient formulation of > Gorthendieck's theory of crystals in characteristic 0. I will explain its > construction and basic properties. The category of D-modules is defined > as sheaves in the deRham stack. This construction has a number of > benefits; for instance, Kashiwara's Lemma and h-descent are easy > consequences of the definition. I will also explain how this approach > compares to more familiar definitions. > > > > > > > >

No seminar on Monday. Nick will continue next Thursday: Thursday (October 18), 4:30 p.m, room E 206. Nick Rozenblyum (NWU). Duality and D-modules via derived algebraic geometry. III > Abstract > > I will describe joint work with D. Gaitsgory formulating the theory of > D-modules using derived algebraic geometry. I will begin with an > overview of Grothendieck-Serre duality in derived algebraic geometry via > the formalism of ind-coherent sheaves. The theory of D-modules will be > built as an extension of this theory. > > A key player in the story is the deRham stack, introduced by Simpson in > the context of nonabelian Hodge theory. It is a convenient formulation > of Grothendieck's theory of crystals in characteristic 0. I will explain > its construction and basic properties. The category of D-modules is > defined as sheaves in the deRham stack. This construction has a number of > benefits; for instance, Kashiwara's Lemma and h-descent are easy > consequences of the definition. I will also explain how this approach > compares to more familiar definitions.

Thursday (October 18), 4:30 p.m, room E 206. Nick Rozenblyum (NWU). Duality and D-modules via derived algebraic geometry. III > Abstract > > I will describe joint work with D. Gaitsgory formulating the theory of > D-modules using derived algebraic geometry. I will begin with an > overview of Grothendieck-Serre duality in derived algebraic geometry via > the formalism of ind-coherent sheaves. The theory of D-modules will be > built as an extension of this theory. > > A key player in the story is the deRham stack, introduced by Simpson in > the context of nonabelian Hodge theory. It is a convenient formulation > of Grothendieck's theory of crystals in characteristic 0. I will explain > its construction and basic properties. The category of D-modules is > defined as sheaves in the deRham stack. This construction has a number of > benefits; for instance, Kashiwara's Lemma and h-descent are easy > consequences of the definition. I will also explain how this approach > compares to more familiar definitions.

Monday (October 22), 4:30 p.m, room E 206. Nick Rozenblyum (NWU). Duality and D-modules via derived algebraic geometry. IV.

No seminar until Mitya Boyarchenko's talk on Nov 8. (So we have plenty of time to think about Nick's talks!) Please note Sarnak's Albert lectures on Oct 31 (Wednesday), Nov 1 (Thursday), Nov 2 (Friday), see http://math.uchicago.edu/research/abstracts/albert_abstracts.shtml

Peter Sarnak's Albert lectures have been moved to Nov 7-9, see http://math.uchicago.edu/research/abstracts/albert_abstracts.shtml As announced before, Mitya Boyarchenko will speak on Nov 8 (Thursday). On Nov 12 (Monday) Jonathan Barlev will begin his series of talks on the spaces of rational maps.

No seminar tomorrow (Monday). The title&abstract of Mitya Boyarchenko's Thursday talks are below. Please note Sarnak's Albert lectures on "Randomness in Number Theory" on Wednesday, Thursday, and Friday, see http://math.uchicago.edu/research/abstracts/albert_abstracts.shtml ************* Thursday (Nov 8), 4:30 p.m, room E 206. Mitya Boyarchenko (University of Michigan) New geometric structures in the local Langlands program. Abstract The problem of explicitly constructing the local Langlands correspondence for GL_n(K), where K is a p-adic field, contains as an important special case the problem of constructing automorphic induction (or "twisted parabolic induction") from certain 1-dimensional characters of L^* (where L is a given Galois extension of K of degree n) to irreducible supercuspidal representations of GL_n(K). Already in 1979 Lusztig proposed a very elegant, but still conjectural, geometric construction of twisted parabolic induction for unramified maximal tori in arbitrary reductive p-adic groups. An analysis of Lusztig's construction and of the Lubin-Tate tower of K leads to interesting new varieties that provide an analogue of Deligne-Lusztig theory for certain families of unipotent groups over finite fields. I will describe the known examples of this phenomenon and their relationship to the local Langlands correspondence. All the necessary background will be provided. Part of the talk will be based on joint work with Jared Weinstein (Boston University).

Thursday (Nov 8), 4:30 p.m, room E 206. Mitya Boyarchenko (University of Michigan) New geometric structures in the local Langlands program. (Sarnak's second Albert lecture is at 3 p.m., so you can easily attend both lectures). ************* Abstract The problem of explicitly constructing the local Langlands correspondence for GL_n(K), where K is a p-adic field, contains as an important special case the problem of constructing automorphic induction (or "twisted parabolic induction") from certain 1-dimensional characters of L^* (where L is a given Galois extension of K of degree n) to irreducible supercuspidal representations of GL_n(K). Already in 1979 Lusztig proposed a very elegant, but still conjectural, geometric construction of twisted parabolic induction for unramified maximal tori in arbitrary reductive p-adic groups. An analysis of Lusztig's construction and of the Lubin-Tate tower of K leads to interesting new varieties that provide an analogue of Deligne-Lusztig theory for certain families of unipotent groups over finite fields. I will describe the known examples of this phenomenon and their relationship to the local Langlands correspondence. All the necessary background will be provided. Part of the talk will be based on joint work with Jared Weinstein (Boston University).

Monday (Nov 12), 4:30 p.m, room E 206. Jonathan Barlev. Models for spaces of rational maps Abstract I will discuss the equivalence between three different models for spaces of rational maps in algebraic geometry. In particular, I will explain the relation between spaces of quasi-maps and the model for the space of rational maps which Gaitsgory uses in his recent contractibility theorem. Categories of D-modules on spaces of rational maps arise in the context of the geometric Langlands program. However, as such spaces are not representable by (ind-)schemes, the construction of such categories relies on the general theory presented in Nick Rozenblyum's talks. I will explain how each of the different models for these spaces exhibit different properties of their categories of D-modules.

Thursday (Nov 15), 4:30 p.m, room E 206. Jonathan Barlev. Models for spaces of rational maps. II. Abstract I will discuss the equivalence between three different models for spaces of rational maps in algebraic geometry. In particular, I will explain the relation between spaces of quasi-maps and the model for the space of rational maps which Gaitsgory uses in his recent contractibility theorem. Categories of D-modules on spaces of rational maps arise in the context of the geometric Langlands program. However, as such spaces are not representable by (ind-)schemes, the construction of such categories relies on the general theory presented in Nick Rozenblyum's talks. I will explain how each of the different models for these spaces exhibit different properties of their categories of D-modules.

No seminar until Thanksgiving. John Francis (NWU) will give his first talk after Thanksgiving (probably on Thursday). ******* Attached is a proof of the contractibility statement in the classical topology (over the complex numbers). Please check. I make there two additional assumptions, which are not really necessary: (a) I assume that the target variety equals {affine space}-{hypersurface}. This implies the statement in the more general setting considered at the seminar (when the target variety is connected and locally isomorphic to an affine space). One uses here the following fact: if a topological space is covered by open sets so that all finite intersections of these subsets are contractible then the whole space is contractible. (b) I assume that K is the field of rational functions. This immediately implies the statement for any finite extension of K. To see this, note that if K' is an extension of K of degree n then (K')^m =K^{mn}. The scientific name for this is "Weil restriction of scalars".

**Attachment:
Contractibility.pdf**

No seminar on Monday (Nov 26). Thursday (Nov 29), 4:30 p.m, room E 206. John Francis (NWU). Factorization homology of topological manifolds. Abstract Factorization homology, a.k.a. topological chiral homology, of Lurie, is a type of homology theory for n-manifolds whose system of coefficients is given by an n-disk, or E_n-, algebra. It was formulated as a topological analogue of the homology of the algebro-geometric factorization algebras of Beilinson & Drinfeld, and it generalizes previous work in topology of Salvatore and Segal. Factorization homology is characterized by a generalization of the Eilenberg-Steenrod axioms. I'll use this to give a short proof of nonabelian Poincare duality and then discuss other calculations, including factorization homology with coefficients in enveloping algebras of Lie algebras -- a topological analogue of Beilinson & Drinfeld's description of chiral homology of chiral enveloping algebras.

Thursday (Nov 29), 4:30 p.m, room E 206. John Francis (NWU). Factorization homology of topological manifolds. Abstract Factorization homology, a.k.a. topological chiral homology, of Lurie, is a type of homology theory for n-manifolds whose system of coefficients is given by an n-disk, or E_n-, algebra. It was formulated as a topological analogue of the homology of the algebro-geometric factorization algebras of Beilinson & Drinfeld, and it generalizes previous work in topology of Salvatore and Segal. Factorization homology is characterized by a generalization of the Eilenberg-Steenrod axioms. I'll use this to give a short proof of nonabelian Poincare duality and then discuss other calculations, including factorization homology with coefficients in enveloping algebras of Lie algebras -- a topological analogue of Beilinson & Drinfeld's description of chiral homology of chiral enveloping algebras.

Thursday (Dec 6), 4:30 p.m, room E 206. John Francis (NWU). Factorization homology of topological manifolds.II. Abstract Factorization homology, a.k.a. topological chiral homology, of Lurie, is a type of homology theory for n-manifolds whose system of coefficients is given by an n-disk, or E_n-, algebra. It was formulated as a topological analogue of the homology of the algebro-geometric factorization algebras of Beilinson & Drinfeld, and it generalizes previous work in topology of Salvatore and Segal. Factorization homology is characterized by a generalization of the Eilenberg-Steenrod axioms. I'll use this to give a short proof of nonabelian Poincare duality and then discuss other calculations, including factorization homology with coefficients in enveloping algebras of Lie algebras -- a topological analogue of Beilinson & Drinfeld's description of chiral homology of chiral enveloping algebras.

Thursday (Dec 6), 4:30 p.m, room E 206. John Francis (NWU). Factorization homology of topological manifolds.II. Abstract Factorization homology, a.k.a. topological chiral homology, of Lurie, is a type of homology theory for n-manifolds whose system of coefficients is given by an n-disk, or E_n-, algebra. It was formulated as a topological analogue of the homology of the algebro-geometric factorization algebras of Beilinson & Drinfeld, and it generalizes previous work in topology of Salvatore and Segal. Factorization homology is characterized by a generalization of the Eilenberg-Steenrod axioms. I'll use this to give a short proof of nonabelian Poincare duality and then discuss other calculations, including factorization homology with coefficients in enveloping algebras of Lie algebras -- a topological analogue of Beilinson & Drinfeld's description of chiral homology of chiral enveloping algebras.

No more meetings of the Geometric Langlands seminar this quarter.

The geometric Langlands seminar does not meet this week. Next Monday (January 14) Beilinson will give an introductory talk on topological cyclic homology, to be followed by T.Goodwillie's talk on the same subject on Thursday January 17. On Jan 21 and 24 Nick Rozenblyum will explain Kevin Costello's approach to the Witten genus. Next speakers: Bhargav Bhatt (Jan 28), Jared Weinstein: February 4,5,7.

Monday (January 14), 4:30 p.m, room E 206. Alexander Beilinson. An introduction to Goodwillie's talk on topological cyclic homology. [Presumably, in his Thursday talk Goodwillie will explain several ways of looking at topological cyclic homology.] Abstract My talk is intended to serve as an introduction to T.Goodwillie's talk on Thursday January 17. No prior knowledge of the subject is assumed. A recent article by Bloch, Esnault, and Kerz about p-adic deformations of algebraic cycles uses topological cyclic homology (TCH) as a principal, if hidden, tool. I will try to explain the main features of TCH theory and discuss the relation of TCH to classical cyclic homology as motivated by the Bloch-Esnault-Kerz work and explained to me by V.Angeltveit and N.Rozenblum. No prior knowledge of the subject is assumed.

Below are: (i) information on Goodwillie's Thursday talk; (ii) a link to an article by Peter May. ******* Thursday (January 17), 4:30 p.m, room E 206. Thomas Goodwillie (Brown University). On topological cyclic homology. Abstract The cyclotomic trace is an important map from algebraic K-theory whose target is a kind of topological cyclic homology. Rationally it can be defined purely algebraically, but integrally its definition uses equivariant stable homotopy theory. I will look at this topic from several points of view. In particular it is interesting to look at the cyclotomic trace in the case of Waldhausen K-theory, where it leads to equivariant constructions on loops in a manifold. ****** Here is the link to Peter May's notes for a 1997 talk: http://www.math.uchicago.edu/~may/TALKS/THHTC.pdf The talk was before anyone was using orthogonal spectra (although in fact Peter May first defined them in a 1980 paper).

Monday (January 21), 4:30 p.m, room E 206. Nick Rozenblyum (NWU). Nonabelian duality, Feynman integration and the Witten genus. Abstract We will describe a formal version of nonabelian duality in derived algebraic geometry, using the Beilinson-Drinfeld theory of chiral algebras. This provides a local-to-global approach to the study of a certain class of moduli spaces -- such as mapping spaces, the moduli space of curves and the moduli space of principal G-bundles. In this context, we will describe a quantization procedure and the associated theory of Feynman integration. As an application, we obtain an algebro-geometric version of Costello's construction of the Witten genus.

Thursday (January 24), 4:30 p.m, room E 206. Nick Rozenblyum (NWU). Nonabelian duality, Feynman integration and the Witten genus. II. Abstract We will describe a formal version of nonabelian duality in derived algebraic geometry, using the Beilinson-Drinfeld theory of chiral algebras. This provides a local-to-global approach to the study of a certain class of moduli spaces -- such as mapping spaces, the moduli space of curves and the moduli space of principal G-bundles. In this context, we will describe a quantization procedure and the associated theory of Feynman integration. As an application, we obtain an algebro-geometric version of Costello's construction of the Witten genus.

Monday (Jan 28), 4:30 p.m, room E 206. Nick Rozenblyum (NWU). Nonabelian duality, Feynman integration and the Witten genus. III. Abstract We will describe a formal version of nonabelian duality in derived algebraic geometry, using the Beilinson-Drinfeld theory of chiral algebras. This provides a local-to-global approach to the study of a certain class of moduli spaces -- such as mapping spaces, the moduli space of curves and the moduli space of principal G-bundles. In this context, we will describe a quantization procedure and the associated theory of Feynman integration. As an application, we obtain an algebro-geometric version of Costello's construction of the Witten genus.

I am resending this message, just in case. ******* Monday (Jan 28), 4:30 p.m, room E 206. Nick Rozenblyum (NWU). Nonabelian duality, Feynman integration and the Witten genus. III. Abstract We will describe a formal version of nonabelian duality in derived algebraic geometry, using the Beilinson-Drinfeld theory of chiral algebras. This provides a local-to-global approach to the study of a certain class of moduli spaces -- such as mapping spaces, the moduli space of curves and the moduli space of principal G-bundles. In this context, we will describe a quantization procedure and the associated theory of Feynman integration. As an application, we obtain an algebro-geometric version of Costello's construction of the Witten genus.

No seminar on Thursday this week. ****** Next week Jared Weinstein (Boston University) will speak at the Langlands seminar on Monday and Thursday. He will also speak at the Number Theory seminar on Tuesday. To the best of my knowledge, his talks will be related to the following works: http://arxiv.org/abs/1207.6424 http://arxiv.org/abs/1211.6357 More details will be announced later.

Monday (Feb 4), 4:30 p.m, room E 206. Jared Weinstein (Boston University). Moduli of formal groups with infinite level structure. I. Prof. Weinstein will also speak at the Langlands seminar on Thursday and at the Number Theory seminar on Tuesday, see http://www.math.uchicago.edu/~reduzzi/NTseminar/ Abstract A formal group is a bi-variate formal power series which mimics the behavior of an abelian group. More generally one can talk about formal $O$-modules, where $O$ is any ring. Suppose $K$ is a local nonarchimedean field with ring of integers $O$ and residue field $k$. For each $n$, there is up to isomorphism a unique formal $O$-module $H$ of height $n$ over the algebraic closure of $k$. In 1974, Drinfeld introduced an ascending family of regular local rings $A_m$ which parameterize deformations of $H$ with level $m$ structure. These rings are implicated in the proof by Harris and Taylor of the local Langlands correspondence for GL_n(K). In this talk, we will discuss the ring $A$ obtained by completing the union of the $A_m$. It turns out that this ring has a very explicit description -- despite not being noetherian, it is somehow simpler than any of the finite level rings $A_m$. These observations generalize to other deformation spaces of p-divisible groups (joint work with Peter Scholze), and suggest the usefulness of working at infinite level in the context of other arithmetic moduli problems.

Thursday (Feb 7), 4:30 p.m, room E 206. Jared Weinstein (Boston University). Moduli of formal groups with infinite level structure. II. Abstract A formal group is a bi-variate formal power series which mimics the behavior of an abelian group. More generally one can talk about formal $O$-modules, where $O$ is any ring. Suppose $K$ is a local nonarchimedean field with ring of integers $O$ and residue field $k$. For each $n$, there is up to isomorphism a unique formal $O$-module $H$ of height $n$ over the algebraic closure of $k$. In 1974, Drinfeld introduced an ascending family of regular local rings $A_m$ which parameterize deformations of $H$ with level $m$ structure. These rings are implicated in the proof by Harris and Taylor of the local Langlands correspondence for GL_n(K). In this talk, we will discuss the ring $A$ obtained by completing the union of the $A_m$. It turns out that this ring has a very explicit description -- despite not being noetherian, it is somehow simpler than any of the finite level rings $A_m$. These observations generalize to other deformation spaces of p-divisible groups (joint work with Peter Scholze), and suggest the usefulness of working at infinite level in the context of other arithmetic moduli problems.

Monday (Feb 11), 4:30 p.m, room E 206. David Kazhdan (Hebrew University). Minimal representations of simply-laced reductive groups. Abstract For any local field F the Weil representation is a representation of M(2n,f), the double cover of the group Sp(2n,F); this remarkable representation is the basis of the Howe duality. In fact, the Weil representation is the "minimal" representation of M(2n,f). I will define the notion of minimal (unitary) representation for reductive groups over local fields, give explicit formulas for spherical vectors for simply-laced groups, describe the space of smooth vectors and the structure of the automorphic functionals.

Thursday (Feb 14), 4:30 p.m, room E 206. David Kazhdan (Hebrew University). Minimal representations of simply-laced reductive groups. II. Abstract For any local field F the Weil representation is a representation of M(2n,f), the double cover of the group Sp(2n,F); this remarkable representation is the basis of the Howe duality. In fact, the Weil representation is the "minimal" representation of M(2n,f). I will define the notion of minimal (unitary) representation for reductive groups over local fields, give explicit formulas for spherical vectors for simply-laced groups, describe the space of smooth vectors and the structure of the automorphic functionals.

Monday (Feb 18), 4:30 p.m, room E 206. Alexander Efimov (Moscow). Homotopy finiteness of DG categories from algebraic geometry. [To understand the talk, it suffices to know standard facts about triangulated and derived categories. In other words, don't be afraid of words like "homotopy finiteness".] Abstract We will explain that for any separated scheme $X$ of finite type over a field $k$ of characteristic zero, its derived category $D^b_{coh}(X)$ (considered as a DG category) is homotopically finitely presented over $k$, confirming a conjecture of Kontsevich. More precisely, we show that $D^b_{coh}(X)$ can be represented as a DG quotient of some smooth and proper DG category $C$ by a subcategory generated by a single object. This category $C$ has a semi-orthogonal decomposition into derived categories of smooth and proper varieties. The construction uses the categorical resolution of singularities of Kuznetsov and Lunts, which in turn uses Hironaka Theorem. A similar result holds also for the 2-periodic DG category $MF_{coh}(X,W)$ of coherent matrix factorizations on $X$ for any potential $W$.

Thursday (Feb 21), 4:30 p.m, room E 206. Alexander Efimov (Moscow). Homotopy finiteness of DG categories from algebraic geometry.II. ******* Here are the references for the results mentioned in Efimov's first talk: B. Toen, M. Vaquie, Moduli of objects in dg-categories, arXiv:math/0503269 Valery A. Lunts, Categorical resolution of singularities, arXiv:0905.4566 Raphael Rouquier, Dimensions of triangulated categories, arXiv:math/0310134 Alexei Bondal, Michel Van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, arXiv:math/0204218 Alexander Kuznetsov, Valery A. Lunts, Categorical resolutions of irrational singularities, arXiv:1212.6170 M. Auslander, Representation dimension of Artin algebras, in Selected works of Maurice Auslander. Part 1. American Mathematical Society, Providence, RI, 1999.

No seminar on Monday (Feb 25). ****** On Thursday (Feb 28) there will be a talk by Alexander Polishchuk (University of Oregon). Title of his talk: Matrix factorizations and cohomological field theories. Abstract This is joint work with Arkady Vaintrob. I will explain how one can use DG categories of matrix factorizations to construct a cohomological field theory associated with a quasihomogeneous polynomial with isolated singularity at zero.

Thursday (Feb 28), 4:30 p.m, room E 206. Alexander Polishchuk (University of Oregon). Matrix factorizations and cohomological field theories. Abstract This is joint work with Arkady Vaintrob. I will explain how one can use DG categories of matrix factorizations to construct a cohomological field theory associated with a quasihomogeneous polynomial with isolated singularity at zero.

Monday (March 4), 4:30 p.m, room E 206. Richard Taylor (IAS). Galois representations for regular algebraic cusp forms. Abstract I will start by reviewing what is expected, and what is known, about the correspondence between algebraic l-adic representations of the absolute Galois group of a number field and algebraic cuspidal automorphic representations of GL(n) over that number field. I will then discuss recent work with Harris, Lan and Thorne constructing l-adic representations for regular algebraic cuspidal automorphic representations of GL(n) over a CM field, without any self-duality assumption on the automorphic representation. Without such an assumption it is believed that these l-adic representations do not occur in the cohomology of any Shimura variety, and we do not know how to construct the corresponding motive (though we believe that a motive should exist). Nonetheless we can construct the l-adic representations as an l-adic limit of motivic l-adic representations.

No more meetings of the Geometric Langlands seminar this quarter.

The geometric Langlands seminar does not meet this week. On next Monday (April 8) Bhargav Bhatt will speak on Derived de Rham cohomology in characteristic 0. After that, on April 15 and 18 Ivan Losev will give lectures on categorifications of Kac-Moody algebras. (There are good reasons to expect his lectures to be understandable!)

Monday (April 8), 4:30 p.m, room E 206. Bhargav Bhatt (IAS). Derived de Rham cohomology in characteristic 0. Abstract Derived de Rham cohomology is a refinement of classical de Rham cohomology of algebraic varieties that works better in the presence of singularities; the difference, roughly, is the replacement of the cotangent sheaf by the cotangent complex. In my talk, I will first recall Illusie's definition of this cohomology theory (both completed and non-completed variants). Then I will explain why the completed variant computes algebraic de Rham cohomology (and hence Betti cohomology) for arbitrary algebraic varieties in characteristic 0; the case of local complete intersection singularities is due to Illusie. As a corollary, one obtains a new filtration on Betti cohomology refining the Hodge-Deligne filtration. Another consequence that will be discussed is that the completed Amitsur complex of a variety also calculates its algebraic de Rham cohomology.

Monday (April 8), 4:30 p.m, room E 206. Bhargav Bhatt (IAS). Derived de Rham cohomology in characteristic 0. Abstract Derived de Rham cohomology is a refinement of classical de Rham cohomology of algebraic varieties that works better in the presence of singularities; the difference, roughly, is the replacement of the cotangent sheaf by the cotangent complex. In my talk, I will first recall Illusie's definition of this cohomology theory (both completed and non-completed variants). Then I will explain why the completed variant computes algebraic de Rham cohomology (and hence Betti cohomology) for arbitrary algebraic varieties in characteristic 0; the case of local complete intersection singularities is due to Illusie. As a corollary, one obtains a new filtration on Betti cohomology refining the Hodge-Deligne filtration. Another consequence that will be discussed is that the completed Amitsur complex of a variety also calculates its algebraic de Rham cohomology.

No seminar on Thursday. Next week Ivan Losev (Northeastern University) will speak on Monday (April 15) and Thursday (April 15). Title of Losev's lectures: Introduction to categorical Kac-Moody actions. Abstract The goal of these lectures is to provide an elementary introduction to categorical actions of Kac-Moody algebras from a representation theoretic perspective. In a naive way (which, of course, appeared first), a categorical Kac-Moody action is a collection of functors on a category that on the level of Grothendieck groups give actions of the Chevalley generators of the Kac-Moody algebra. Such functors were first observed in the representation theory of symmetric groups in positive characteristic and then for the BGG category O of gl(n). Analyzing the examples, in 2004 Chuang and Rouquier gave a formal definition of a categorical sl(2)-action. Later (about 2008) Rouquier and Khovanov-Lauda extended this definition to arbitrary Kac-Moody Lie algebras. Categorical Kac-Moody actions are very useful in Representation theory and (potentially, at least) in Knot theory. Their usefulness in Representation theory is three-fold. First, they allow to obtain structural results about the categories of interest (branching rules for the symmetric groups obtained by Kleshchev, or derived equivalences between different blocks constructed by Chuang and Rouquier in order to prove the Broue abelian defect conjecture). Second, categories with Kac-Moody actions are often uniquely determined by the "type of an action", sometimes this gives character formulas. Third, the categorification business gives rise to new interesting classes of algebras that were not known before: the KLR (Khovanov-Lauda-Rouquier) algebras. Potential applications to Knot theory include categorical (hence stronger) versions of quantum knot invariants, this area is very much still in development. I will start from scratch and try to keep the exposition elementary, in particular I will only consider Kac-Moody algebras of type A, i.e., sl(n) and \hat{sl(n)}. The most essential prerequisite is a good understanding of the standard categorical language (e.g., functor morphisms). Familiarity with classical representation theoretic objects such as affine Hecke algebras or BGG categories O is also useful although these will be recalled. A preliminary plan is as follows: 0) Introduction. 1) Examples: symmetric groups/type A Hecke algebras, BGG categories O. 2) Formal definition of a categorical action. 3) More examples (time permitting): representations of GL. 4) Consequences of the definition: divided powers, categorifications of reflections, categorical Serre relations, crystals. 5) Yet some more examples: cyclotomic Hecke algebras. 6) Structural results: minimal categorifications and their uniqueness, filtrations, (time permitting) actions on highest weight categories, tensor products. Here are some important topics related to categorical Kac-Moody actions that will not be discussed: a) Categorical actions in other types and those of quantum groups. b) Categorification of the algebras U(n),U(g), etc. c) Connections to categorical knot invariants. a) is described in reviews http://arxiv.org/abs/1301.5868 by Brundan and (a more advanced text) http://arxiv.org/abs/1112.3619 by Rouquier. The latter also deals with b). A more basic review for b) is http://arxiv.org/abs/1112.3619 by Lauda dealing with the sl_2 case and also introducing diagrammatic calculus. I am not aware of any reviews on c), a connection to Reshetikhin-Turaev invariants was established in full generality by Webster in http://arxiv.org/abs/1001.2020, http://arxiv.org/abs/1005.4559.

Monday (April 15), 4:30 p.m, room E 206. Ivan Losev (Northeastern University) Introduction to categorical Kac-Moody actions.I Abstract The goal of this lecture and the one on April 18 is to provide an elementary introduction to categorical actions of Kac-Moody algebras from a representation theoretic perspective. In a naive way (which, of course, appeared first), a categorical Kac-Moody action is a collection of functors on a category that on the level of Grothendieck groups give actions of the Chevalley generators of the Kac-Moody algebra. Such functors were first observed in the representation theory of symmetric groups in positive characteristic and then for the BGG category O of gl(n). Analyzing the examples, in 2004 Chuang and Rouquier gave a formal definition of a categorical sl(2)-action. Later (about 2008) Rouquier and Khovanov-Lauda extended this definition to arbitrary Kac-Moody Lie algebras. Categorical Kac-Moody actions are very useful in Representation theory and (potentially, at least) in Knot theory. Their usefulness in Representation theory is three-fold. First, they allow to obtain structural results about the categories of interest (branching rules for the symmetric groups obtained by Kleshchev, or derived equivalences between different blocks constructed by Chuang and Rouquier in order to prove the Broue abelian defect conjecture). Second, categories with Kac-Moody actions are often uniquely determined by the "type of an action", sometimes this gives character formulas. Third, the categorification business gives rise to new interesting classes of algebras that were not known before: the KLR (Khovanov-Lauda-Rouquier) algebras. Potential applications to Knot theory include categorical (hence stronger) versions of quantum knot invariants, this area is very much still in development. I will start from scratch and try to keep the exposition elementary, in particular I will only consider Kac-Moody algebras of type A, i.e., sl(n) and \hat{sl(n)}. The most essential prerequisite is a good understanding of the standard categorical language (e.g., functor morphisms). Familiarity with classical representation theoretic objects such as affine Hecke algebras or BGG categories O is also useful although these will be recalled. A preliminary plan is as follows: 0) Introduction. 1) Examples: symmetric groups/type A Hecke algebras, BGG categories O. 2) Formal definition of a categorical action. 3) More examples (time permitting): representations of GL. 4) Consequences of the definition: divided powers, categorifications of reflections, categorical Serre relations, crystals. 5) Yet some more examples: cyclotomic Hecke algebras. 6) Structural results: minimal categorifications and their uniqueness, filtrations, (time permitting) actions on highest weight categories, tensor products. Here are some important topics related to categorical Kac-Moody actions that will not be discussed: a) Categorical actions in other types and those of quantum groups. b) Categorification of the algebras U(n),U(g), etc. c) Connections to categorical knot invariants. a) is described in reviews http://arxiv.org/abs/1301.5868 by Brundan and (a more advanced text) http://arxiv.org/abs/1112.3619 by Rouquier. The latter also deals with b). A more basic review for b) is http://arxiv.org/abs/1112.3619 by Lauda dealing with the sl_2 case and also introducing diagrammatic calculus. I am not aware of any reviews on c), a connection to Reshetikhin-Turaev invariants was established in full generality by Webster in http://arxiv.org/abs/1001.2020, http://arxiv.org/abs/1005.4559.

Today, 4:30 p.m, room E 206. Ivan Losev (Northeastern University) Introduction to categorical Kac-Moody actions.I Abstract The goal of this lecture and the one on April 18 is to provide an elementary introduction to categorical actions of Kac-Moody algebras from a representation theoretic perspective. In a naive way (which, of course, appeared first), a categorical Kac-Moody action is a collection of functors on a category that on the level of Grothendieck groups give actions of the Chevalley generators of the Kac-Moody algebra. Such functors were first observed in the representation theory of symmetric groups in positive characteristic and then for the BGG category O of gl(n). Analyzing the examples, in 2004 Chuang and Rouquier gave a formal definition of a categorical sl(2)-action. Later (about 2008) Rouquier and Khovanov-Lauda extended this definition to arbitrary Kac-Moody Lie algebras. Categorical Kac-Moody actions are very useful in Representation theory and (potentially, at least) in Knot theory. Their usefulness in Representation theory is three-fold. First, they allow to obtain structural results about the categories of interest (branching rules for the symmetric groups obtained by Kleshchev, or derived equivalences between different blocks constructed by Chuang and Rouquier in order to prove the Broue abelian defect conjecture). Second, categories with Kac-Moody actions are often uniquely determined by the "type of an action", sometimes this gives character formulas. Third, the categorification business gives rise to new interesting classes of algebras that were not known before: the KLR (Khovanov-Lauda-Rouquier) algebras. Potential applications to Knot theory include categorical (hence stronger) versions of quantum knot invariants, this area is very much still in development. I will start from scratch and try to keep the exposition elementary, in particular I will only consider Kac-Moody algebras of type A, i.e., sl(n) and \hat{sl(n)}. The most essential prerequisite is a good understanding of the standard categorical language (e.g., functor morphisms). Familiarity with classical representation theoretic objects such as affine Hecke algebras or BGG categories O is also useful although these will be recalled. A preliminary plan is as follows: 0) Introduction. 1) Examples: symmetric groups/type A Hecke algebras, BGG categories O. 2) Formal definition of a categorical action. 3) More examples (time permitting): representations of GL. 4) Consequences of the definition: divided powers, categorifications of reflections, categorical Serre relations, crystals. 5) Yet some more examples: cyclotomic Hecke algebras. 6) Structural results: minimal categorifications and their uniqueness, filtrations, (time permitting) actions on highest weight categories, tensor products. Here are some important topics related to categorical Kac-Moody actions that will not be discussed: a) Categorical actions in other types and those of quantum groups. b) Categorification of the algebras U(n),U(g), etc. c) Connections to categorical knot invariants. a) is described in reviews http://arxiv.org/abs/1301.5868 by Brundan and (a more advanced text) http://arxiv.org/abs/1112.3619 by Rouquier. The latter also deals with b). A more basic review for b) is http://arxiv.org/abs/1112.3619 by Lauda dealing with the sl_2 case and also introducing diagrammatic calculus. I am not aware of any reviews on c), a connection to Reshetikhin-Turaev invariants was established in full generality by Webster in http://arxiv.org/abs/1001.2020, http://arxiv.org/abs/1005.4559.

Thursday (April 18), 4:30 p.m, room E 206. Ivan Losev. Introduction to categorical Kac-Moody actions.II.

No seminar on Monday (Apr 22) and Thursday (Apr 25). The next meeting is on FRIDAY April 26 (4:30 p.m, room E 206). (I do realize that Friday is not a very good day for a seminar, but unfortunately, the speaker was unable to speak on another day.) Friday (April 26), 4:30 p.m, room E 206. Roman Bezrukavnikov (MIT). Towards character sheaves for loop groups Abstract I will describe a joint project with D. Kazhdan and Y. Varshavsky whose aim is to develop the theory of character sheaves for loop groups and apply it to the theory of endoscopy for reductive $p$-adic groups. The project started from an attempt to understand the relation of Lusztig's classification of character sheaves (discussed in an earlier talk by the speaker in this seminar) to local Langlands conjectures. I will discuss results (to appear shortly) on a geometric proof of the result by Kazhdan--Varshavsky and De Backer -- Reeder on stable combinations of characters in a generic depth zero L-packet, and a proof of the unramified case of the stable center conjecture. Time permitting, I will describe a general approach to relating local geometric Langlands duality to endoscopy. Character sheaves on loop groups are also the subject of two recent papers by Lusztig.

Losev's notes of his talks are here: http://www.math.uchicago.edu/~mitya/langlands/LosevNotes.pdf ******* Recall that the next meeting of the seminar is on FRIDAY: Friday (April 26), 4:30 p.m, room E 206. Roman Bezrukavnikov (MIT). Towards character sheaves for loop groups Abstract I will describe a joint project with D. Kazhdan and Y. Varshavsky whose aim is to develop the theory of character sheaves for loop groups and apply it to the theory of endoscopy for reductive $p$-adic groups. The project started from an attempt to understand the relation of Lusztig's classification of character sheaves (discussed in an earlier talk by the speaker in this seminar) to local Langlands conjectures. I will discuss results (to appear shortly) on a geometric proof of the result by Kazhdan--Varshavsky and De Backer -- Reeder on stable combinations of characters in a generic depth zero L-packet, and a proof of the unramified case of the stable center conjecture. Time permitting, I will describe a general approach to relating local geometric Langlands duality to endoscopy. Character sheaves on loop groups are also the subject of two recent papers by Lusztig.

No seminar on Monday (April 29). Dennis Gaitsgory will speak on Thursday (May 2) and Monday (May 6). The title of his talk will be announced soon.

Thursday (May 2), 4:30 p.m, room E 206. Dennis Gaitsgory (Harvard). Eisenstein part of Geometric Langlands correspondence.I. Abstract Geometric Eisenstein series, Eis_!, is a functor D-mod(Bun_T)->D-mod(Bun-G), given by pull-push using the stack Bun_B is an intermediary. Spectral Eisenstein series Eis_{spec} is a functor QCoh(LocSys_{\check{T})->IndCoh(LocSys_{\check{G}}) given by pull-push using the stack LocSys_{\check{B}} is an intermediary. One of the expected key properties of the (still conjectural) Geometric Langlands equivalence is that it intertwines Eis_! and Eis_{spec}. This implies that the algebras of endomorphisms of Eis_! and Eis_{spec}, viewed as functors out of D-mod(Bun_T)=QCoh(LocSys_{\check{T}) are isomorphic (here the equivalence D-mod(Bun_T)=QCoh(LocSys_{\check{T}) is Geometric Langlands for the torus, which is given by the Fourier-Mukai transform). Vice versa, establishing the isomorphism of the above algebras of endomorphisms is equivalent to establishing the Eisenstein part of the Geometric Langlands equivalence. In these talks, we will indicate a strategy toward the proof of this isomorphism. We will reduce the problem from being global to one which is local (in particular, on the geometric side, instead of Bun_G we will be dealing with the affine Grassmannian). We will show that the local problem is equivalent to a factorizable version of Bezrukavnikov's theory of Langlands duality for various categories of D-modules on the affine Grassmannian.

Tomorrow (i.e., Thursday, May 2), 4:30 p.m, room E 206. Dennis Gaitsgory (Harvard). Eisenstein part of Geometric Langlands correspondence.I. Abstract Geometric Eisenstein series, Eis_!, is a functor D-mod(Bun_T)->D-mod(Bun-G), given by pull-push using the stack Bun_B is an intermediary. Spectral Eisenstein series Eis_{spec} is a functor QCoh(LocSys_{\check{T})->IndCoh(LocSys_{\check{G}}) given by pull-push using the stack LocSys_{\check{B}} is an intermediary. One of the expected key properties of the (still conjectural) Geometric Langlands equivalence is that it intertwines Eis_! and Eis_{spec}. This implies that the algebras of endomorphisms of Eis_! and Eis_{spec}, viewed as functors out of D-mod(Bun_T)=QCoh(LocSys_{\check{T}) are isomorphic (here the equivalence D-mod(Bun_T)=QCoh(LocSys_{\check{T}) is Geometric Langlands for the torus, which is given by the Fourier-Mukai transform). Vice versa, establishing the isomorphism of the above algebras of endomorphisms is equivalent to establishing the Eisenstein part of the Geometric Langlands equivalence. In these talks, we will indicate a strategy toward the proof of this isomorphism. We will reduce the problem from being global to one which is local (in particular, on the geometric side, instead of Bun_G we will be dealing with the affine Grassmannian). We will show that the local problem is equivalent to a factorizable version of Bezrukavnikov's theory of Langlands duality for various categories of D-modules on the affine Grassmannian.

Monday (May 6), 4:30 p.m, room E 206. Dennis Gaitsgory (Harvard). Eisenstein part of Geometric Langlands correspondence.II. Abstract Geometric Eisenstein series, Eis_!, is a functor D-mod(Bun_T)->D-mod(Bun-G), given by pull-push using the stack Bun_B is an intermediary. Spectral Eisenstein series Eis_{spec} is a functor QCoh(LocSys_{\check{T})->IndCoh(LocSys_{\check{G}}) given by pull-push using the stack LocSys_{\check{B}} is an intermediary. One of the expected key properties of the (still conjectural) Geometric Langlands equivalence is that it intertwines Eis_! and Eis_{spec}. This implies that the algebras of endomorphisms of Eis_! and Eis_{spec}, viewed as functors out of D-mod(Bun_T)=QCoh(LocSys_{\check{T}) are isomorphic (here the equivalence D-mod(Bun_T)=QCoh(LocSys_{\check{T}) is Geometric Langlands for the torus, which is given by the Fourier-Mukai transform). Vice versa, establishing the isomorphism of the above algebras of endomorphisms is equivalent to establishing the Eisenstein part of the Geometric Langlands equivalence. In these talks, we will indicate a strategy toward the proof of this isomorphism. We will reduce the problem from being global to one which is local (in particular, on the geometric side, instead of Bun_G we will be dealing with the affine Grassmannian). We will show that the local problem is equivalent to a factorizable version of Bezrukavnikov's theory of Langlands duality for various categories of D-modules on the affine Grassmannian.

Thursday (May 9), 4:30 p.m, room E 206. Dustin Clausen (MIT). Arithmetic Duality in Algebraic K-theory. Abstract Let R be any commutative ring classically considered in algebraic number theory (global field, local field, ring of integers...). We will give a uniform definition of a ``compactly supported G-theory'' spectrum G_c(R) associated to R, supposed to be dual to the algebraic K-theory K(R). Then, for every prime $\ell$ invertible in R, we will construct a functorial $\ell$-adic pairing implementing this duality. Finally, using work of Thomason connecting algebraic K-theory to Galois theory, we will explain how these pairings allow to give a uniform construction of the various Artin maps associated to such rings R, one by which the Artin reciprocity law becomes tautological. The crucial input is a simple homotopy-theoretic connection between tori, real vector spaces, and spheres, which we hope to explain.

Monday (May 13), 4:30 p.m, room E 206. Takako Fukaya. On non-commutative Iwasawa theory. Abstract Iwasawa theory studies a mysterious connection between algebraic objects (ideal class groups, etc.) and analytic objects (p-adic Riemann zeta functions etc.) in a p-adic way, considering certain p-adic infinite towers of Galois extensions of number fields. Historically, people first used infinite Galois extensions whose Galois group is abelian. However, in recent years, non-commutative Iwasawa theory, which considers infinite Galois extensions whose Galois group is non-commutative has been developed. We will first review ``commutative Iwasawa theory (usual Iwasawa theory)", then introduce the history of non-commutative Iwasawa theory, and the results obtained recently.

No more meetings of the Langlands seminar this quarter.

As usual, the seminar will meet on Mondays and/or Thursdays in room E206 at 4:30 p.m. We will begin with a series of talks by Beilinson on his recent work (the title and abstract are below). In particular, he will give a proof of the results of the article http://arxiv.org/abs/1203.2776 (by Bloch, Esnault, and Kerz), which is more understandable and elementary than the original one. The first meeting is on October 10 (Thursday). Alexander Beilinson. Relative continuous K-theory and cyclic homology. Abstract Let X be a smooth proper scheme over the ring of p-adic integers Z_p. Suppose we have a class of a vector bundle v \in K_0 (X/p) on the closed fiber. How can one check if c comes from a class in K_0 (X)? A necessary condition is that the Chern class ch(v) in the crystalline cohomology of X/p (which is the same as de Rham cohomology of X) lies in the middle term of the Hodge filtration. A variant of the deformational Hodge conjecture says that, up to torsion, this condition is sufficient as well. This conjecture remains a mystery, but in a recent work "p-adic deformation of algebraic cycle classes" Bloch, Esnault, and Kerz proved that (subject to some conditions on X) the conjecture is valid if we replace K_0 (X) by the projective limit of groups K_0 (X/p^n). In this series of talks I will explain a p-adic version of Goodwillie's theorem which identifies the relative continuous K-theory of a p-adic associative algebra with its continuous cyclic homology, and that implies the Bloch-Esnault-Kerz theorem. I will explain the background material, so no prior knowledge of the subject is needed.

No seminar on Monday. Thursday (Oct 10), 4:30 p.m, room E 206. Alexander Beilinson will give his first talk on Relative continuous K-theory and cyclic homology. Abstract Let X be a smooth proper scheme over the ring of p-adic integers Z_p. Suppose we have a class of a vector bundle v \in K_0 (X/p) on the closed fiber. How can one check if c comes from a class in K_0 (X)? A necessary condition is that the Chern class ch(v) in the crystalline cohomology of X/p (which is the same as de Rham cohomology of X) lies in the middle term of the Hodge filtration. A variant of the deformational Hodge conjecture says that, up to torsion, this condition is sufficient as well. This conjecture remains a mystery, but in a recent work "p-adic deformation of algebraic cycle classes" Bloch, Esnault, and Kerz proved that (subject to some conditions on X) the conjecture is valid if we replace K_0 (X) by the projective limit of groups K_0 (X/p^n). In this series of talks I will explain a p-adic version of Goodwillie's theorem which identifies the relative continuous K-theory of a p-adic associative algebra with its continuous cyclic homology, and that implies the Bloch-Esnault-Kerz theorem. I will explain the background material, so no prior knowledge of the subject is needed.

Thursday (Oct 10), 4:30 p.m, room E 206. Alexander Beilinson will give his first talk on Relative continuous K-theory and cyclic homology. Abstract Let X be a smooth proper scheme over the ring of p-adic integers Z_p. Suppose we have a class of a vector bundle v \in K_0 (X/p) on the closed fiber. How can one check if c comes from a class in K_0 (X)? A necessary condition is that the Chern class ch(v) in the crystalline cohomology of X/p (which is the same as de Rham cohomology of X) lies in the middle term of the Hodge filtration. A variant of the deformational Hodge conjecture says that, up to torsion, this condition is sufficient as well. This conjecture remains a mystery, but in a recent work "p-adic deformation of algebraic cycle classes" Bloch, Esnault, and Kerz proved that (subject to some conditions on X) the conjecture is valid if we replace K_0 (X) by the projective limit of groups K_0 (X/p^n). In this series of talks I will explain a p-adic version of Goodwillie's theorem which identifies the relative continuous K-theory of a p-adic associative algebra with its continuous cyclic homology, and that implies the Bloch-Esnault-Kerz theorem. I will explain the background material, so no prior knowledge of the subject is needed.

Thursday (Oct 10), 4:30 p.m, room E 206. Alexander Beilinson will give his first talk on Relative continuous K-theory and cyclic homology. Abstract Let X be a smooth proper scheme over the ring of p-adic integers Z_p. Suppose we have a class of a vector bundle v \in K_0 (X/p) on the closed fiber. How can one check if c comes from a class in K_0 (X)? A necessary condition is that the Chern class ch(v) in the crystalline cohomology of X/p (which is the same as de Rham cohomology of X) lies in the middle term of the Hodge filtration. A variant of the deformational Hodge conjecture says that, up to torsion, this condition is sufficient as well. This conjecture remains a mystery, but in a recent work "p-adic deformation of algebraic cycle classes" Bloch, Esnault, and Kerz proved that (subject to some conditions on X) the conjecture is valid if we replace K_0 (X) by the projective limit of groups K_0 (X/p^n). In this series of talks I will explain a p-adic version of Goodwillie's theorem which identifies the relative continuous K-theory of a p-adic associative algebra with its continuous cyclic homology, and that implies the Bloch-Esnault-Kerz theorem. I will explain the background material, so no prior knowledge of the subject is needed.

Monday (Oct 14), 4:30 p.m, room E 206. Alexander Beilinson. Relative continuous K-theory and cyclic homology. II. Abstract Let X be a smooth proper scheme over the ring of p-adic integers Z_p. Suppose we have a class of a vector bundle v \in K_0 (X/p) on the closed fiber. How can one check if c comes from a class in K_0 (X)? A necessary condition is that the Chern class ch(v) in the crystalline cohomology of X/p (which is the same as de Rham cohomology of X) lies in the middle term of the Hodge filtration. A variant of the deformational Hodge conjecture says that, up to torsion, this condition is sufficient as well. This conjecture remains a mystery, but in a recent work "p-adic deformation of algebraic cycle classes" Bloch, Esnault, and Kerz proved that (subject to some conditions on X) the conjecture is valid if we replace K_0 (X) by the projective limit of groups K_0 (X/p^n). In this series of talks I will explain a p-adic version of Goodwillie's theorem which identifies the relative continuous K-theory of a p-adic associative algebra with its continuous cyclic homology, and that implies the Bloch-Esnault-Kerz theorem.

No seminar this Thursday. Alexander Beilinson will continue on Monday (Oct 21).

Monday (Oct 21), 4:30 p.m, room E 206. Alexander Beilinson. Relative continuous K-theory and cyclic homology. III. Abstract Let X be a smooth proper scheme over the ring of p-adic integers Z_p. Suppose we have a class of a vector bundle v \in K_0 (X/p) on the closed fiber. How can one check if c comes from a class in K_0 (X)? A necessary condition is that the Chern class ch(v) in the crystalline cohomology of X/p (which is the same as de Rham cohomology of X) lies in the middle term of the Hodge filtration. A variant of the deformational Hodge conjecture says that, up to torsion, this condition is sufficient as well. This conjecture remains a mystery, but in a recent work "p-adic deformation of algebraic cycle classes" Bloch, Esnault, and Kerz proved that (subject to some conditions on X) the conjecture is valid if we replace K_0 (X) by the projective limit of groups K_0 (X/p^n). In this series of talks I will explain a p-adic version of Goodwillie's theorem which identifies the relative continuous K-theory of a p-adic associative algebra with its continuous cyclic homology, and that implies the Bloch-Esnault-Kerz theorem.

Monday (Oct 21), 4:30 p.m, room E 206. Alexander Beilinson. Relative continuous K-theory and cyclic homology. III. Abstract Let X be a smooth proper scheme over the ring of p-adic integers Z_p. Suppose we have a class of a vector bundle v \in K_0 (X/p) on the closed fiber. How can one check if c comes from a class in K_0 (X)? A necessary condition is that the Chern class ch(v) in the crystalline cohomology of X/p (which is the same as de Rham cohomology of X) lies in the middle term of the Hodge filtration. A variant of the deformational Hodge conjecture says that, up to torsion, this condition is sufficient as well. This conjecture remains a mystery, but in a recent work "p-adic deformation of algebraic cycle classes" Bloch, Esnault, and Kerz proved that (subject to some conditions on X) the conjecture is valid if we replace K_0 (X) by the projective limit of groups K_0 (X/p^n). In this series of talks I will explain a p-adic version of Goodwillie's theorem which identifies the relative continuous K-theory of a p-adic associative algebra with its continuous cyclic homology, and that implies the Bloch-Esnault-Kerz theorem.

No seminar on Thursday. Beilinson will continue on Monday (Oct 28).

No seminar on Monday (Oct 28); Beilinson's talk has been CANCELED because quite unexpectedly, he has to go to Moscow (his mother-in-law died). **** Next Thursday (Oct 31) Steve Zelditch (NWU) will give his first talk on Berezin-Toeplitz quantization. Title of his talk: Quantization and Toeplitz operators. Abstract One of the basic settings of geometric quantization is a Kahler manifold (M, J, \omega), polarized by a Hermitian holomorphic line bundle $(L, h) \to (M, \omega)$. The metric h induces inner products on the spaces $H^0(M, L^k)$ of holomorphic sections of the k-th power of L. A basic principle is that 1/k plays the role of Planck's constant, and one has semi-classical asymptotics as k goes to infinity. The purpose of my first lecture is to introduce the Szego kernels $\Pi_k$ in this context and to explain why the semi-classical asymptotics exist. Toeplitz operators are of the form $\Pi_k M_f \Pi_k$ where $M_f$ is multiplication by $f \in C^{\infty}(M)$, and one gets a * product on the smooth functions by composing operators. There is a more general formalism for almost complex symplectic manifolds and in other settings.

Thursday (Oct 31), 4:30 p.m, room E 206. Steve Zelditch (NWU) will give his first talk on Quantization and Toeplitz operators. Abstract One of the basic settings of geometric quantization is a Kahler manifold (M, J, \omega), polarized by a Hermitian holomorphic line bundle $(L, h) \to (M, \omega)$. The metric h induces inner products on the spaces $H^0(M, L^k)$ of holomorphic sections of the k-th power of L. A basic principle is that 1/k plays the role of Planck's constant, and one has semi-classical asymptotics as k goes to infinity. The purpose of my first lecture is to introduce the Szego kernels $\Pi_k$ in this context and to explain why the semi-classical asymptotics exist. Toeplitz operators are of the form $\Pi_k M_f \Pi_k$ where $M_f$ is multiplication by $f \in C^{\infty}(M)$, and one gets a * product on the smooth functions by composing operators. There is a more general formalism for almost complex symplectic manifolds and in other settings.

No seminar on Monday November 4. ***** The next meeting is on Thursday (Nov 7) at an UNUSUAL time, namely at 4:00 p.m. (in the usual room E 206). Steve Zelditch (NWU) will give his second talk on Quantization and Toeplitz operators. Attached is a PDF file with Zelditch's notes of his first talk and the beginning of the second one. ***** Let me also tell you that on Monday November 11 Danny Calegari will give an introductory talk "Fundamental groups of Kahler manifolds".

**Attachment:
Zelditch.pdf**

The next meeting is on Thursday (Nov 7) at an UNUSUAL time, namely at 4:00 p.m. (in the usual room E 206). Steve Zelditch (NWU) will give his second talk on Quantization and Toeplitz operators.

Monday (Nov 11), 4:30 p.m, room E 206. Danny Calegari. Fundamental groups of Kahler manifolds (an introduction) Abstract I will try to explain some of what is known and not known about fundamental groups of (closed) Kahler manifolds (hereafter "Kahler groups"), especially concentrating on the constraints that arise for geometric reasons, where "geometry" here is understood in the sense of a geometric group theorist; so (for example), some of the tools I will discuss include L^2 cohomology, Bieri-Neumann-Strebel invariants, and the theory of harmonic maps to trees. One reason to be interested in such groups is because nonsingular projective varieties (over the complex numbers) are Kahler, so in principle, constraints on Kahler groups (and their linear representations) have implications for understanding local systems on projective varieties (but I will not talk about this). Most of what I want to discuss is classical, and has been well-known for over 20 years, but I hope to discuss at least two interesting recent developments: (1) an elementary construction (due to Panov-Petrunin) to show that every finitely presented group arises as the fundamental group of a compact complex 3-fold (typically not projective!); (2) a theorem of Delzant that a solvable Kahler group contains a nilpotent group with finite index (the corresponding fact for fundamental groups of nonsingular projective varieties is due to Arapura and Nori, and their proof is very different). This talk should be accessible to graduate students.

The next meeting is on Thursday (Nov 14) at an UNUSUAL time, namely at 4:00 p.m. (in the usual room E 206). Steve Zelditch (NWU) will give his third talk on Quantization and Toeplitz operators. (Danny Calegary will finish his talk on \pi_1 of Kahler manifolds on Monday, Nov 18).

Attached is a file with Steve Zelditch's notes of his second and third lecture on "Quantization and Toeplitz operators" (The third lecture is today at 4:00 p.m.)

**Attachment:
Zelditch lectures 2-3.pdf**

Monday (Nov 18), 4:30 p.m, room E 206. Danny Calegari. Fundamental groups of Kahler manifolds. II

The next meeting is on Thursday (Nov 21) at an UNUSUAL time, namely at 4:00 p.m. (in the usual room E 206). Steve Zelditch (NWU) will give his last talk on Quantization and Toeplitz operators. [On Monday (Nov 25) Kazuya Kato will speak on "Heights of motives".]

Monday (Nov 25), 4:30 p.m, room E 206. Kazuya Kato. Heights of motives. Abstract The height of a rational number a/b (a, b integers which are coprime) is defined as max(|a|, |b|). A rational number with small (resp. big) height is a simple (resp. complicated) number. Though the notion height is so naive, height has played fundamental roles in number theory. There are important variants of this notion. In 1983, when Faltings proved Mordell conjecture formulated in 1921, Faltings first proved Tate conjecture for abelian varieties (it was also a great conjecture) by defining heights of an abelian varieties, and then he deduced Mordell conjecture from the latter conjecture. In this talk, after I explain these things, I will explain that the heights of abelian varieties by Faltings are generalized to heights of motives. (Motive is thought of as a kind of generalization of abelian variety.) This generalization of height is related to open problems in number theory. If we can prove finiteness of the number of motives of bounded heights, we can prove important conjectures in number theory such as general Tate conjecture and Mordell-Weil type conjectures in many cases.

Monday (Nov 25), 4:30 p.m, room E 206. Kazuya Kato. Heights of motives. Abstract The height of a rational number a/b (a, b integers which are coprime) is defined as max(|a|, |b|). A rational number with small (resp. big) height is a simple (resp. complicated) number. Though the notion height is so naive, height has played fundamental roles in number theory. There are important variants of this notion. In 1983, when Faltings proved Mordell conjecture formulated in 1921, Faltings first proved Tate conjecture for abelian varieties (it was also a great conjecture) by defining heights of an abelian varieties, and then he deduced Mordell conjecture from the latter conjecture. In this talk, after I explain these things, I will explain that the heights of abelian varieties by Faltings are generalized to heights of motives. (Motive is thought of as a kind of generalization of abelian variety.) This generalization of height is related to open problems in number theory. If we can prove finiteness of the number of motives of bounded heights, we can prove important conjectures in number theory such as general Tate conjecture and Mordell-Weil type conjectures in many cases.

No more meetings of the seminar this quarter.

The first meeting of the seminar is on Jan 9. Thursday (Jan 9), 4:30 p.m, room E 206. Dima Arinkin (Univ. of Wisconsin). Ind-coherent sheaves, relative D-modules, and the Langlands conjecture. I. Abstract In its `categorical' version, the geometric Langlands conjecture predicts an equivalence between two categories of a very different nature. One of them is the derived category of D-modules on the moduli stack of principal bundles on a curve. The other is a certain category of ind-coherent sheaves on the moduli stack of local systems, which is a certain extension of the derived category of quasi-coherent sheaves. Of these two categories, the former is more familiar: its objects can be viewed as geometric counterparts of automorphic forms. The category can be studied using the Fourier transform, which yields a certain additional structure on it. Roughly speaking, the category embeds into a larger category (that of `Fourier coefficients'), which admits a natural filtration indexed by conjugacy classes of parabolic subgroups. In a joint project with D.Gaitsgory, we construct a similar structure on the other side of the Langlands conjecture. Let LS(G) be the stack of G-local systems, where G is a reductive group. For any parabolic subgroup P of G, we consider LS(P) as a stack over LS(G). The conjugacy classes of parabolic subgroups form an ordered set, and the corresponding stacks LS(P) fit into a diagram over LS(G). Our main result is the embedding of the category of ind-coherent sheaves on LS(G) into the category of relative D-modules on this diagram. The result reduces to a purely classical, but seemingly new, property of the topological (spherical) Bruhat-Tits building of G. In my talk, I plan to review the formalism of ind-coherent sheaves and the role it plays in the categorical Langlands conjecture. I will show how relative D-modules appear in the study of ind-coherent sheaves and how the topological Bruhat-Tits building enters the picture.

Thursday (Jan 9), 4:30 p.m, room E 206. Dima Arinkin (Univ. of Wisconsin). Ind-coherent sheaves, relative D-modules, and the Langlands conjecture. I. Abstract In its `categorical' version, the geometric Langlands conjecture predicts an equivalence between two categories of a very different nature. One of them is the derived category of D-modules on the moduli stack of principal bundles on a curve. The other is a certain category of ind-coherent sheaves on the moduli stack of local systems, which is a certain extension of the derived category of quasi-coherent sheaves. Of these two categories, the former is more familiar: its objects can be viewed as geometric counterparts of automorphic forms. The category can be studied using the Fourier transform, which yields a certain additional structure on it. Roughly speaking, the category embeds into a larger category (that of `Fourier coefficients'), which admits a natural filtration indexed by conjugacy classes of parabolic subgroups. In a joint project with D.Gaitsgory, we construct a similar structure on the other side of the Langlands conjecture. Let LS(G) be the stack of G-local systems, where G is a reductive group. For any parabolic subgroup P of G, we consider LS(P) as a stack over LS(G). The conjugacy classes of parabolic subgroups form an ordered set, and the corresponding stacks LS(P) fit into a diagram over LS(G). Our main result is the embedding of the category of ind-coherent sheaves on LS(G) into the category of relative D-modules on this diagram. The result reduces to a purely classical, but seemingly new, property of the topological (spherical) Bruhat-Tits building of G. In my talk, I plan to review the formalism of ind-coherent sheaves and the role it plays in the categorical Langlands conjecture. I will show how relative D-modules appear in the study of ind-coherent sheaves and how the topological Bruhat-Tits building enters the picture.

Monday (Jan 13), 4:30 p.m, room E 206. Dima Arinkin (Univ. of Wisconsin). Ind-coherent sheaves, relative D-modules, and the Langlands conjecture. II. Abstract In its `categorical' version, the geometric Langlands conjecture predicts an equivalence between two categories of a very different nature. One of them is the derived category of D-modules on the moduli stack of principal bundles on a curve. The other is a certain category of ind-coherent sheaves on the moduli stack of local systems, which is a certain extension of the derived category of quasi-coherent sheaves. Of these two categories, the former is more familiar: its objects can be viewed as geometric counterparts of automorphic forms. The category can be studied using the Fourier transform, which yields a certain additional structure on it. Roughly speaking, the category embeds into a larger category (that of `Fourier coefficients'), which admits a natural filtration indexed by conjugacy classes of parabolic subgroups. In a joint project with D.Gaitsgory, we construct a similar structure on the other side of the Langlands conjecture. Let LS(G) be the stack of G-local systems, where G is a reductive group. For any parabolic subgroup P of G, we consider LS(P) as a stack over LS(G). The conjugacy classes of parabolic subgroups form an ordered set, and the corresponding stacks LS(P) fit into a diagram over LS(G). Our main result is the embedding of the category of ind-coherent sheaves on LS(G) into the category of relative D-modules on this diagram. The result reduces to a purely classical, but seemingly new, property of the topological (spherical) Bruhat-Tits building of G. In my talk, I plan to review the formalism of ind-coherent sheaves and the role it plays in the categorical Langlands conjecture. I will show how relative D-modules appear in the study of ind-coherent sheaves and how the topological Bruhat-Tits building enters the picture.

No seminar on Thursday (Jan 16). On Monday (Jan 20) Dmitry Tamarkin (NWU) will give his first talk on Microlocal theory of sheaves and its applications to symplectic topology. Abstract I will start with explaining some basics of the Kashiwara-Schapira microlocal theory of sheaves on manifolds. This theory associates to any sheaf $S$ on a smooth manifold $M$ a homogeneous (a.k.a conic) subset of $T^*M$ called the singular support of $S$. Using a 'conification' trick, one can associate to any sheaf $F$ on $M\times R$ (satisfying certain conditions) a (not-necessarily homogeneous) closed subset of $T^*M$, Given a Hamiltonian symplectomorphism of $T^*M$ (equal to identity outside of a compact), one constructs an endofunctor on an appropriate full category of sheaves on $M\times R$, which transforms microsupports in the obvious way. This allows one to solve some non-displaceability questions in symplectic topology.

Monday (Jan 20), 4:30 p.m, room E 206. Dmitry Tamarkin (NWU). Microlocal theory of sheaves and its applications to symplectic topology. I. Abstract I will start with explaining some basics of the Kashiwara-Schapira microlocal theory of sheaves on manifolds. This theory associates to any sheaf $S$ on a smooth manifold $M$ a homogeneous (a.k.a conic) subset of $T^*M$ called the singular support of $S$. Using a 'conification' trick, one can associate to any sheaf $F$ on $M\times R$ (satisfying certain conditions) a (not-necessarily homogeneous) closed subset of $T^*M$, Given a Hamiltonian symplectomorphism of $T^*M$ (equal to identity outside of a compact), one constructs an endofunctor on an appropriate full category of sheaves on $M\times R$, which transforms microsupports in the obvious way. This allows one to solve some non-displaceability questions in symplectic topology.

No seminar on Thursday (Jan 23). Tamarkin will continue on Monday, January 27.

Monday (Jan 27), 4:30 p.m, room E 206. Dmitry Tamarkin (NWU). Microlocal theory of sheaves and its applications to symplectic topology. II. Abstract I will start with explaining some basics of the Kashiwara-Schapira microlocal theory of sheaves on manifolds. This theory associates to any sheaf $S$ on a smooth manifold $M$ a homogeneous (a.k.a conic) subset of $T^*M$ called the singular support of $S$. Using a 'conification' trick, one can associate to any sheaf $F$ on $M\times R$ (satisfying certain conditions) a (not-necessarily homogeneous) closed subset of $T^*M$, Given a Hamiltonian symplectomorphism of $T^*M$ (equal to identity outside of a compact), one constructs an endofunctor on an appropriate full category of sheaves on $M\times R$, which transforms microsupports in the obvious way. This allows one to solve some non-displaceability questions in symplectic topology.

Monday (Feb 3), 4:30 p.m, room E 206. Dmitry Tamarkin (NWU). Microlocal theory of sheaves and its applications to symplectic topology. III.

Monday (Feb 3), 4:30 p.m, room E 206. Dmitry Tamarkin (NWU). Microlocal theory of sheaves and its applications to symplectic topology. III.

No seminar on Thursday. Nikita Nekrasov (Simons Center at Stony Brook) will speak on Monday (Feb 10). Title of his talk: Geometric definition of the (q_1, q_2)-characters, and instanton fusion. Abstract I will give a geometric definition of a one-parametric deformation of q-characters of the quantum affine and toroidal algebras, and discuss their applications to the calculation of the instanton partition functions of quiver gauge theories.

Monday (Feb 10), 4:30 p.m, room E 206. Nikita Nekrasov (Simons Center for Geometry and Physics at Stony Brook). Geometric definition of the (q_1, q_2)-characters, and instanton fusion. Abstract I will give a geometric definition of a one-parametric deformation of q-characters of the quantum affine and toroidal algebras, and discuss their applications to the calculation of the instanton partition functions of quiver gauge theories.

Thursday (Feb 13), 4:30 p.m, room E 206. Luc Illusie (Paris-Sud). Around the Thom-Sebastiani theorem. Abstract For germs of holomorphic functions $f : \mathbf{C}^{m+1} \to \mathbf{C}$, $g : \mathbf{C}^{n+1} \to \mathbf{C}$ having an isolated critical point at 0 with value 0, the classical Thom-Sebastiani theorem describes the vanishing cycles group $\Phi^{m+n+1}(f \oplus g)$ (and its monodromy) as a tensor product $\Phi^m(f) \otimes \Phi^n(g)$, where $$(f \oplus g)(x,y) = f(x) + g(y), x = (x_0,...,x_m), y = (y_0,...,y_n).$$ In this talk and in the subsequent one(s) I will discuss algebraic variants and generalizations of this result over fields of any characteristic, where the tensor product is replaced by a certain local convolution product, as suggested by Deligne. The main theorem is a Kunneth formula for $R\Psi$ in the framework of Deligne's theory of nearby cycles over general bases, of which I will review the basics. At the end, I will discuss questions logically independent of this, pertaining to the comparison between convolution and tensor product in the tame case.

No seminar on Monday. Luc Illusie will continue his talk on Thursday (Feb 20). As mentioned in the yesterday talk, the key example of blow-up is explained in Section 9 of Orgogozo's article available at http://arxiv.org/abs/math/0507475 Oriented products are reviewed in Expos\'e XI from the book available at http://www.math.polytechnique.fr/~orgogozo/travaux_de_Gabber/ Sabbah's example of "hidden blow-up" is contained in the following article: Sabbah, Claude Morphismes analytiques stratifi\'es sans \'eclatement et cycles \'evanescents. C. R. Acad. Sci. Paris Ser. I Math. 294 (1982), no. 1, 39-41.

Thursday (Feb 20), 4:30 p.m, room E 206. Luc Illusie (Paris-Sud). Around the Thom-Sebastiani theorem. II.

No seminar on Monday. Luc Illusie will finish his talk on Thursday (Feb 27). The article by Laumon mentioned today is available here: http://www.numdam.org/numdam-bin/item?id=PMIHES_1987__65__131_0 The article by N.Katz with the proof of the Gabber-Katz theorem is here: http://www.numdam.org/item?id=AIF_1986__36_4_69_0 Relevant for Illusie's talk is the first part, in which Katz introduces a certain category of "special" finite etale coverings of the multiplicative group over a field of characteristic p; he shows that the category of such special coverings is equivalent to the category of all finite etale coverings of the punctured formal neighbourhood of infinity.

Thursday (Feb 27), 4:30 p.m, room E 206. Luc Illusie (Paris-Sud). Around the Thom-Sebastiani theorem. III.

No seminar on Monday. Spencer Bloch will give Albert lectures on Friday, Monday, and Tuesday, see http://math.uchicago.edu/research/abstracts/albert_abstracts.shtml On Thursday (March 6) Dima Tamarkin will speak. Title of his talk: On Laplace transform Abstract: I will review the papers 'Integral kernels and Laplace transform' by Kashiwara-Schapira '97 and 'On Laplace transform' by d'Agnolo '2013. Both papers aim at describing Laplace transform images of various spaces of complex-analytic functions of tempered growth. In order to work with such spaces, a technique of ind-sheaves is used; the answers are given in terms of the Fourier-Sato transform and its non-homogeneous generalizations.

Today (March 6), 4:30 p.m, room E 206. Dmitry Tamarkin (NWU). On Laplace transform. Abstract I will review the papers 'Integral kernels and Laplace transform' by Kashiwara-Schapira (1997) and 'On Laplace transform' by d'Agnolo (2013). Both papers aim at describing Laplace transform images of various spaces of complex-analytic functions of tempered growth. In order to work with such spaces, a technique of ind-sheaves is used; the answers are given in terms of the Fourier-Sato transform and its non-homogeneous generalizations.

No more seminars this quarter. Tamarkin will explain d'Agnolo's work in spring.

No seminar this week. The first meeting is on April 7 (i.e., next Monday). Dmitry Tamarkin will speak on D'Agnolo's article "On the Laplace transform for tempered holomorphic functions".

Monday (April 7), 4:30 p.m, room E 206. Dmitry Tamarkin (NWU). Laplace transform: non-homogeneous case. Abstract I am going to review d'Agnolo's paper "On the Laplace transform of tempered holomorphic functions", see http://arxiv.org/abs/1207.5278 His article focuses on defining the Laplace transform for certain spaces of regular functions in several complex variables. This is a generalization of the Kaschiwara-Schapira paper "Integral transforms with exponential kernels and Laplace transform" (1997), which answers a similar question for the spaces of tempered functions on homogeneous open subsets (with respect to dilations of the complex space). Here is one of the simplest corollaries of d'Agnolo's result. Let U be an open pre-compact sub-analytic convex subset of a complex vector space V. Let V' be the dual complex space and let h_A be the function on V' defined as follows: h_A(y) is the infimum of Re(x,y) where x runs through A. Let O^t(U) be the space of tempered holomorphic functions on $U$. Let B^{p,q} be the space of (p,q)-forms on V' that grow (along with the derivatives) no faster than a polynomial times e^{-h_A}. d'Agnolo's construction provides an identification of O^t(U) with the quotient of B^{n,n} by the delta bar image of B^{n,n-1}. I am also planning to discuss a couple of other applications of d'Agnolo's result.

No seminar on Thursday (April 10) and Monday (April 14). On April 17 (Thursday) Xinwen Zhu (NWU) will give his first talk on "Cycles on modular varieties via geometric Satake" (this is a more detailed version of the talk that he gave in June 2013 at the number theory seminar at UofC).

Thursday (April 17), 4:30 p.m, room E 206. Xinwen Zhu (NWU). Cycles on modular varieties via geometric Satake. I. Abstract I will first describe certain conjectural Tate classes in the etale cohomology of the special fibers of modular varieties (Shimura varieities and the moduli space of Shtukas). According to the Tate conjecture, there should exist corresponding algebraic cycles. Then I will use ideas from geometric Satake to construct these conjectural cycles. This is based on a joint work with Liang Xiao. The construction consists of two parts. The first part is a parametrization of the irreducible components of certain affine Deligne-Lusztig varieties (and its mixed characteristic analogue). The Mirkovic-Vilonen theory plays an important role here. By the Rapoport-Zink uniformization, they provide the conjectural cycles. The second part is to calculate the intersection matrix of these cycles (still work in progress). Using the generalization of some recent ideas of V. Lafforgue, we reduce this calculation to certain intersection numbers of cycles in the affine Grassmannian, which again can be understood via geometric Satake.

No seminar on Monday. Xinwen Zhu will give his next talk on Thursday April 24.

Thursday (April 24), 4:30 p.m, room E 206. Xinwen Zhu (NWU). Cycles on modular varieties via geometric Satake. II. Abstract I will first describe certain conjectural Tate classes in the etale cohomology of the special fibers of modular varieties (Shimura varieities and the moduli space of Shtukas). According to the Tate conjecture, there should exist corresponding algebraic cycles. Then I will use ideas from geometric Satake to construct these conjectural cycles. This is based on a joint work with Liang Xiao. The construction consists of two parts. The first part is a parametrization of the irreducible components of certain affine Deligne-Lusztig varieties (and its mixed characteristic analogue). The Mirkovic-Vilonen theory plays an important role here. By the Rapoport-Zink uniformization, they provide the conjectural cycles. The second part is to calculate the intersection matrix of these cycles (still work in progress). Using the generalization of some recent ideas of V. Lafforgue, we reduce this calculation to certain intersection numbers of cycles in the affine Grassmannian, which again can be understood via geometric Satake.

No seminar next week. Dima Arinkin will speak on the Monday after next week (i.e., on May 5).

Monday (May 5), 4:30 p.m, room E 206. Dima Arinkin (Univ. of Wisconsin). Cohomology of line bundles on completely integrable systems. (The talk is introductory in nature and will be accessible to non-specialists). Abstract Let A be an abelian variety. The Fourier-Mukai transform gives an equivalence between the derived category of quasicoherent sheaves on A and the derived category of the dual abelian variety. The key step in the construction of this equivalence is the computation of the cohomology of A with coefficients in a topologically trivial line bundle. In my talk, I will provide a generalization of this result to (algebraic) completely integrable systems. Generically, an integrable system can be viewed as a family of (Lagrangian) abelian varieties; however, special fibers may be singular. We will show that the cohomology of fibers with coefficients in topologically trivial line bundles are given by the same formula (even if fibers are singular). The formula implies a `partial' Fourier-Mukai transform for completely integrable systems.

No seminar on Thursday May 8 and Monday May 12. Zhiwei Yun (Stanford) will speak on Thursday May 15.

Thursday (May 15), 4:30 p.m, room E 206. Zhiwei Yun (Stanford). Rigid automorphic representations and rigid local systems. Abstract We define what it means for an automorphic representation of a reductive group over a function field to be rigid. Under the Langlands correspondence, we expect them to correspond to rigid local systems. In general, rigid automorphic representations are easier to come up with than rigid local systems, and the Langlands correspondence between the two can be realized using techniques from the geometric Langlands program. Using this observation we construct several new families of rigid local systems, with applications to questions about motivic Galois groups and the inverse Galois problem over Q.

Monday (May 19), 4:30 p.m, room E 206. Alexander Goncharov (Yale). Hodge correlators and open string Hodge theory. Abstract Thanks to the work of Simpson, (which used results of Hitchin and Donaldson) we have an action of the multiplicative group of C on semisimple complex local systems on a compact Kahler manifold. We define Hodge correlators for semisimple complex local systems on a compact Kahler manifold, and show that they can be organized into an "open string theory data". Precisely, the category of semisimple local systems on a Kahler manifold gives rise to a BV algebra. Given a family of Kahler manifolds over a base B, these BV algebras form a variation (of pure twistor structures) on B. The Hodge correlators are organized into a solution of the quantum Master equation on B for this variation. Here are two special cases of this construction when the base B is a point. 1. Consider the genus zero part of the Hodge correlators. We show that it encodes a homotopy action of the twistor-Hodge Galois group by A-infinity autoequivalences of the category of smooth complexes on X. It extends the Simpson C^* action on semisimple local systems. It can be thought of as the Hodge analog (for smooth complexes) of the Galois group action on the etale site. 2. The simplest possible Hodge correlators on modular curves deliver Rankin-Selberg integrals for the special values of L-functions of modular forms at integral points, which, thanks to Beilinson, are known to be the regulators of motivic zeta-elements. We suggest that there is a similar open string structure on the category of all holonomic D-modules.

No more meetings of the seminar this year. Note that this week there is a conference at NWU on "Representation Theory, Integrable Systems and Quantum Fields", see http://www.math.northwestern.edu/emphasisyear/

As usual, the seminar will meet on Mondays and/or Thursdays in room E206 at 4:30 p.m. The first meeting is on October 9 (Thursday). We will begin with talks by Gaitsgory (Oct 9 and possibly Oct 13) and by Bezrukavnikov (Oct 16 and possibly Oct 20).

October 9 (Thursday), 4:30 p.m, room E 206. Dennis Gaitsgory (Harvard). The Tamagawa number conjecture for function fields. I. Abstract This is a joint work with Jacob Lurie. In the case of the function field of a curve X, the Tamagawa number conjecture can be reformulated as the formula for the weighted sum of isomorphism classes of G-bundles on X. During the talk on Thursday we will show how this formula follows from the Atiyah-Bott formula for the cohomology of the moduli space Bun_G of G-bundles on X. On Monday we will show how to deduce the Atiyah-Bott formula from another local-to-global result, namely the so-called non-Abelian Poincare duality (the latter says that Bun_G is uniformized by the affine Grassmannian of G with contractible fibers). The deduction {non-Abelian Poincare duality}-->{Atiyah-Bott formula} will be based on performing Verdier duality on the Ran space of X. This is a non-trivial procedure, and most of the talk will be devoted to explanations of how to make it work.

Thursday (October 9), 4:30 p.m, room E 206. Dennis Gaitsgory (Harvard). The Tamagawa number conjecture for function fields. I. Abstract This is a joint work with Jacob Lurie. In the case of the function field of a curve X, the Tamagawa number conjecture can be reformulated as the formula for the weighted sum of isomorphism classes of G-bundles on X. During the talk on Thursday we will show how this formula follows from the Atiyah-Bott formula for the cohomology of the moduli space Bun_G of G-bundles on X. On Monday we will show how to deduce the Atiyah-Bott formula from another local-to-global result, namely the so-called non-Abelian Poincare duality (the latter says that Bun_G is uniformized by the affine Grassmannian of G with contractible fibers). The deduction {non-Abelian Poincare duality}-->{Atiyah-Bott formula} will be based on performing Verdier duality on the Ran space of X. This is a non-trivial procedure, and most of the talk will be devoted to explanations of how to make it work.

Monday (October 13), 4:30 p.m, room E 206. Dennis Gaitsgory (Harvard). The Tamagawa number conjecture for function fields. II. Abstract We will show how to deduce the Atiyah-Bott formula from another local-to-global result, namely the so-called non-Abelian Poincare duality (the latter says that Bun_G is uniformized by the affine Grassmannian of G with contractible fibers). The deduction {non-Abelian Poincare duality}-->{Atiyah-Bott formula} will be based on performing Verdier duality on the Ran space of X. This is a non-trivial procedure, and most of the talk will be devoted to explanations of how to make it work.

Gaitsgory's article is attached. The next meeting is on Thursday (Oct 16), 4:30 p.m, room E 206. Roman Bezrukavnikov (MIT). Geometry of second adjointness for p-adic groups Abstract Basic operations in representation theory of reductive p-adic groups are functors of parabolic induction and restriction (also known as Jacquet functor). It is clear from the definitions that the induction functor is right adjoint to the Jacquet functor. It was discovered by Casselman and Bernstein in (or around) 1970's that the two functors satisfy also another, less obvious adjointness. I will describe a joint work with D.Kazhdan devoted to a geometric construction of this adjointness. We will show that it comes from a map on spaces of functions which is formally similar to (but is not known to be formally related to) nearby cycles for D-modules.

**Attachment:
Denis on Tamagawa.pdf**

Bernstein's pre-print on second adjointness and his lectures on representations of p-adic groups can be found at http://www.math.uchicago.edu/~mitya/langlands.html ____________________________________________ No seminar on Monday. ____________________________________________ Thursday (Oct 23), 4:30 p.m, room E 206. Amnon Yekutiel (Ben Gurion University). Local Beilinson-Tate Operators. Abstract In 1968 Tate introduced a new approach to residues on algebraic curves, based on a certain ring of operators that acts on the completion at a point of the function field of the curve. This approach was generalized to higher dimensional algebraic varieties by Beilinson in 1980. However Beilinson's paper had very few details, and his operator-theoretic construction remained cryptic for many years. Currently there is a renewed interest in the Beilinson-Tate approach to residues in higher dimensions (by Braunling, Wolfson and others). This current work also involves n-dimensional Tate spaces and is related to chiral algebras. In this talk I will discuss my recent paper arXiv:1406.6502, with same title as the talk. I introduce a variant of Beilinson's operator-theoretic construction. I consider an n-dimensional topological local field (TLF) K, and define a ring of operators E(K) that acts on K, which I call the ring of local Beilinson-Tate operators. My definition is of an analytic nature (as opposed to the original geometric definition of Beilinson). I study various properties of the ring E(K). In particular I show that E(K) has an n-dimensional cubical decomposition, and this gives rise to a residue functional in the style of Beilinson-Tate. I conjecture that this residue functional coincides with the residue functional that I had constructed in 1992 (itself an improved version of the residue functional of Parshin-Lomadze). Another conjecture is that when the TLF K arises as the Beilinson completion of an algebraic variety along a maximal chain of points, then the ring of operators E(K) that I construct, with its cubical decomposition (the depends only on the TLF structure of K), coincides with the cubically decomposed ring of operators that Beilinson constructed in his original paper (and depends on the geometric input). In the talk I will recall the necessary background material on semi-topological rings, high dimensional TLFs, the TLF residue functional and the Beilinson completion operation (all taken from Asterisque 208).

Thursday (Oct 23), 4:30 p.m, room E 206. Amnon Yekutieli (Ben Gurion University). Local Beilinson-Tate Operators. Abstract In 1968 Tate introduced a new approach to residues on algebraic curves, based on a certain ring of operators that acts on the completion at a point of the function field of the curve. This approach was generalized to higher dimensional algebraic varieties by Beilinson in 1980. However Beilinson's paper had very few details, and his operator-theoretic construction remained cryptic for many years. Currently there is a renewed interest in the Beilinson-Tate approach to residues in higher dimensions (by Braunling, Wolfson and others). This current work also involves n-dimensional Tate spaces and is related to chiral algebras. In this talk I will discuss my recent paper arXiv:1406.6502, with same title as the talk. I introduce a variant of Beilinson's operator-theoretic construction. I consider an n-dimensional topological local field (TLF) K, and define a ring of operators E(K) that acts on K, which I call the ring of local Beilinson-Tate operators. My definition is of an analytic nature (as opposed to the original geometric definition of Beilinson). I study various properties of the ring E(K). In particular I show that E(K) has an n-dimensional cubical decomposition, and this gives rise to a residue functional in the style of Beilinson-Tate. I conjecture that this residue functional coincides with the residue functional that I had constructed in 1992 (itself an improved version of the residue functional of Parshin-Lomadze). Another conjecture is that when the TLF K arises as the Beilinson completion of an algebraic variety along a maximal chain of points, then the ring of operators E(K) that I construct, with its cubical decomposition (the depends only on the TLF structure of K), coincides with the cubically decomposed ring of operators that Beilinson constructed in his original paper (and depends on the geometric input). In the talk I will recall the necessary background material on semi-topological rings, high dimensional TLFs, the TLF residue functional and the Beilinson completion operation (all taken from Asterisque 208).

Monday (Oct 27), 4:30 p.m, room E 206. Adam Gal (Tel Aviv University). Self-adjoint Hopf categories and Heisenberg categorification. Abstract We use the language of higher category theory to define what we call a "symmetric self-adjoint Hopf" (SSH) structure on a semisimple abelian category, which is a categorical analog of Zelevinsky's positive self-adjoint Hopf algebras. As a first result, we obtain a categorical analog of the Heisenberg double and its Fock space action, which is constructed in a canonical way from the SSH structure.

No seminar on Thursday. ______________________________ Monday (Nov 3), 4:30 p.m, room E 206. Francis Brown (IHES). Periods, iterated integrals and modular forms. Abstract It is conjectured that there should be a Galois theory of certain transcendental numbers called periods. Using this as motivation, I will explain how the notion of motivic periods gives a setting in which this can be made to work. The goal is then to use geometry to compute the Galois action on interesting families of (motivic) periods. I will begin with the projective line minus three points, whose periods are multiple zeta values, and try to work up to the upper half plane modulo SL_2(Z), whose periods correspond to multiple versions of L-values of modular forms.

Monday (Nov 3), 4:30 p.m, room E 206. Francis Brown (IHES). Periods, iterated integrals and modular forms. Abstract It is conjectured that there should be a Galois theory of certain transcendental numbers called periods. Using this as motivation, I will explain how the notion of motivic periods gives a setting in which this can be made to work. The goal is then to use geometry to compute the Galois action on interesting families of (motivic) periods. I will begin with the projective line minus three points, whose periods are multiple zeta values, and try to work up to the upper half plane modulo SL_2(Z), whose periods correspond to multiple versions of L-values of modular forms.

Thursday (Nov 6), 4:30 p.m, room E 206. Francis Brown will continue on Thursday Nov 6 (4:30 p.m, room E 206).

Monday (Nov 10), 4:30 p.m, room E 206. Sam Raskin (MIT). Chiral principal series categories. I. Abstract We will discuss geometric Langlands duality for unramified principal series categories. This generalizes (in a roundabout way) some previous work in local geometric Langlands to the setting where points in a curve are allowed to move and collide. Using this local theory, we obtain applications to the global geometric program, settling a conjecture of Gaitsgory in the theory of geometric Eisenstein series.

Thursday (Nov 13), 4:30 p.m, room E 206. Sam Raskin (MIT). Chiral principal series categories. II.

Attached is Sam Raskin's write-up on "D-modules in infinite type", which could help you understand his yesterday talk. As I said, Sam will give his second talk on Thursday (Nov 13), 4:30 p.m, room E 206.

**Attachment:
D-modules in infinite type.pdf**

1. Sam Raskin's notes of his talks are attached. 2. No seminar next week. 3. Afterward, Keerthi Madapusi Pera will give several talks. I asked him to us some "fairy tales" about Shimura varieties which appear as quotients of the symmetric space SO(2,n)/{SO(2)\times SO(n)}. (Here "fairy tale" means "an understandable talk for non-experts about something truly mysterious mathematical objects".) Keerthi will speak on (some of) the following dates: Nov 24, Dec 1, Dec 4. The date of his first talk and the title&abstract of his series of talks will be announced later.

**Attachment:
Sam Raskin's notes.pdf**

Monday (Nov 24), 4:30 p.m, room E 206. Keerthi Madapusi Pera. Orthogonal Shimura varieties and their canonical models. I. Abstract The protagonists of the talk are arithmetic quotients of certain real semi-algebraic Grassmannians associated with quadratic spaces of signature (n,2). They are natural generalizations of the modular curves: the upper half plane can be seen as a real Grassmannian of signature (1,2). In certain cases, these spaces are also closely related to the moduli spaces for K3 surfaces. Quite miraculously, it turns out that these spaces are quasi-projective algebraic varieties defined over the rational numbers, and even the integers. One reason this is surprising is that they are not known to be the solution to any natural moduli problem. However, due to the work of many people, beginning with Deligne, we can say quite a bit about them by using the 'motivic' properties of cohomological cycles on abelian varieties. This talk will mainly be a leisurely explication of this last sentence.

Monday (Dec 1), 4:30 p.m, room E 206. Keerthi Madapusi Pera. Orthogonal Shimura varieties and their canonical models. II.

Monday (Dec 1), 4:30 p.m, room E 206. Keerthi Madapusi Pera. Orthogonal Shimura varieties and their canonical models. II.

No more meetings of the seminar this quarter.

The first meeting is on Thursday (Jan 8), 4:30 p.m, room E 206. Jacob Lurie will give two unrelated talks on Thursday (Jan 8) and Monday (Jan 12). The titles and abstracts are below. ****** Thursday (Jan 8), 4:30 p.m, room E 206. Jacob Lurie (Harvard). Roots of Unity in Stable Homotopy Theory Abstract In classical algebraic geometry, there is often a stark difference between the behavior of fields of characteristic zero (such as the complex numbers) and fields of characteristic p (such as finite fields). For example, the equation x^p = 1 has p distinct solutions over the field of complex numbers, but only one solution over any field of characteristic p. In this talk, I'll introduce the subject of K(n)-local stable homotopy theory, which in some sense interpolates between characteristic zero and characteristic p, and describe the curious behavior of roots of unity in this intermediate regime. ****** Monday (Jan 12), 4:30 p.m, room E 206. Jacob Lurie (Harvard). Rotation Invariance in Algebraic K-Theory Abstract For any triangulated category C, one can introduce an abelian group K_0(C) which is freely generated by symbols [X] where X is an object of C, subject to the relation [X] = [X'] + [X''] whenever there is a distinguished triangle X' -> X -> X''. This relation immediately implies that the double suspension map from C to itself induces the identity map from K_0(C) to K_0(C). In this talk, I will describe a "delooping" of this observation, which asserts that the formation of algebraic K-theory is equivariant with respect to a certain action of the circle group U(1).

Thursday (Jan 8), 4:30 p.m, room E 206. Jacob Lurie (Harvard). Roots of Unity in Stable Homotopy Theory Abstract In classical algebraic geometry, there is often a stark difference between the behavior of fields of characteristic zero (such as the complex numbers) and fields of characteristic p (such as finite fields). For example, the equation x^p = 1 has p distinct solutions over the field of complex numbers, but only one solution over any field of characteristic p. In this talk, I'll introduce the subject of K(n)-local stable homotopy theory, which in some sense interpolates between characteristic zero and characteristic p, and describe the curious behavior of roots of unity in this intermediate regime.

Monday (Jan 12), 4:30 p.m, room E 206. Jacob Lurie (Harvard). Rotation Invariance in Algebraic K-Theory Abstract For any triangulated category C, one can introduce an abelian group K_0(C) which is freely generated by symbols [X] where X is an object of C, subject to the relation [X] = [X'] + [X''] whenever there is a distinguished triangle X' -> X -> X''. This relation immediately implies that the double suspension map from C to itself induces the identity map from K_0(C) to K_0(C). In this talk, I will describe a "delooping" of this observation, which asserts that the formation of algebraic K-theory is equivariant with respect to a certain action of the circle group U(1).

No seminar on Thursday (Jan 15) and Monday (Jan 19). On Thursday next week (i.e., on Jan 22) there will be a talk by Carlos Simpson.

Thursday (Jan 22), 4:30 p.m, room E 206. Carlos Simpson (University of Nice Sophia Antipolis). Constructing two-dimensional buildings. Abstract This reports on work in progress with Katzarkov, Noll and Pandit. We would like to generalize the leaf-space tree of a quadratic differential, to spectral curves for higher-rank Higgs bundles. Our current work concerns $SL_3$. In this case the corresponding buildings have dimension two. Given a spectral curve corresponding to multivalued differential $(\phi _1,\phi _2,\phi _3)$ we propose a construction by a successive series of cut and paste steps, of a universal pre-building. The distance function in this pre-building calculates the exponent for any WKB problem with limiting spectral curve $\phi$. The construction is conditioned on non-existence of BPS states in the Gaiotto-Moore-Neitzke spectral network.

No seminar on Monday. ********* Thursday (Jan 29), 4:30 p.m, room E 206. Simon Schieder (Harvard). The Drinfeld-Lafforgue-Vinberg degeneration of the stack of G-bundles. Abstract We study the singularities of the Drinfeld-Lafforgue-Vinberg compactification of the moduli stack of G-bundles on a smooth projective curve for a reductive group G. The definition of this compactification is due to Drinfeld and relies on the Vinberg semigroup of G. We will mostly focus on the case G=SL_2; in this case the compactification can alternatively be viewed as a canonical one-parameter degeneration of the moduli stack of SL_2-bundles. We then study the singularities of this one-parameter degeneration via the associated nearby cycles construction. Time permitting, we might sketch a generalization to the case of an arbitrary reductive group G and the relation to Langlands duality.

Thursday (Jan 29), 4:30 p.m, room E 206. Simon Schieder (Harvard). The Drinfeld-Lafforgue-Vinberg degeneration of the stack of G-bundles. Abstract We study the singularities of the Drinfeld-Lafforgue-Vinberg compactification of the moduli stack of G-bundles on a smooth projective curve for a reductive group G. The definition of this compactification is due to Drinfeld and relies on the Vinberg semigroup of G. We will mostly focus on the case G=SL_2; in this case the compactification can alternatively be viewed as a canonical one-parameter degeneration of the moduli stack of SL_2-bundles. We then study the singularities of this one-parameter degeneration via the associated nearby cycles construction. Time permitting, we might sketch a generalization to the case of an arbitrary reductive group G and the relation to Langlands duality.

No seminar on Monday. ********* Thursday (Feb 5), 4:30 p.m, room E 206. Ivan Mirkovic (Univ. of Massachusetts, Amherst): Loop Grassmannians from the point of view of local spaces. Abstract The loop Grassmannians of reductive groups will be reconstructed in the setting of “local spaces” over a curve. The structure of a local space is a version of the fundamental structure of a factorization space introduced and developed by Beilinson and Drinfeld. The weakening of the requirements formalizes some well known examples of “almost factorization spaces'' . The change of emphases leads to new constructions. The main example will be generalizations of loop Grassmannians corresponding to quadratic forms Q on based lattices. The quadratic form corresponding to the loop Grassmannian of a simply connected group G is essentially the "basic level" of G.

Thursday (Feb 5), 4:30 p.m, room E 206. Ivan Mirkovic (Univ. of Massachusetts, Amherst): Loop Grassmannians from the point of view of local spaces. Abstract The loop Grassmannians of reductive groups will be reconstructed in the setting of “local spaces” over a curve. The structure of a local space is a version of the fundamental structure of a factorization space introduced and developed by Beilinson and Drinfeld. The weakening of the requirements formalizes some well known examples of “almost factorization spaces'' . The change of emphases leads to new constructions. The main example will be generalizations of loop Grassmannians corresponding to quadratic forms Q on based lattices. The quadratic form corresponding to the loop Grassmannian of a simply connected group G is essentially the "basic level" of G.

Ivan Mirkovic will continue his talk on Monday (Feb 9), 4:30 p.m, room E 206. The talk will recall the so called zastava spaces which appear in several places in the geometric representation theory.. The goal is to make them accessible by comparing different points of view and emphasizing the examples and the visual intuition given by the corresponding polytopes. (Ivan's first talk on generalizing loop Grassmannians tried to introduce the characters of the story and one these was the zastava. However, the two talks are independent of each other.) PS. Ivan's web page http://people.math.umass.edu/~mirkovic/ now contains a section "NOTES on Loop Grassmannians, Zastava Spaces.Semiinfinite Grassmannians". Some of these may be helpful (but not necessary). 1. The various definitions of Zastava spaces are compared in - Zastava Spaces <http://people.math.umass.edu/%7Emirkovic/xx.LoopGrassmannians/BC.ZastavaSpaces.pdf> 2. The Zastva spaces are the Beilinson-Drinfeld deformations of intersections of the opposite semiinfinite orbits in Loop Grassmannians. The semiinfinite orbits and their intersections contain the information on the negative part of the enveloping algebra of the Langlands dual of our reductive group. This is not essential for understanding the zastava spaces but is a useful part of the landscape. - Loop Grassmannian construction of the negative part of the Enveloping Algebra for the Langlands dual group. <http://www.math.umass.edu/%7Emirkovic/xx.LoopGrassmannians/ALGEBRAS/LoopGrassmannianConstruction.of.NegativeEnvelopingAlgebra.pdf> 3. The zastava spaces appeared in the paper Smiinfinite Flags I and 2 with Finkelberg, Feigin, Kuznetsov. These papers also contain mujch more -- the computation of the intersection cohomology of zastava spaces, a construction of a skeleton of the semiinfinite Grassmannian, a construction of the enveloping algebra of the Langlands dual group etc. This is surveyed in > - > <http://people.math.umass.edu/%7Emirkovic/xx.LoopGrassmannians/BC.ZastavaSpaces.pdf> - Notes on the papers Semiinfinite Flags I and II. > <http://people.math.umass.edu/%7Emirkovic/xx.LoopGrassmannians/SemiinfiniteFlagsPapers.Notes.pdf>

Ivan Mirkovic will continue his talk on Zastava spaces on Thursday (Feb 12), 4:30 p.m, room E 206.

Ivan Mirkovic will continue his talk on Zastava spaces on Monday (Feb 16), 4:30 p.m, room E 206.

No seminar on Thursday. On Monday (Feb 23) Ngo Bao Chau will begin to speak on his recent work. Title: Local unramified L-factor and singularity in a reductive monoid. Abstract We are interested in the (unramified) test function on a reductive group over a non-archimedean local field which gives rise to a local (unramified) factor of Langlands' automorphic L-function. Langlands' automorphic L-function depends on an algebraic representation of the dual group. In the case of the standard representation of GL(n), this test function is essentially the characteristic function of the space of matrices with integral coefficients, according to Godement-Jacquet. For a general representation, the test function is related to singularity of a reductive monoid which was constructed by Braverman and Kazhdan.

Monday (Feb 23), 4:30 p.m, room E 206. Ngo Bao Chau. Local unramified L-factor and singularity in a reductive monoid Abstract We are interested in the (unramified) test function on a reductive group over a non-archimedean local field which gives rise to a local (unramified) factor of Langlands' automorphic L-function. Langlands' automorphic L-function depends on an algebraic representation of the dual group. In the case of the standard representation of GL(n), this test function is essentially the characteristic function of the space of matrices with integral coefficients, according to Godement-Jacquet. For a general representation, the test function is related to singularity of a reductive monoid which was constructed by Braverman and Kazhdan.

Monday (March 2), 4:30 p.m, room E 206. Ngo Bao Chau. Local unramified L-factor and singularity in a reductive monoid. II. Abstract We are interested in the (unramified) test function on a reductive group over a non-archimedean local field which gives rise to a local (unramified) factor of Langlands' automorphic L-function. Langlands' automorphic L-function depends on an algebraic representation of the dual group. In the case of the standard representation of GL(n), this test function is essentially the characteristic function of the space of matrices with integral coefficients, according to Godement-Jacquet. For a general representation, the test function is related to singularity of a reductive monoid which was constructed by Braverman and Kazhdan.

Monday (March 2), 4:30 p.m, room E 206. Ngo Bao Chau. Local unramified L-factor and singularity in a reductive monoid. II. Abstract We are interested in the (unramified) test function on a reductive group over a non-archimedean local field which gives rise to a local (unramified) factor of Langlands' automorphic L-function. Langlands' automorphic L-function depends on an algebraic representation of the dual group. In the case of the standard representation of GL(n), this test function is essentially the characteristic function of the space of matrices with integral coefficients, according to Godement-Jacquet. For a general representation, the test function is related to singularity of a reductive monoid which was constructed by Braverman and Kazhdan.

No more meetings of the seminar this quarter. ********** Jochen Heinloth from University of Essen will give a series of lectures starting on this Friday, March 6, 2-3PM, room E203. The topic of his lectures will be: An introduction to the P=W conjecture and related conjectures of Hausel. Abstract I will try to explain the work of Hausel and Rodriguez-Villegas and de Cataldo-Hausel-Migliorini resulting in a series of conjectures on the global geometry of moduli spaces of Higgs bundles. The starting point will be the different algebraic structures on the manifold underlying the moduli space of Higgs bundles on a curve. Hausel and Rodriguez-Villegas managed to do point counting arguments in one of the complex structures (the character variety) and using this, they found interesting properties of the cohomology that are reminiscent of properties of intersection cohomology. This finally led de Cataldo-Hausel-Migliorini to propose the P=W conjecture which they could prove in some cases.

No meetings of the seminar during the first week of the quarter. First meeting: Monday April 6 (4:30 p.m, room E 206). On April 6 Nick Rozenblyum will give the first talk in a series devoted to his joint work with David Ayala and John Francis. Title: Higher categories and manifold topology. I. Abstract Over the past few decades, there has been a fruitful interplay between manifold topology and (higher) category theory. I will give an overview of some of these connections, and discuss joint work with David Ayala and John Francis, which describes higher categories in terms of the topology of stratified manifolds. This approach provides a precise dictionary between manifold topology and higher category theory, and makes numerous connections between the two manifest.

Monday (April 6), 4:30 p.m, room E 206. Nick Rozenblyum. Higher categories and manifold topology. This is the first talk in a series. Most probably, Nick will also speak on Thursday April 9. The next talks will be given by David Ayala, see http://www.math.uchicago.edu/calendar?calendar=Geometric%20Langlands Abstract Over the past few decades, there has been a fruitful interplay between manifold topology and (higher) category theory. I will give an overview of some of these connections, and discuss joint work with David Ayala and John Francis, which describes higher categories in terms of the topology of stratified manifolds. This approach provides a precise dictionary between manifold topology and higher category theory, and makes numerous connections between the two manifest.

Thursday (Apr 9), 4:30 p.m, room E 206. Nick Rozenblyum. Higher categories and manifold topology.II.

No seminar on Monday (Apr 13). On Thursday Apr 16 David Ayala will give his first talk. Title: Higher categories are sheaves on manifolds. Abstract This project is an effort to merge higher algebra/category theory and differential topology. As an outcome, information flows in both directions: coherent constructions of manifold and embedding invariants from higher algebraic/categorical data, such as that of a representation of a quantum group lending to knot invariants; deformations of higher algebraic/categorical parameters indexed by manifolds, such as Hochschild (co)homology. The talks will be framed by one main result, and a couple formal applications thereof. The main construction is factorization homology with coefficients in higher categories. The body of the talks will focus on essential aspects of our definitions that facilitate the coherent cancelations that support our main result. This is joint work with John Francis and Nick Rozenblyum.

Thursday (Apr 16), 4:30 p.m, room E 206. David Ayala (Montana State University). Higher categories are sheaves on manifolds. I. Abstract This project is an effort to merge higher algebra/category theory and differential topology. As an outcome, information flows in both directions: coherent constructions of manifold and embedding invariants from higher algebraic/categorical data, such as that of a representation of a quantum group lending to knot invariants; deformations of higher algebraic/categorical parameters indexed by manifolds, such as Hochschild (co)homology. The talks will be framed by one main result, and a couple formal applications thereof. The main construction is factorization homology with coefficients in higher categories. The body of the talks will focus on essential aspects of our definitions that facilitate the coherent cancelations that support our main result. This is joint work with John Francis and Nick Rozenblyum.

Monday (Apr 20), 4:30 p.m, room E 206. David Ayala. Higher categories are sheaves on manifolds. II.

Thursday (April 23), 4:30 p.m, room E 206. Bhargav Bhatt (University of Michigan). Integral p-adic Hodge theory. Abstract Let X be a smooth and proper scheme over the ring of integers in a p-adic field. Classical p-adic Hodge theory relates the etale and de Rham cohomologies of X: the theories are naturally identified after extending scalars to a suitable ring of periods constructed by Fontaine. This isomorphism is compatible with Galois actions, and thus plays a crucial role in our understanding of Galois representations. This identification, however, neglects all torsion phenomena as the period ring is a Q-algebra. In my talk, I will briefly recall this rational story, and then describe a new "comparison" between these two cohomology theories that works integrally: we will realize the de Rham cohomology of X as a specialization of the etale cohomology, integrally, over a 2-parameter base. As an application, we deduce the optimal result relating torsion in the two theories: the torsion in de Rham cohomology is an upper bound for the torsion in etale cohomology (and the inequality can be strict). This inequality can be used to explain some of the "pathologies" in de Rham cohomology in characteristic p. (Based on joint work in progress with Morrow and Scholze.)

No seminar next week.

1. No seminar this week. 2. Joseph Bernstein will be visiting our department starting from Wednesday May 6. He will give a seminar talk on Monday May 11 and at least one talk after that (on May 18 and maybe May 21). Yiannis Sakellaridis will be visiting us on May 13-16; he will speak on May 14. The titles and abstracts can be found at http://www.math.uchicago.edu/calendar?calendar=Geometric%20Langlands 3. A mysterious theorem is formulated on p.2 of the article http://arxiv.org/pdf/math/0701615v3.pdf by Kumar, Lusztig, and Dipendra Prasad. As explained there, the theorem is proved in Jantzen's Ph.D. thesis (1973). An equivalent formulation is given in the Corollary (also on p.2). DOES ANYBODY KNOW A CONCEPTUAL EXPLANATION of the result? (E.g., is there any categorical statement behind it?) The result is as follows. Let G be a connected simply connected almost-simple group equipped with a pinning (English translation of Bourbaki's "epinglage"). Let \sigma be a nontrivial automorphism of the Dynkin diagram, then \sigma acts on G. Let G^\sigma denote the subgroup of fixed points. Let G_\sigma denote the simply connected group whose root system is dual to that of G^\sigma . The Corollary on p.2 says that the \sigma-characters of G are equal to the characters of G_\sigma . (This is mysterious because there is no apparent relation between G and G_\sigma and also because passing to the dual root system is just a formal combinatorial operation).

Monday (May 11), 4:30 p.m, room E 206. Joseph Bernstein (Tel Aviv University). Stacks in Representation Theory. (What is a representation of an algebraic group?) Abstract I will discuss a new approach to representation theory of algebraic groups. In the usual approach one starts with an algebraic group G over some local (or finite) field F, considers the group G(F) of its F-points as a topological group and studies some category Rep (G(F)) of continuous representations of the group G(F). I will argue that more correct objects to study are some kind of sheaves on the stack BG corresponding to the group G. I will show that this point of view naturally requires to change the formulation of some basic problems in Representation Theory. In particular, this approach might explain the appearance of representations of all pure forms of a group G in Vogan's formulation of Langlands' correspondence.

Thursday (May 14), 4:30 p.m, room E 206. Yiannis Sakellaridis (Rutgers University). Spectral decomposition on homogeneous spaces. Abstract I will present results from my joint work with Venkatesh, Delorme and Harinck on harmonic analysis on homogeneous spaces. These results have been established for spherical homogeneous spaces over p-adic fields, but most of their analogs exist for automorphic functions, and the talk will attempt to cover those as well (especially in the function field case, where technicalities due to Archimedean places do not arise). The general structure of these results is the following: For a given G-space X, there are "simpler" G-spaces X_i (of the same dimension but with more symmetries, i.e. non-trivial groups of G-automorphisms) such that functions on X decompose into a "discrete modulo automorphisms" part plus a homomorphic image of the "discrete modulo automorphisms" part of the spaces X_i. There are smooth and L^2 versions of this story, and for the former the word "discrete" should be replaced by "cuspidal". The talk will emphasize general principles (largely based on ideas of Joseph Bernstein) that give rise to the same kind of decomposition irrespective of the space, as well as points in the method that have still not been clarified enough.

SPECIAL SEMINAR on Tuesday (May 19), 1:30 p.m, room E 206. Pham H. Tiep (University of Arizona). Bounding character values of finite groups of Lie type. Abstract Let G be a finite group of Lie type. In spite of many foundational results on complex representation theory of G, several questions on character values still remain open. One such question, essential for various applications, including random walks and word maps on finite simple groups, is to obtain sharp bounds on |\chi(g)| for all elements g in G and for all irreducible complex characters \chi of G. In the case of symmetric groups, this problem has been solved by Larsen and Shalev. We will discuss recent progress on this problem for finite groups of Lie type, obtained in joint work with R. Bezrukavnikov, M. Liebeck, and A. Shalev.

Monday (May 18), 4:30 p.m, room E 206. Joseph Bernstein (Tel Aviv University). Convexity and subconvexity bounds for Automorphic Periods (joint work with A. Reznikov). Abstract In my lecture I will discuss basic problems related to bounds on periods of automorphic functions. One of my goals is to discuss the insight into these bounds given by relation of periods to special values of L-functions (for example this predicts some convexity and subconvexity bounds on periods coming from L-function theory). I describe a method based on Representation Theory of real groups that allows to analyze such bounds. I will concentrate on two very concrete problems. Let Y be a compact Riemannian surface of constant curvature -1. A Maass form is a function f on Y that is an eigenfunction of the Laplace operator D. Problem 1 (Fourier expansion). Fix a closed geodesic C in Y , restrict some Maass form f to the circle C and consider Fourier coefficients a_k of this function. How to give bounds on these coefficients when k tends to infinity ? Problem 2 (Triple product). Let p be a product of two Maass forms. How to give bound on the scalar product <p, f> of the function p with a Maass form f in terms of the eigenvalue of f when this eigenvalue tends to infinity ?

Monday (May 18), 4:30 p.m, room E 206. Joseph Bernstein (Tel Aviv University). Convexity and subconvexity bounds for Automorphic Periods (joint work with A. Reznikov). Abstract In my lecture I will discuss basic problems related to bounds on periods of automorphic functions. One of my goals is to discuss the insight into these bounds given by relation of periods to special values of L-functions (for example this predicts some convexity and subconvexity bounds on periods coming from L-function theory). I describe a method based on Representation Theory of real groups that allows to analyze such bounds. I will concentrate on two very concrete problems. Let Y be a compact Riemannian surface of constant curvature -1. A Maass form is a function f on Y that is an eigenfunction of the Laplace operator D. Problem 1 (Fourier expansion). Fix a closed geodesic C in Y , restrict some Maass form f to the circle C and consider Fourier coefficients a_k of this function. How to give bounds on these coefficients when k tends to infinity ? Problem 2 (Triple product). Let p be a product of two Maass forms. How to give bound on the scalar product <p, f> of the function p with a Maass form f in terms of the eigenvalue of f when this eigenvalue tends to infinity ?

SPECIAL SEMINAR on Tuesday (May 19), 1:30 p.m, room E 206. Pham H. Tiep (University of Arizona). Bounding character values of finite groups of Lie type. Abstract Let G be a finite group of Lie type. In spite of many foundational results on complex representation theory of G, several questions on character values still remain open. One such question, essential for various applications, including random walks and word maps on finite simple groups, is to obtain sharp bounds on |\chi(g)| for all elements g in G and for all irreducible complex characters \chi of G. In the case of symmetric groups, this problem has been solved by Larsen and Shalev. We will discuss recent progress on this problem for finite groups of Lie type, obtained in joint work with R. Bezrukavnikov, M. Liebeck, and A. Shalev.

Thursday (May 21), 4:30 p.m, room E 206. Joseph Bernstein (Tel Aviv University). Convexity and subconvexity bounds of Automorphic Periods. II.

No more meetings of the seminar in this quarter.

As usual, the seminar will meet on Mondays and/or Thursdays in room E206 at 4:30 p.m. We begin on Thursday Oct 1 with Peter Sarnak’s talk on the very important notion of analytic conductor in the theory of automorphic forms, see http://www.math.uchicago.edu/calendar?calendar=Geometric%20Langlands for more details. On Thursday Oct 8 Laurent Fargues will begin his mini-course “Geometrization of the local Langlands correspondence”, in which he will explain his new and very exciting conjectures. (As far as I understand, they are in the spirit of the global unramified geometric Langalnds conjecture, but instead of usual curves he considers the “curve” that he and Fontaine had associated to an arbitrary non-Archimedean local field E. Because of that, he ends up with conjectures that would imply the local Langlands conjecture for E.) Presumably, Fargues will give 8 lectures. The title and abstract of his course can be found at http://www.math.uchicago.edu/calendar?calendar=Geometric%20Langlands

Thursday (Oct 1), 4:30 p.m, room E 206. Peter Sarnak (IAS). The analytic conductor in the theory of automorphic forms. Abstract The analytic conductor of an automorphic cusp form on GL(n) over a number field is a measure of its complexity: especially in connection with the corresponding L-function. We review some of the definitions, properties and central role played by the conductor. If time permits we discuss some recent applications to fast computations of epsilon factors and the Mobius function.

No seminar on Monday (Oct.5). On Thursday (Oct.8) Laurent Fargues will begin his mini-course on Geometrization of the local Langlands correspondence

Thursday (October 8), 4:30 p.m, room E 206. Laurent Fargues (Institut de Mathematiques de Jussieu) will begin his mini-course on Geometrization of the local Langlands correspondence I will explain a recent conjecture giving a geometrization of the local Langlands correspondence over a non-archimedean local field. The purpose is to explain the precise statement of the conjecture and evidences for it. For this I will introduce the objects that appear, in particular the curve defined and studied in our joint work with Fontaine, the structure of G-bundles on this curve and the basic properties of the associated stack of G-bundles. Presumably, there will be 8 lectures in the mini-course.

No seminar on Monday Oct 12. Laurent Fargues will give his second lecture next Thursday (Oct 15). I will inform you when the notes of his first lecture become available. Program for the rest of October: Laurent Fargues will speak on Oct 15, 22, 26. Bhargav Bhatt will give a talk on Oct 19. Information on the Oberwolfach workshop mentioned by Fargues is at www.mfo.de/occasion/1614/www_view

Laurent Fargues says that to understand his yesterday lecture, one can read one of the following articles available at his homepage http://webusers.imj-prg.fr/~laurent.fargues/Publications.html 1. Factorization of analytic functions in mixed characteristic 2. Vector bundles and p-adic Galois representations 3. Vector bundles on curves and p-adic Hodge theory The first article is the most elementary, and Fargues says that already that article covers the material of his yesterday lecture. (The third article is the most advanced.)

The notes of the first lecture by Fargues are at http://math.uchicago.edu/~seanpkh/farguesLL/notes.pdf The notes were made by Sean Howe. He says: > As the lectures continue I'll update that file. Already in the first > lecture there are probably some typos I've missed; I'll correct them as I > or others find them.

Laurent Fargues will give his second lecture on Thursday (Oct 15), 4:30 p.m, room E 206.

Monday (Oct 19), 4:30 p.m, room E 206. Bhargav Bhatt (University of Michigan). The Witt vector affine Grassmannian. Abstract The affine Grassmannian is an ind-variety over a field k that parametrizes k[[t]]-lattices in a fixed k((t))-vector space. I will explain the construction of a p-adic analog, i.e., an ind-scheme over F_p that parametrizes Z_p-lattices in a fixed Q_p-vector space. This builds on recent work of X. Zhu. The construction takes place in the world of algebraic geometry with perfect schemes, and a large portion of the talk will be devoted to explaining certain surprisingly nice features of this world. (This talk is based on joint work with Peter Scholze.)

Monday (Oct 19), 4:30 p.m, room E 206. Bhargav Bhatt (University of Michigan). The Witt vector affine Grassmannian. Abstract The affine Grassmannian is an ind-variety over a field k that parametrizes k[[t]]-lattices in a fixed k((t))-vector space. I will explain the construction of a p-adic analog, i.e., an ind-scheme over F_p that parametrizes Z_p-lattices in a fixed Q_p-vector space. This builds on recent work of X. Zhu. The construction takes place in the world of algebraic geometry with perfect schemes, and a large portion of the talk will be devoted to explaining certain surprisingly nice features of this world. (This talk is based on joint work with Peter Scholze.)

Laurent Fargues will give his third lecture on Thursday (Oct 22), 4:30 p.m, room E 206.

The notes at http://math.uchicago.edu/~seanpkh/farguesLL/notes.pdf now include the second lecture by Fargues. Fargues will give his third lecture on Thursday (usual time and place).

Laurent Fargues will give his next lecture on MONDAY (Oct 26), 4:30 p.m, room E 206.

No seminar on Thursday. The notes from Fargues' Thursday lecture are now online at http://math.uchicago.edu/~seanpkh/farguesLL/notes.pdf Laurent Fargues will give his next lecture on Monday (Nov 2), 4:30 p.m, room E 206.

Laurent Fargues will give his next lecture on Monday (Nov 2), 4:30 p.m, room E 206. ******* Many of us know the name of Andrei Zelevinsky, who worked in representation theory (including the Langlands program) and combinatorics. He died a few years ago, when he was only 60. I just learned that Northeastern University has established a prestigious post-doctoral position named in Zelevinsky's memory. More information (in particular, instructions how to donate) can be found at http://avzel.blogspot.com/2015/10/andrei-zelevinsky-research-instructor.html See also http://zelevinsky.com/Zelevinsky_Fund_Letter.pdf Andrei was a very good mathematician and a very good man. (We first met at a mathematical olympiad when he was 16 and I was 15. At that time he was very impressed by Vilenkin's book on combinatorics. Later combinatorics became a major area of his research...)

The notes of Fargues' Monday lecture are now online at http://math.uchicago.edu/~seanpkh/farguesLL/notes.pdf Laurent Fargues will give his next lecture tomorrow, i.e., Monday (Nov 2), 4:30 p.m, room E 206.

Laurent Fargues has to CANCEL his lecture today.

Laurent Fargues will give his next lecture on Thursday (Nov 5), 4:30 p.m, room E 206.

Laurent Fargues will give his next lecture on Monday (Nov 9), 4:30 p.m, room E 206.

Laurent Fargues will give his next lecture on Thursday (Nov 12), 4:30 p.m, room E 206.

Monday (Nov 16) and Thursday (Nov 19), 4:30 p.m, room E 206. Akshay Venkatesh (Stanford University). Motivic cohomology and the cohomology of arithmetic groups. Abstract The cohomology of an arithmetic group, e.g. SL_n(Z), carries a Hecke action. The same system of Hecke eigenvalues will usually occur in multiple cohomological degrees. This suggests the existence of extra endomorphisms of the cohomology that commute with the Hecke operators but shift cohomological degree. (In the Shimura case these are provided by "Lefschetz operators" but the situation in general is much more interesting.) I will explain a conjecture that, in fact, the motivic cohomology of the associated motives act on the cohomology and provide these extra endomorphisms. (According to the Langlands program, a Hecke eigenclass occurring in cohomology should give a system of motives, indexed by representations of the dual group. The "associated" motives we need are the ones associated to the adjoint representation of the dual group.) This structure should exist at the level of cohomology with Q-coefficients, but I don't know how to construct it. However, one can construct the corresponding action on cohomology with real or p-adic coefficients, using the corresponding regulator map on the motivic cohomology, and then try to get evidence that it preserves Q-structures. LECTURE ONE: I will explain the overall conjecture and how to construct the action with real coefficients. This is joint work with Prasanna. In particular, we are able to give evidence that the action preserves Q-structures, basically by relating it to the theory of periods of automorphic forms (e.g. things like the Gan--Gross--Prasad conjecture), and also by using analytic torsion. LECTURE TWO: I will explain the story with p-adic coefficients. The story here is algebraically richer; there are two related way of constructing extra endomorphisms of the cohomology. One is via a derived version of the Hecke algebra and one is via a derived version of Mazur's Galois deformation ring (joint with Soren Galatius). Here there is no evidence, at present, that this preserves Q-structures.

Thursday (Nov 19), 4:30 p.m, room E 206. Akshay Venkatesh (Stanford University). Motivic cohomology and the cohomology of arithmetic groups. II. Abstract The cohomology of an arithmetic group, e.g. SL_n(Z), carries a Hecke action. The same system of Hecke eigenvalues will usually occur in multiple cohomological degrees. This suggests the existence of extra endomorphisms of the cohomology that commute with the Hecke operators but shift cohomological degree. (In the Shimura case these are provided by "Lefschetz operators" but the situation in general is much more interesting.) I will explain a conjecture that, in fact, the motivic cohomology of the associated motives act on the cohomology and provide these extra endomorphisms. (According to the Langlands program, a Hecke eigenclass occurring in cohomology should give a system of motives, indexed by representations of the dual group. The "associated" motives we need are the ones associated to the adjoint representation of the dual group.) This structure should exist at the level of cohomology with Q-coefficients, but I don't know how to construct it. However, one can construct the corresponding action on cohomology with real or p-adic coefficients, using the corresponding regulator map on the motivic cohomology, and then try to get evidence that it preserves Q-structures. LECTURE ONE: I will explain the overall conjecture and how to construct the action with real coefficients. This is joint work with Prasanna. In particular, we are able to give evidence that the action preserves Q-structures, basically by relating it to the theory of periods of automorphic forms (e.g. things like the Gan--Gross--Prasad conjecture), and also by using analytic torsion. LECTURE TWO: I will explain the story with p-adic coefficients. The story here is algebraically richer; there are two related way of constructing extra endomorphisms of the cohomology. One is via a derived version of the Hecke algebra and one is via a derived version of Mazur's Galois deformation ring (joint with Soren Galatius). Here there is no evidence, at present, that this preserves Q-structures.

Laurent Fargues will speak on Monday (Nov 23), 4:30 p.m, room E 206. Notes from the 5th and 6th lectures by Fargues are now available at http://math.uchicago.edu/~seanpkh/farguesLL/notes.pdf Notes from the 7th lecture are expected to appear relatively soon (maybe on Sunday).

The notes of Fargues' November 12 lecture are now online at http://math.uchicago.edu/~seanpkh/farguesLL/notes.pdf He will speak today at the usual time and place.

Laurent Fargues will continue tomorrow (i.e., Tuesday Nov 24), 1:30 p.m, room E 202.

No more meetings of the seminar this quarter. Happy Thanksgiving!

The first meeting of the seminar is on Thursday (Jan 7), 4:30 p.m, room E 206. Alexander Beilinson will begin his series of talks on The singular support and characteristic cycle of etale sheaves. Abstract Singular support and characteristic cycle are fundamental notions of the theory of D-modules; they were rendered into the setting of constructible sheaves on (real or complex) analytic manifolds by Kashiwara and Shapira in their book on microlocal theory of sheaves. This series of talks treats the case of etale sheaves on varieties over a field of finite characteristic studied recently by T.Saito, http://lanl.arxiv.org/abs/1510.03018 and myself, http://lanl.arxiv.org/abs/1505.06768. In dimension one the story amounts to the classical Grothendieck-Ogg-Shafarevich formula. In the first talk I will remind, as a warm-up, the classical D-module and Kashiwara-Shapira pictures, formulate the main results, discuss a peculiar old observation by Deligne about non-integrability of characteristics, and introduce the main tool of Brylinski's Radon transform. If time permits, I will give a proof of the basic upper estimate on the dimension of the singular support.

Thursday (Jan 7), 4:30 p.m, room E 206. Alexander Beilinson will begin his series of talks on The singular support and characteristic cycle of etale sheaves. Abstract Singular support and characteristic cycle are fundamental notions of the theory of D-modules; they were rendered into the setting of constructible sheaves on (real or complex) analytic manifolds by Kashiwara and Shapira in their book on microlocal theory of sheaves. This series of talks treats the case of etale sheaves on varieties over a field of finite characteristic studied recently by T.Saito, http://lanl.arxiv.org/abs/1510.03018 and myself, http://lanl.arxiv.org/abs/1505.06768. In dimension one the story amounts to the classical Grothendieck-Ogg-Shafarevich formula. In the first talk I will remind, as a warm-up, the classical D-module and Kashiwara-Shapira pictures, formulate the main results, discuss a peculiar old observation by Deligne about non-integrability of characteristics, and introduce the main tool of Brylinski's Radon transform. If time permits, I will give a proof of the basic upper estimate on the dimension of the singular support.

Deligne's letter mentioned by Sasha is at http://math.uchicago.edu/~drinfeld/Deligne's_letter_SingSupp.pdf ****** Next talk: Monday (Jan 11), 4:30 p.m, room E 206. Daniil Rudenko (Moscow). Goncharov conjectures and functional equations for polylogarithms. Abstract Classical polylogarithms and functional equations which these functions satisfy have been studied since the beginning of the 19th century. Nevertheless, the structure of these equations is still understood very poorly. I will explain an approach to this subject, based on the link between polylogarithms and mixed Tate motives. A substantial part of the talk will be devoted to the explanation of this link, provided by Goncharov Conjectures. After that, I will present some results about functional equations which can be proved unconditionally. If time permits, I will finish with another application of this circle of ideas to scissor congruence theory.

A few years ago I wrote some notes for myself, which can be found here: http://www.math.uchicago.edu/~drinfeld/Cotangent_notes-2011/Notes-2011.pdf They are closely related to Sasha's Thursday talk. A brief explanation of the subject of my notes and the relation with Sasha's talk can be found here: http://www.math.uchicago.edu/~drinfeld/Cotangent_notes-2011/Read_me.pdf ****** As already announced, on Monday (Jan 11) Daniil Rudenko will speak on Goncharov conjectures and functional equations for polylogarithms.

Thursday (Jan 7), 4:30 p.m, room E 206. Alexander Beilinson will give his second talk on The singular support and characteristic cycle of etale sheaves.

Thursday (Jan 14), 4:30 p.m, room E 206. Alexander Beilinson will give his second talk on The singular support and characteristic cycle of etale sheaves.

No seminar on Monday. Beilinson will continue on Thursday (Jan 21).

Beilinson will continue on Monday (Jan 25).

Sasha will speak on Monday (as announced before). Here is Sean Howe's message about his notes of Fargues' lectures. -------------------------------------------------------------------------- Subject: Full notes available From: "Sean Howe" <seanpkh@gmail.com> Date: Sat, January 23, 2016 1:12 pm To: "Vladimir G. Drinfeld" <drinfeld@math.uchicago.edu> -------------------------------------------------------------------------- Hi all, The full notes from Fargues' lectures last quarter are now available on my website: http://math.uchicago.edu/~seanpkh/farguesLL/notes.pdf I am sorry for the very long delay in posting the final two lectures; I had some unresolved questions about the last lecture and had trouble finding the time to understand them over winter break. The notes are a preliminary version still -- eventually I will add more references, clean up typos, and most importantly fix some remaining issues with the final section on local-global compatibility (which is still very rough -- I've put a warning about this in the notes). But I thought it'd be better to get them up now and then worry about that in the future! Please let me know of any changes that should be made and I will try to incorporate them in a more timely fashion. Thanks! Best, Sean

Beilinson will continue on Thursday (Jan 28).

No seminar on Monday. The next meeting is on Thursday. The title and abstract of the talk are below. ("Schober" is a German word; one of its meanings is "haystack". My guess is that the Schobers from Kapranov's talk are "kind of stacks".) Thursday (Feb 4), 4:30 p.m, room E 206. Mikhail Kapranov (Kavli Institute, Japan). Perverse Schobers. Abstract I will explain the project of developing a theory of "perverse sheaves of triangulated categories". One motivation for it is the desire of introducing coefficients in the definition of Fukaya categories (which are categorical analogs of (co)homology with constant coefficients). Since perverse sheaves are complexes of sheaves and not just sheaves, their categorical analogs are not obvious. Nevertheless in several interesting cases the definition can be made, and we can make the first steps in the cohomological formalism. The talk is based on joint work with T. Dyckerhoff, V. Schechtman and Y. Soibelman.

Thursday (Feb 4), 4:30 p.m, room E 206. Mikhail Kapranov (Kavli Institute, Japan). Perverse Schobers. Abstract I will explain the project of developing a theory of "perverse sheaves of triangulated categories". One motivation for it is the desire of introducing coefficients in the definition of Fukaya categories (which are categorical analogs of (co)homology with constant coefficients). Since perverse sheaves are complexes of sheaves and not just sheaves, their categorical analogs are not obvious. Nevertheless in several interesting cases the definition can be made, and we can make the first steps in the cohomological formalism. The talk is based on joint work with T.Dyckerhoff, V.Schechtman and Y.Soibelman.

No seminar on Monday. Thursday (Feb 11), 4:30 p.m, room E 206. Mikhail Kapranov (Kavli Institute, Japan). Homotopy Lie algebras associated to secondary polytopes. Abstract Motivated by the work of Gaiotto-Moore-Witten on the "algebra of infrared", we construct a homotopy Lie algebra associated to the secondary polytope of a finite set A of points in the n-dimensional Euclidean space. While the construction can be made for any n (and leads to E_n-algebras), the case of "physical" interest is when A consists of critical values of a holomorphic Morse function. The talk is based on joint work with M. Kontsevich and Y. Soibelman.

Thursday (Feb 11), 4:30 p.m, room E 206. Mikhail Kapranov (Kavli Institute, Japan). Homotopy Lie algebras associated to secondary polytopes. Abstract Motivated by the work of Gaiotto-Moore-Witten on the "algebra of infrared", we construct a homotopy Lie algebra associated to the secondary polytope of a finite set A of points in the n-dimensional Euclidean space. While the construction can be made for any n (and leads to E_n-algebras), the case of "physical" interest is when A consists of critical values of a holomorphic Morse function. The talk is based on joint work with M. Kontsevich and Y. Soibelman.

Monday (Feb 15), 4:30 p.m, room E 206. Kiran Kedlaya (UCSD). Introduction to F-isocrystals. I Prof. Kedlaya kindly agreed to give two introductory lectures on this important subject (the second one on March 4). Abstract Let X be a variety over a field of characteristic p>0. The notion of l-adic local system on X has not one but two p-adic analogs, called "convergent F-isocrystal" and "overconvergent F-isocrystal". I will start from scratch and give an overview of the theory of F-isocrystals, paying close attention to analogies from the l-adic case.

Kiran Kedlaya will continue on March 4 (Friday) at 4:00 p.m. in room E202. No meetings until then. Kedlaya's notes on F-isocrystals are here: http://kskedlaya.org/papers/isocrystals.pdf

Kiran Kedlaya will continue on March 4 (Friday) at 4:00 p.m. in room E202. (Unusual day, time, and room!) Kedlaya's notes on F-isocrystals are updated: http://kskedlaya.org/papers/isocrystals.pdf

Kiran Kedlaya will speak tomorrow (Friday) at 4:00 p.m. in room E202. (Unusual day, time, and room!)

No seminar on Monday. Thursday (March 10), 4:30 p.m, room E 206. Dima Arinkin (Univ. of Wisconsin). Classical limit of the geometric Langlands correspondence. Abstract The classical limit of the geometric Langlands correspondence is the conjectural Fourier-Mukai equivalence between the Hitchin fibrations for a reductive group G and its dual. There was a significant progress on this statement when G=GL(n) and the Hitchin fibers are identified with compacitified Jacobians of spectral curves. Unfortunately, the methods are specific to the group GL(n), and much less is known about the case of general G. In my talk, I plan to review the current state of the area, and then sketch a new approach (work in progress joint with R.Fedorov). The approach is based on studying the classical limit of the Hecke `algebra', which turns out to be a much richer object than its usual (`quantum') counterpart.

Thursday (March 10), 4:30 p.m, room E 206. Dima Arinkin (Univ. of Wisconsin). Classical limit of the geometric Langlands correspondence. Abstract The classical limit of the geometric Langlands correspondence is the conjectural Fourier-Mukai equivalence between the Hitchin fibrations for a reductive group G and its dual. There was a significant progress on this statement when G=GL(n) and the Hitchin fibers are identified with compacitified Jacobians of spectral curves. Unfortunately, the methods are specific to the group GL(n), and much less is known about the case of general G. In my talk, I plan to review the current state of the area, and then sketch a new approach (work in progress joint with R.Fedorov). The approach is based on studying the classical limit of the Hecke `algebra', which turns out to be a much richer object than its usual (`quantum') counterpart.

No more meetings this quarter.

No meetings this week and the next one. On April 11 (Monday) Kazuya Kato will begin his series of talks.

Monday (April 11), 4:30 p.m, room E 206. Kazuya Kato. On compactifications of the moduli spaces of Drinfeld modules. I. Abstract The main subject in my talk is to construct (1) toroidal compactifications of the moduli spaces of Drinfeld modules. This is similar to the well known (2) toroidal compactifications of the moduli spaces of polarized abelian vareities. But there is a big difference as is explained below. This is a joint work with T. Fukaya and R. Sharifi. We are studying the analogue of Sharifi conjecture (it is a refined Iwasawa theory) for GL(n) of a function field, and then the nice toroidal boundary of the moduli space of Drinfeld modules became necessary. The moduli space of polarized abelian varieties has two kinds of compactification, Satake-Baily-Borel compactification and toroidal compactifications. For the moduli space of Drinfeld modules, an analogue of the Satake-Baily-Borel compactification was constructed by Kapranov and Pink. The analogy is very strong here. Pink wrote a short paper in 1994 on toroidal compactifications of the moduli space of Drinfeld modules. But the details are not yet published. For the toroidal compactification, there is a big difference between (1) and (2). These toroidal compactifications treat degenerations of Drinfeld modules and of polarized abelian varieties, respectively. In the degeneration, the local monodromy of a degenerating polarized abelian variety has length of unipotence two, but the local monodromy of a degenerating Drinfeld module can have bigger length of unipotence. In my talk, 1. I give an overview of the analytic theory and explain how such difference of (1) and (2) appears. 2. I explain the theory of degeneration of Drinfeld modules (theory of iterated Tate uniformizations, where the iteration is necessary to treat the larger length of the unipotence). To my regret, these 1 and 2 are the best things which I can explain well now. We have not yet completed a paper on the construction of the toroidal compactifications and we are not yet perfectly sure that the proofs are OK. I hope to explain the construction in a next opportunity.

No seminar on Thursday. Kazuya Kato will continue on Monday April 18.

Monday (April 18), 4:30 p.m, room E 206. Kazuya Kato. On compactifications of the moduli spaces of Drinfeld modules. II. Abstract An elliptic curve over \C is presented as \C/L where L is a \Z-lattice of rank 2. Tate discovered that an elliptic curve over a local field K has the presentation K^\times/q^{\Z} as the quotient of the multiplicative group of K by a \Z-lattice q^{\Z} of rank 1 if the elliptic curve has multiplicative reduction. This was generalized by Raynaud, Mumford, Faltings-Chai to higher dimensional abelian varieties over the quotient field of a complete local normal integral domain of higher dimension. In his first paper on Drinfeld modules, Drinfeld proved that the Drinfeld module over a local filed K has a similar presentation K/L as the quotient of the additive group K by a certain lattice L. I will explain how to generalize this theory of Drinfeld to the higher dimensional base case. This is a theory of degeneration of Drinfeld modules. This is important for the toroidal compactification of the moduli space of Drinfeld modules. The q of Tate in the case of an elliptic curve is the best coordinate function of the compactified modular curve at the cusp, but q is not an algebraic function. It is an analytic function or a function which appears after the completion. The best coordinate functions of the toroidal compactification at the boundary are not algebraic, and they appear to classify the lattice L which appears after the completion. There is a big difference from the case of abelian variety. This difference is due to the fact that the length of the unipotence of the local monodromy is two for abelian varieties but can be bigger for Drinfeld modules.

No seminar on Thursday April 21.

Thursday (April 28), 4:30 p.m, room E 206. Tsao-Hsien Chen (NWU) Quantization of Hitchin integrable system via positive characteristic. Abstract In their work "Quantization of Hitchin's integrable system and Hecke eigensheaves", Beilinson and Drinfeld give a construction of an automorphic D-module corresponding to a local system which carries an additional structure of an oper. In my talk, I will explain a new proof of this result, in the case of G=GL(n), via positive characteristic method. This talk is based on joint work with R.Bezrukavnikov, R.Travkin and X.Zhu.

No seminar on Monday. **** Thursday (May 5), 4:30 p.m, room E 206. Alexander Beilinson. The characteristic cycle of an etale sheaf. I. Abstract In his recent article "The characteristic cycle and the singular support of a constructible sheaf" (arXiv:1510.03018) Takeshi Saito developed the theory of characteristic cycle that generalizes the theory of Kashiwara-Shapira (from their book "Sheaves on manifolds") to varieties over a field of arbitrary characteristic. For sheaves on a curve the characteristic cycle amounts to the Artin conductor. One of the central results of the theory is the global Euler characteristic formula; for a curve this is the classical formula of Grothendieck-Ogg-Shafarevich. In the talks I will explain the principal ideas of Saito's theory and sketch the proofs. They continue my winter talks about the singular support; all necessary facts will be reminded.

Thursday (May 5), 4:30 p.m, room E 206. Alexander Beilinson. The characteristic cycle of an etale sheaf. I. Abstract In his recent article "The characteristic cycle and the singular support of a constructible sheaf" (arXiv:1510.03018) Takeshi Saito developed the theory of characteristic cycle that generalizes the theory of Kashiwara-Shapira (from their book "Sheaves on manifolds") to varieties over a field of arbitrary characteristic. For sheaves on a curve the characteristic cycle amounts to the Artin conductor. One of the central results of the theory is the global Euler characteristic formula; for a curve this is the classical formula of Grothendieck-Ogg-Shafarevich. In the talks I will explain the principal ideas of Saito's theory and sketch the proofs. They continue my winter talks about the singular support; all necessary facts will be reminded.

Presumably, Beilinson will continue on May 16. Next week there will be two talks by Yun (Monday and Thursday). ***** Monday (May 9), 4:30 p.m, room E 206. Zhiwei Yun (Stanford). Intersection numbers and higher derivatives of L-functions for function fields. I. Abstract In joint work with Wei Zhang, we prove a higher derivative analogue of the Waldspurger formula and the Gross-Zagier formula in the function field setting under some unramifiedness assumptions. Our formula relates the self-intersection number of certain cycles on the moduli of Drinfeld Shtukas for GL(2) to higher derivatives of automorphic L-functions for GL(2). In the first talk I will give motivation and state the main results, giving all the necessary definitions. In the second talk (Thursday May 12) I will sketch the geometric ideas in the proof.

Thursday (May 12), 4:30 p.m, room E 206. Zhiwei Yun (Stanford). Intersection numbers and higher derivatives of L-functions for function fields. II.

Monday (May 16), 4:30 p.m, room E 206. Alexander Beilinson. The characteristic cycle of an etale sheaf. II. Abstract I will sketch the proofs of the theorems on characteristic cycles of etale sheaves on varieties over a field of arbitrary characteristic.

Thursday (May 19), 4:30 p.m, room E 206. Alexander Beilinson. The characteristic cycle of an etale sheaf. III.

Monday (May 23), 4:30 p.m, room E 206. Sam Raskin (MIT). Single variable calculus and local geometric Langlands Abstract The moduli of (possibly irregular) formal connections in one variable (up to gauge transformations) is an infinite-dimensional space that "feels finite-dimensional," e.g., it has finite-dimensional tangent spaces. However, it is not so clear how this perception is actually reflected in the global geometry of this space. Previous works have focused on explicit parametrization of this space. As we will recall during the talk, this approach has significant limitations, and is insufficient to say anything about the global geometry. But we will instead find that this space appears kinder through the lens of homological (or more poetically, noncommutative) geometry, exhibiting better features than all its close relatives. Finally, we will discuss how these results lend credence to the existence of a de Rham Langlands program incorporating arbitrary singularities (the usual story is unramified, or at worst has Iwahori ramification).

Thursday (May 26), 4:30 p.m, room E 206. Sam Raskin (MIT). W-algebras and Whittaker categories Abstract Affine W-algebras are a somewhat complicated family of (topological) associative algebras associated with a semisimple Lie algebra, quantizing functions on the algebraic loop space of Kostant's slice. They have attracted a great deal of attention because of Feigin-Frenkel's duality theorem for them, which identifies W-algebras for a Lie algebra and for its Langlands dual through a subtle construction. The purpose of this talk is threefold: 1) to introduce a natural family of (discrete) modules for the affine W-algebra, 2) to prove an analogue of Skryabin's equivalence in this setting, realizing the category of (discrete) modules over the W-algebra in a more natural way, and 3) to explain how these constructions help understand Whittaker categories in the more general setting of local geometric Langlands. These three points all rest on a simple geometric observation, which provides a family of affine analogues of Bezrukavnikov-Braverman-Mirkovic.

No more meetings of the seminar this quarter.

No meetings this week. The first meeting is on Oct. 3 (Monday), 4:30 p.m, room E 206. I will discuss some recent results by K.Kedlaya and me, see http://arxiv.org/abs/1604.00660 Let X be a variety over F_p . Fix a prime \ell different from p, an algebraic closure \overline{Q_\ell}, and a p-adic valuation v of the subfield of algebraic numbers in \overline{Q_\ell} normalized so that v(p)=1. Let M be a \overline{Q_\ell}-sheaf on X such that for every closed point x in X the eigenvalues of the geometric Frobenius acting on M_x are algebraic numbers. Applying the valuation v to these numbers and dividing by the degree of v over F_p, we get a collection of rational numbers, which are called slopes of M at x. We proved that if X is a smooth curve and M is an irreducible local system then for almost all x the gaps between two consecutive slopes of M at x are not greater than 1. If M has rank 2 then one can even replace “almost all” by “all”, but in the rank 3 case this is false. I will say almost nothing about the proof (which is based on the theory of F-isocrystals).

Monday (Oct 3), 4:30 p.m, room E 206. V.Drinfeld. Slopes of irreducible local systems. Abstract I will discuss some recent results by K.Kedlaya and me, see http://arxiv.org/abs/1604.00660 Let X be a variety over F_p . Fix a prime \ell different from p, an algebraic closure \overline{Q_\ell}, and a p-adic valuation v of the subfield of algebraic numbers in \overline{Q_\ell} normalized so that v(p)=1. Let M be a \overline{Q_\ell}-sheaf on X such that for every closed point x in X the eigenvalues of the geometric Frobenius acting on M_x are algebraic numbers. Applying the valuation v to these numbers and dividing by the degree of x over F_p, we get a collection of rational numbers, which are called slopes of M at x. We proved that if X is a smooth curve and M is an irreducible local system then for almost all x the gaps between two consecutive slopes of M at x are not greater than 1. If M has rank 2 then one can even replace “almost all” by “all”, but in the rank 3 case this is false. I will say almost nothing about the proof (which is based on the theory of F-isocrystals).

No seminar on Thursday (Oct 6). Monday (Oct 10), 4:30 p.m, room E 206. Ben Davison (IST, Vienna). Purity and surjectivity for some symplectic stacks. Abstract I will explain a special case of a general procedure, called dimensional reduction, for identifying the cohomology of the stack-theoretic zero locus of a moment map with the vanishing cycle cohomology of the stack of objects in a 3-Calabi-Yau category, focussing almost entirely on down to earth cases coming from the theory of representations of quivers. I will also introduce another useful feature of this theory, which is the observation that the map from the stack of representations to the coarse moduli space can be "approximated by proper maps" - this amounts to a nice (module-theoretic) partial compactification of Totaro's construction for calculating equivariant cohomology. I will explain how these two features together give rise to surprising purity and surjectivity results.

Monday (Oct 10), 4:30 p.m, room E 206. Ben Davison (IST, Vienna). Purity and surjectivity for some symplectic stacks. Abstract I will explain a special case of a general procedure, called dimensional reduction, for identifying the cohomology of the stack-theoretic zero locus of a moment map with the vanishing cycle cohomology of the stack of objects in a 3-Calabi-Yau category, focussing almost entirely on down to earth cases coming from the theory of representations of quivers. I will also introduce another useful feature of this theory, which is the observation that the map from the stack of representations to the coarse moduli space can be "approximated by proper maps" - this amounts to a nice (module-theoretic) partial compactification of Totaro's construction for calculating equivariant cohomology. I will explain how these two features together give rise to surprising purity and surjectivity results.

Thursday (Oct 13), 4:30 p.m, room E 206. David Nadler (Berkeley). Arboreal singularities of Lagrangian subvarieties. I. Abstract Arboreal singularities are a class of singularities of Lagrangian subvarieties of symplectic manifolds. They have several elementary characterizations and are essentially combinatorial objects. It turns out that any Lagrangian singularity admits a deformation to a nearby Lagrangian subvariety with arboreal singularities. Moreover, the deformation is non-characteristic in the sense that categories of "Lagrangian branes", for example in the form of microlocal sheaves, are invariant under it. This yields an elementary method for calculating them which can be applied in situations of interest in mirror symmetry. At a more basic level, Lagrangian subvarieties with arboreal singularities offer a higher dimensional analogue of ribbon graphs from which one can hope to recover the ambient symplectic manifold itself. The talks will be based primarily on the papers: arXiv:1309.4122, arXiv:1507.01513, arXiv:1507.08735, and time permitting, work in progress with Eliashberg and Starkston.

Monday (Oct 17), 4:30 p.m, room E 206. David Nadler (Berkeley). Arboreal singularities of Lagrangian subvarieties. II. Abstract Arboreal singularities are a class of singularities of Lagrangian subvarieties of symplectic manifolds. They have several elementary characterizations and are essentially combinatorial objects. It turns out that any Lagrangian singularity admits a deformation to a nearby Lagrangian subvariety with arboreal singularities. Moreover, the deformation is non-characteristic in the sense that categories of "Lagrangian branes", for example in the form of microlocal sheaves, are invariant under it. This yields an elementary method for calculating them which can be applied in situations of interest in mirror symmetry. At a more basic level, Lagrangian subvarieties with arboreal singularities offer a higher dimensional analogue of ribbon graphs from which one can hope to recover the ambient symplectic manifold itself. The talks will be based primarily on the papers: arXiv:1309.4122, arXiv:1507.01513, arXiv:1507.08735, and time permitting, work in progress with Eliashberg and Starkston.

NB: the next meeting is on *Friday* (not Thursday)! Friday (Oct 21), 4:30 p.m, room E 206. Dennis Gaitsgory (Harvard). Metaplectic geometric Langlands. Abstract We will explain what it is the geometric counterpart of a metaplectic extension (in the global or local cases), the construction of the metaplectic spectral group and metaplectic geometric Satake. The goal of the talk is to state the metaplectic spectral decomposition conjecture, which is a metaplectic analog of the statement that D(Bun_G) is acted on by the monoidal category QCoh(LocSys_{G^L}). This is a joint work with S. Lysenko.

NB: the next meeting is on *Friday* (not Thursday)! Friday (Oct 21), 4:30 p.m, room E 206. Dennis Gaitsgory (Harvard). Metaplectic geometric Langlands. Abstract We will explain what it is the geometric counterpart of a metaplectic extension (in the global or local cases), the construction of the metaplectic spectral group and metaplectic geometric Satake. The goal of the talk is to state the metaplectic spectral decomposition conjecture, which is a metaplectic analog of the statement that D(Bun_G) is acted on by the monoidal category QCoh(LocSys_{G^L}). This is a joint work with S. Lysenko.

Monday (Oct 24), 4:30 p.m, room E 206. Dennis Gaitsgory (Harvard). Quantum geometric Langlands. Abstract We will state the quantum global unramified geometric Langlands conjecture about the equivalence of the twisted D(Bun_G) and D(Bun_{G^L}) for a matching pair of twistings. We will relate this to the metaplectic spectral decomposition conjecture from the previous talk. We will explain how as an ingredient of the quantum global conjecture we have the quantum local equivalence, between the twisted Whittaker category on the affine Grassmannian for G, and the Kazhdan-Lusztig category for G^L.

No seminar on Thursday Oct 27. On Monday Oct 31 and Thursday Nov 3 there will be talks by Xinwen Zhu (Caltech). Title of his talks: Towards a p-adic non-abelian Hodge theory. Abstract I will first review the non-abelian Hodge theory for complex manifolds and describe a conjectural p-adic analogue. Then I will discuss what we know so far and the following consequence of (the known part of) the theory: Let L be a p-adic local system on a (geometrically) connected algebraic variety over a number field. If its stalk at one point, regarded as a p-adic Galois representation, is geometric in the sense of Fontaine-Mazur, then the stalk at every point is geometric. This is based on a joint work with Ruochuan Liu.

Monday (Oct 31), 4:30 p.m, room E 206. Xinwen Zhu (Caltech). Towards a p-adic non-abelian Hodge theory.I. Abstract I will first review the non-abelian Hodge theory for complex manifolds and describe a conjectural p-adic analogue. Then I will discuss what we know so far and the following consequence of (the known part of) the theory: Let L be a p-adic local system on a (geometrically) connected algebraic variety over a number field. If its stalk at one point, regarded as a p-adic Galois representation, is geometric in the sense of Fontaine-Mazur, then the stalk at every point is geometric. This is based on a joint work with Ruochuan Liu.

Thursday (Nov 3), 4:30 p.m, room E 206. Xinwen Zhu (Caltech). Towards a p-adic non-abelian Hodge theory.II. Abstract I will first review the non-abelian Hodge theory for complex manifolds and describe a conjectural p-adic analogue. Then I will discuss what we know so far and the following consequence of (the known part of) the theory: Let L be a p-adic local system on a (geometrically) connected algebraic variety over a number field. If its stalk at one point, regarded as a p-adic Galois representation, is geometric in the sense of Fontaine-Mazur, then the stalk at every point is geometric. This is based on a joint work with Ruochuan Liu.

Monday (Nov 7), 4:30 p.m, room E 206. Bhargav Bhatt (University of Michigan). The direct summand conjecture and its derived variant. Abstract In the late 60's, Hochster formulated the direct summand conjecture (DSC) in commutative algebra, which is the following innocuous looking assertion: a finite extension A --> B of commutative rings admits an A-module splitting if A is regular and noetherian. A few years later, Hochster himself proved the DSC when the ring contains a field; this and related ideas eventually had a significant impact on the development of the theory of F-singularities. In the mixed characteristic setting, the case of dimension <= 3 was settled by Heitmann in the 90's. The general case, however, remained wide open until extremely recently, when it was resolved beautifully by Andr\'e using the theory of perfectoid spaces. In this talk, I'll discuss a proof of the DSC that is related to, but simplifies, Andr\'e’s proof. I will also explain why similar ideas help establish a derived variant of the DSC put forth by de Jong; the latter roughly states that regular rings have rational singularities. One of my main goals in this talk to explain why passing from a mixed characteristic ring to a perfectoid extension is a useable analog of the passage to the perfection (direct limit over Frobenius) in characteristic p. The relevant background from perfectoid geometry will be explained.

No seminar on Thursday. ****** Monday (Nov 14), 4:30 p.m, room E 206. Victor Ginzburg. Point counting on moduli spaces via factorization sheaves.I. Abstract Given a linear category over a finite field such that the moduli space of its objects is a smooth Artin stack (and some additional conditions) we give a formula for the number of absolutely indecomposable objects of the category and also a similar expression for a certain stacky exponential sum over the set of all objects of the category. Our formulas involve the geometry of the cotangent bundle on the moduli stack. The first formula was inspired by the work of Hausel, Letellier, and Rodriguez-Villegas. It provides a new approach for counting absolutely indecomposable quiver representations, vector bundles with parabolic structure on a projective curve, and irreducible l-adic local systems (via a result of Deligne). Our second formula resembles formulas appearing in the theory of Donaldson-Thomas invariants.

Monday (Nov 14), 4:30 p.m, room E 206. Victor Ginzburg. Point counting on moduli spaces via factorization sheaves.I. Abstract Given a linear category over a finite field such that the moduli space of its objects is a smooth Artin stack (and some additional conditions) we give a formula for the number of absolutely indecomposable objects of the category and also a similar expression for a certain stacky exponential sum over the set of all objects of the category. Our formulas involve the geometry of the cotangent bundle on the moduli stack. The first formula was inspired by the work of Hausel, Letellier, and Rodriguez-Villegas. It provides a new approach for counting absolutely indecomposable quiver representations, vector bundles with parabolic structure on a projective curve, and irreducible l-adic local systems (via a result of Deligne). Our second formula resembles formulas appearing in the theory of Donaldson-Thomas invariants.

Monday (Nov 14), 4:30 p.m, room E 206. Victor Ginzburg. Point counting on moduli spaces via factorization sheaves.I. Abstract Given a linear category over a finite field such that the moduli space of its objects is a smooth Artin stack (and some additional conditions) we give a formula for the number of absolutely indecomposable objects of the category and also a similar expression for a certain stacky exponential sum over the set of all objects of the category. Our formulas involve the geometry of the cotangent bundle on the moduli stack. The first formula was inspired by the work of Hausel, Letellier, and Rodriguez-Villegas. It provides a new approach for counting absolutely indecomposable quiver representations, vector bundles with parabolic structure on a projective curve, and irreducible l-adic local systems (via a result of Deligne). Our second formula resembles formulas appearing in the theory of Donaldson-Thomas invariants.

The article by Ginzburg,Travkin,and Dobrovolska is at http://math.uchicago.edu/~drinfeld/Ginzburg-Travkin-Galya.pdf No seminar on Thursday. Ginzburg will give his second talk on Monday (Nov 21), 4:30 p.m, room E 206.

Monday (Nov 21), 4:30 p.m, room E 206. Victor Ginzburg. Point counting on moduli spaces via factorization sheaves.II. Abstract Given a linear category over a finite field such that the moduli space of its objects is a smooth Artin stack (and some additional conditions) we give a formula for the number of absolutely indecomposable objects of the category and also a similar expression for a certain stacky exponential sum over the set of all objects of the category. Our formulas involve the geometry of the cotangent bundle on the moduli stack. The first formula was inspired by the work of Hausel, Letellier, and Rodriguez-Villegas. It provides a new approach for counting absolutely indecomposable quiver representations, vector bundles with parabolic structure on a projective curve, and irreducible l-adic local systems (via a result of Deligne). Our second formula resembles formulas appearing in the theory of Donaldson-Thomas invariants.

No more meetings this quarter. Happy Thanksgiving!

The first meeting is on Jan 9 (Monday), 4:30 p.m, room E 206. Yiannis Sakellaridis (Rutgers) will begin his series of lectures “Construction of automorphic L-functions”. The abstract of the whole course is below. As far as I understand, the main keyword of this course is “spherical variety”. As far as I understand, the course is mostly on the classical (rather than geometric) Langlands program, but in the case of function fields there should definitely be a geometric version (with sheaves rather than functions). Abstract L-functions are among the most mysterious objects in number theory. In particular, they are defined in a very abstract way, but any time we can say something useful about them we use a geometric or harmonic-analytic way to represent them. In this series of lectures I will survey different methods for producing L-functions out of automorphic forms, such as by period and Rankin-Selberg integrals, including conjectures and explicit calculations which have some geometric significance that has not been completely understood.

Monday (Jan 9), 4:30 p.m, room E 206. Yiannis Sakellaridis (Rutgers). Construction of automorphic L-functions.I. This is the first lecture of a course. Here is the abstract of the whole course: Abstract L-functions are among the most mysterious objects in number theory. In particular, they are defined in a very abstract way, but any time we can say something useful about them we use a geometric or harmonic-analytic way to represent them. In this series of lectures I will survey different methods for producing L-functions out of automorphic forms, such as by period and Rankin-Selberg integrals, including conjectures and explicit calculations which have some geometric significance that has not been completely understood.

The next meeting is on Monday Jan 16. (Sakellaridis will give his second lecture.)

Sakellaridis will give his second lecture on Monday (Jan 16), 4:30 p.m, room E 206. Note that our buildings WILL BE LOCKED this Monday because of the holiday. So if you have a UofC card please be sure to have it with you. If you don't then please arrive 10 minutes before the seminar, and somebody will let you in.

The next meeting is on Monday Jan 23. (Sakellaridis will give his next lecture.)

Sakellaridis will give his next lecture on Monday (Jan 23), 4:30 p.m, room E 206.

The next meeting is on Monday Jan 30. (Sakellaridis will give his next lecture.)

Sakellaridis will give his next lecture on Monday (Jan 30), 4:30 p.m, room E 206.

The next meeting is on Monday Feb 6. (Sakellaridis will give his next lecture.)

Sakellaridis will give his next lecture on Monday (Feb 6), 4:30 p.m, room E 206.

The next meeting is on Monday Feb 13. (Sakellaridis will give his next lecture.)

Sakellaridis will give his lecture today at 4:30 p.m, room E 206.

Sakellaridis will give his next lecture on Monday (Feb 20), 4:30 p.m, room E 206.

Sakellaridis will give his lecture tomorrow (Monday) at 4:30 p.m, room E 206.

Sakellaridis will give his next lecture on Monday (Feb 27), 4:30 p.m, room E 206.

Sakellaridis will give his next lecture on Monday (Feb 27), 4:30 p.m, room E 206.

Sakellaridis will give his next lecture on Monday (March 6), 4:30 p.m, room E 206.

Sakellaridis will give his next lecture on tomorrow (Monday March 6), 4:30 p.m, room E 206.

No more meetings of the seminar this quarter.

The first meeting is on April 3 (Monday), 4:30 p.m, room E 206. Beilinson will give a talk; the title and abstract are below. ******** As already announced, Yiannis Sakellaridis will be giving a course on the Relative Trace Formula every Wednesday 2:30-4:30 (room E308) starting from this Wednesday, March 29. ******** Title of Beilinson’s talk on April 3: Wild ramification and the Euler characteristic. Abstract Let F be a constructible sheaf on a variety X over an algebraically closed field k. If char k = 0 then the global Euler characteristic \chi (X,F) (as well as its local avatar, the characteristic cycle CC(F)) is determined by the constructible function x \mapsto rk (F_x ), x\in X(k). This is no longer true if char k > 0: indeed, if X is a curve then the Grothendieck-Ogg-Shafarevich formula for \chi (X,F) includes extra local terms - the conductors that measure the wild ramification. I will talk about the works "Wild ramification determines the characteristic cycle" by T.Saito and Y.Yatagawa (arXiv:1604.01513) and "Wild ramification and restriction to curves" by H.Kato (arXiv:1611.07642), which treat the case dim X > 1. In particular they show that \chi (X,F) (and CC(F)) are determined by the conductors of the pullback of F to every curve. No prior knowledge of the subject (in particular, what is CC(F)) is assumed.

Today (Monday), 4:30 p.m, room E 206. A. Beilinson. Wild ramification and the Euler characteristic. Abstract Let F be a constructible sheaf on a variety X over an algebraically closed field k. If char k = 0 then the global Euler characteristic \chi (X,F) (as well as its local avatar, the characteristic cycle CC(F)) is determined by the constructible function x \mapsto rk (F_x ), x\in X(k). This is no longer true if char k > 0: indeed, if X is a curve then the Grothendieck-Ogg-Shafarevich formula for \chi (X,F) includes extra local terms - the conductors that measure the wild ramification. I will talk about the works "Wild ramification determines the characteristic cycle" by T.Saito and Y.Yatagawa (arXiv:1604.01513) and "Wild ramification and restriction to curves" by H.Kato (arXiv:1611.07642), which treat the case dim X > 1. In particular they show that \chi (X,F) (and CC(F)) are determined by the conductors of the pullback of F to every curve. No prior knowledge of the subject (in particular, what is CC(F)) is assumed.

No seminar on Thursday.

April 10 (Monday), 4:30 p.m, room E 206. Ngo Bao Chau. Perverse sheaves on formal arc spaces I will try to explain the foundational work of Bouthier and Kazhdan on the concept of perverse sheaves on formal arc spaces. We expect this theory to have applications in harmonic analysis over non-Archimedean local fields. I will discuss some examples.

As already announced, today at 4:30 p.m. (room E206) Ngo Bao Chau will speak on "Perverse sheaves on formal arc spaces" (after a work by Bouthier-Kazhdan). ******* Tomorrow Nicolas Templier will give a talk at the NT seminar (see below). His talk is related to the geometric Langlands program. Presumably, he will not finish his talk on Tuesday; in this case he will continue on Thursday at the geometric Langlands seminar (4:30 p.m., room E206). Tuesday April 11, 1:30pm - 3:00pm in Eckhart 206 Nicolas Templier (Cornell): Mirror symmetry for minuscule flag varieties. We prove cases of Rietsch mirror conjecture that the Dubrovin-Givental quantum connection for projective homogeneous varieties is isomorphic to the pushforward D-module attached to Berenstein-Kazhdan geometric crystals. The idea is to recognize the quantum connection as Galois and the geometric crystal as automorphic. The isomorphism then comes from global rigidity results where a Hecke eigenform is determined by its local ramification. We reveal relations with the works of Gross, Frenkel-Gross, Heinloth-Ngo-Yun and Zhu on Kloosterman sheaves. The talk will keep the algebraic geometry prerequisite knowledge to a minimum by introducing the above concepts of "mirror" and "crystal" with the examples of CP^1, projective spaces and quadrics. Work with Thomas Lam.

Thursday (April 13), 4:30 p.m, room E 206. Nicolas Templier (Cornell) will finish the talk on Mirror symmetry for minuscule flag varieties that he started on Tuesday at the NT seminar. ********* On Monday (April 17) Ngo Bao Chau will continue his talk on formal arc spaces.

Monday (April 17), 4:30 p.m, room E 206. Ngo Bao Chau will finish his talk on formal arc spaces. The notes of his talks are available at https://math.uchicago.edu/~ngo/Weierstrass.pdf He may keep updating them.

Thursday (April 20), 4:30 p.m, room E 206. V.Drinfeld. A stacky approach to formal arcs. I. The abstract is below. On the other hand, the following 2-page file can serve as a bridge between Ngo's talks and mine: http://math.uchicago.edu/~drinfeld/Smooth%20groupoids/Bridge%20between%202%20series%20of%20talks.pdf (Reading that file is not necessary, but it can help.) Abstract As a complement to Ngo’s talks (and without relying on them), I will explain the geometric ideas behind the computations in my old work https://arxiv.org/pdf/math/0203263.pdf on formal arcs. This will probably take 2 talks. Main points: (i) a finite type model for the formal neighborhood of a formal arc in a variety X is provided by the space of maps from the affine line to X/G, where G is a smooth groupoid acting on X; (ii) usually G does not come from a group action (but there are important cases when it does); (iii) for a given X there is plenty of smooth groupoids acting on X; one can construct them using “affine blow-ups”; (iv) there is a useful notion of the Lie algebroid of a smooth groupoid.

Monday (April 24), 4:30 p.m, room E 206. I will give my second (and hopefully last) talk on the stacky approach to formal arcs. In this talk I will try to explain how to work with smooth groupoids. Here are some related materials (no need to read them now; my Monday talk will be a kind of introduction to these texts): Smooth groupoids on smooth manifolds are discussed in the article https://arxiv.org/abs/math/0611259.pdf by Crainic-Fernandes. In particular, they discuss the notion of Lie algebraic of a Lie groupoid. The “Newton groupoid” (the one which is secretly used in my proof of the Grinberg-Kazhdan theorem) is defined in the following notes: http://math.uchicago.edu/~drinfeld/Smooth%20groupoids/Newton%20groupoid.pdf There could be mistakes there. The Newton groupoid is the one that I suggested to guess at the end of my previous write-up, see http://math.uchicago.edu/~drinfeld/Smooth%20groupoids/Bridge%20between%202%20series%20of%20talks.pdf The video of my yesterday's talk is at https://youtu.be/0bG7V4oavxY My notes of that talk are at http://math.uchicago.edu/~drinfeld/Smooth%20groupoids/Talk1.pdf

The notes of my yesterday talk are available at http://math.uchicago.edu/~drinfeld/Smooth%20groupoids/Talk2.pdf The Newton groupoid I defined yesterday is the one that was denoted by \Gamma_2 at the end of the file http://math.uchicago.edu/~drinfeld/Smooth%20groupoids/Bridge%20between%202%20series%20of%20talks.pdf (I didn't give the definition of \Gamma_2 there). ******** No seminar on Thursday April 27 and Monday May 1. After that, there will be talks by Geordie Williamson on May 4,8,11. Title: "Studying the decomposition theorem with integral coefficients". The abstract can be at the seminar calendar http://math.uchicago.edu/research/calendar/ (see the description of Williamson's talk on May 4).

Thursday (May 4), 4:30 p.m, room E 206. Geordie Williamson (Sydney). Studying the decomposition theorem with integral coefficients. I. Abstract I will explain an approach (via intersection form) to understanding if the decomposition theorem holds with coefficients in characteristic p fields and the integers. I am interested in these questions primarily with applications to modular representation in mind, but they are also fascinating questions by themselves. My lectures will cover the following topics: 1) de Cataldo and Migliorini's Hodge theoretic proof of the decomposition theorem (for the direct image of the constant sheaf); 2) torsion in integral intersection cohomology and obstacles to attempting to carry out the above proof over the integers; 3) what we know and don't know about torsion in the intersection cohomology of Schubert varieties; 4) applications to modular representation theory of algebraic groups (Lusztig conjecture, Finkelberg-Mirkovic conjecture). 5) optional final topic: parity sheaves, p-canonical basis, tilting modules, new character formulas (joint work with Riche and Achar-Makisumi-Riche). For students wishing to prepare for the lectures, the following links might be useful: Part 1) will roughly following my Seminaire Bourbaki lecture: http://front.math.ucdavis.edu/1603.09235 https://www.youtube.com/watch?v=ZneRJzVPZ2M&list=PL9kd4mpdvWcAejNNqq7IGPTyMjrfv09WA&index=2 Background for parts 2) - 3) can be found in the following paper (see also a lecture in Stonybrook): http://front.math.ucdavis.edu/1512.08295 http://scgp.stonybrook.edu/video_portal/video.php?id=2423 Part 4) will roughly follow my Takagi lectures (especially the second part on constructible sheaves): https://arxiv.org/abs/1610.06261 https://www.youtube.com/watch?v=i0XUCkZ0TEo&feature=youtu.be https://www.youtube.com/watch?v=hdNmPqiEn2Y&feature=youtu.be Part 5) (if it is given) will follow the following two papers: https://arxiv.org/abs/0906.2994 http://front.math.ucdavis.edu/1512.08296

Monday (May 8), 4:30 p.m, room E 206. Geordie Williamson (Sydney). Studying the decomposition theorem with integral coefficients. II. Abstract I will explain an approach (via intersection form) to understanding if the decomposition theorem holds with coefficients in characteristic p fields and the integers. I am interested in these questions primarily with applications to modular representation in mind, but they are also fascinating questions by themselves. My lectures will cover the following topics: 1) de Cataldo and Migliorini's Hodge theoretic proof of the decomposition theorem (for the direct image of the constant sheaf); 2) torsion in integral intersection cohomology and obstacles to attempting to carry out the above proof over the integers; 3) what we know and don't know about torsion in the intersection cohomology of Schubert varieties; 4) applications to modular representation theory of algebraic groups (Lusztig conjecture, Finkelberg-Mirkovic conjecture). 5) optional final topic: parity sheaves, p-canonical basis, tilting modules, new character formulas (joint work with Riche and Achar-Makisumi-Riche). For students wishing to prepare for the lectures, the following links might be useful: Part 1) will roughly following my Seminaire Bourbaki lecture: http://front.math.ucdavis.edu/1603.09235 https://www.youtube.com/watch?v=ZneRJzVPZ2M&list=PL9kd4mpdvWcAejNNqq7IGPTyMjrfv09WA&index=2 Background for parts 2) - 3) can be found in the following paper (see also a lecture in Stonybrook): http://front.math.ucdavis.edu/1512.08295 http://scgp.stonybrook.edu/video_portal/video.php?id=2423 Part 4) will roughly follow my Takagi lectures (especially the second part on constructible sheaves): https://arxiv.org/abs/1610.06261 https://www.youtube.com/watch?v=i0XUCkZ0TEo&feature=youtu.be https://www.youtube.com/watch?v=hdNmPqiEn2Y&feature=youtu.be Part 5) (if it is given) will follow the following two papers: https://arxiv.org/abs/0906.2994 http://front.math.ucdavis.edu/1512.08296

Thursday (May 11), 4:30 p.m, room E 206. Geordie Williamson (Sydney). Studying the decomposition theorem with integral coefficients. III. Abstract I will explain an approach (via intersection form) to understanding if the decomposition theorem holds with coefficients in characteristic p fields and the integers. I am interested in these questions primarily with applications to modular representation in mind, but they are also fascinating questions by themselves. My lectures will cover the following topics: 1) de Cataldo and Migliorini's Hodge theoretic proof of the decomposition theorem (for the direct image of the constant sheaf); 2) torsion in integral intersection cohomology and obstacles to attempting to carry out the above proof over the integers; 3) what we know and don't know about torsion in the intersection cohomology of Schubert varieties; 4) applications to modular representation theory of algebraic groups (Lusztig conjecture, Finkelberg-Mirkovic conjecture). 5) optional final topic: parity sheaves, p-canonical basis, tilting modules, new character formulas (joint work with Riche and Achar-Makisumi-Riche). For students wishing to prepare for the lectures, the following links might be useful: Part 1) will roughly following my Seminaire Bourbaki lecture: http://front.math.ucdavis.edu/1603.09235 https://www.youtube.com/watch?v=ZneRJzVPZ2M&list=PL9kd4mpdvWcAejNNqq7IGPTyMjrfv09WA&index=2 Background for parts 2) - 3) can be found in the following paper (see also a lecture in Stonybrook): http://front.math.ucdavis.edu/1512.08295 http://scgp.stonybrook.edu/video_portal/video.php?id=2423 Part 4) will roughly follow my Takagi lectures (especially the second part on constructible sheaves): https://arxiv.org/abs/1610.06261 https://www.youtube.com/watch?v=i0XUCkZ0TEo&feature=youtu.be https://www.youtube.com/watch?v=hdNmPqiEn2Y&feature=youtu.be Part 5) (if it is given) will follow the following two papers: https://arxiv.org/abs/0906.2994 http://front.math.ucdavis.edu/1512.08296

Monday (May 15), 4:30 p.m, room E 206. Luc Illusie (Paris-Sud). Revisiting the de Rham-Witt complex. I. Abstract The de Rham-Witt complex was constructed by Spencer Bloch in the mid 1970's as a tool to analyze the crystalline cohomology of proper smooth schemes over a perfect field of characteristic p >0, with its action of Frobenius, and describe its relations with other types of cohomology, like Hodge cohomology or Serre's Witt vector cohomology. Since then many developments have occurred. These lectures are meant as an introduction to the theory and contain no new material. After briefly recalling the history of the subject, I will explain the main construction and in the case of a polynomial algebra give its simple description by the so-called complex of integral forms. I will then describe the local structure of the de Rham-Witt complex for smooth schemes over a perfect field and its application to the calculation of crystalline cohomology. In the proper smooth case, I will discuss the slope spectral sequence and the main finiteness properties of the cohomology of the de Rham-Witt complex in terms of coherent complexes over the Raynaud ring. I will briefly mention a few complements (logarithmic Hodge-Witt sheaves, Hyodo-Kato log de Rham-Witt complex, Langer-Zink relative variants), and make a tentative list of open problems.

Thursday (May 18), 4:30 p.m, room E 206. Luc Illusie (Paris-Sud). Revisiting the de Rham-Witt complex. II. Abstract The de Rham-Witt complex was constructed by Spencer Bloch in the mid 1970's as a tool to analyze the crystalline cohomology of proper smooth schemes over a perfect field of characteristic p >0, with its action of Frobenius, and describe its relations with other types of cohomology, like Hodge cohomology or Serre's Witt vector cohomology. Since then many developments have occurred. These lectures are meant as an introduction to the theory and contain no new material. After briefly recalling the history of the subject, I will explain the main construction and in the case of a polynomial algebra give its simple description by the so-called complex of integral forms. I will then describe the local structure of the de Rham-Witt complex for smooth schemes over a perfect field and its application to the calculation of crystalline cohomology. In the proper smooth case, I will discuss the slope spectral sequence and the main finiteness properties of the cohomology of the de Rham-Witt complex in terms of coherent complexes over the Raynaud ring. I will briefly mention a few complements (logarithmic Hodge-Witt sheaves, Hyodo-Kato log de Rham-Witt complex, Langer-Zink relative variants), and make a tentative list of open problems.

Monday (May 22), 4:30 p.m, room E 206. Luc Illusie (Paris-Sud). Revisiting the de Rham-Witt complex. III.

Illusie will give his last talk on Thursday (May 25), 4:30 p.m, room E 206. Here are the references to [BMS], i.e., to the following e-print: https://arxiv.org/pdf/1602.03148.pdf The statement that H^i(eta_fC) = H^i(C)/(Ker f : H^i(C) -> H^i(C)) is [BMS, Lemma 6.4]. The statement that (eta_fC^{.})/f -> H^{.}(C^{.}/f) is compatible with d on the lhs and Bockstein on the rhs, and is a quasi-isomorphism, is [BMS, Prop. 6.12].

No more meetings this quarter.