Winter 2020

• • Mar 102020

  Niles Johnson (Ohio State)

  Algebraic K-theory for 2-categories

  Quillen recognized the higher algebraic K-groups of a commutative ring R as homotopy groups of a certain topological space. This has both conceptual and computational advantages, which we review. We then describe how a 2-categorical extension of this theory leads to a new model for K_3(R), together with more general applications. We will give a mild sampling of key technical details and close with some of the problems we're currently working on. The work we present is joint with Gurski-Osorno, Fontes, and Fontes-Gurski.

• • Feb 252020

  Michael Horst (Ohio State)

  A Categorification of Group Cohomology

  We present a categorification of group cohomology by way of Picard categories, which are to be thought of as categories that behave like Abelian groups. After briefly reviewing background information on Picard categories and free Picard categories, we will present a framework for a cohomology theory taking values not in Abelian groups but in Picard categories. We will motive the study of this theory by way of computing specific examples. The remainder of our time will involve discussing the theoretical underpinnings of this theory, including the particularities of the 2-dimensional algebra required for it.

• • Feb 42020

  Jocelyne Ishak (Vanderbilt)

  Rigidity of the K(1)-local stable homotopy category

  In some cases, it is sufficient to work in the homotopy category Ho(C) associated to a model category C, but looking at the homotopy level alone does not provide us with higher order structure information. Therefore, we investigate the question of rigidity: If we just had the structure of the homotopy category, how much of the underlying model structure can we recover?This question has been investigated during the last decade, and some examples have been studied, but there are still a lot of open questions regarding this subject. Starting with the stable homotopy category Ho(Sp), that is the homotopy category of spectra, it has been proved to be rigid by S. Schwede. Moreover, the E(1)-local stable homotopy category, for p=2, has been shown to be rigid by C. Roitzheim. In this talk, we investigate a new case of rigidity, which is the localization of spectra with respect to the Morava K-theory K(1), at p=2.

• • Jan 282020

  Prasit Bhattacharya (University of Virginia)

  Revisiting Stable Adams conjecture

  The Adams conjecture, perhaps one of the most celebrated result in the subject of stable homotopy theory, was resolved Quillen and Sullivan independently in 70s. Essentially, the Adams conjecture claims that that the Adams operation ψq:BU → BU composed with the J-homomorphism J: BU → BGL1(S) is homotopic to  J after localizing away from q. Here GL1(S) denotes the space of units of the sphere spectrum. The stable version of Adams conjecture states that J can be deformed to J ∘ ψq  via the space of infinite loop space maps from BU1/q → BGL1(S)1/q. The stable version of Adams conjecture was resolved by Friedlander in 1980 and remains the only accepted proof. However, we suspect that there might be an error in Friedlander’s proof. In this talk, I will revisit the work of Friedlander, explain the error and a possible correction. This work is joint with N. Kitchloo.

• • Jan 212020

  Jonathan Campbell (Duke)

  New Approaches to Hilbert's Generalized Third Problem

  In this talk I'll explain how one might attack Hilbert's Generalized Third Problem via homotopy theory, and describe recent progress in this direction. Two n-dimensional polytopes, P, Q are said to be scissors congruent if one can cut P along a finite number of hyperplanes, and re-assemble the pieces into Q. The scissors congruence problem, aka Hilbert's Generalized Third Problem, asks: when can we do this? what obstructs this? In two dimensions, two polygons are scissors congruent if and only if they have the same area. In three dimensions, there is volume and another invariant, the Dehn Invariant. In higher dimensions, very little is known --- but the problem is known to have deep connections to motives, values of zeta functions, the weight filtration in algebraic K-theory, and regulator maps. I'll give a leisurely introduction to this very classical problem, and explain some new results.

• • Jan 142020

  Montek Singh Gill (University of Michigan)

  Stabilizations of E_\infty operads and p-adic stable homotopy theory

  I will describe a new class of dg operads, the stable operads, which are in particular sense, stabilizations of E_\infty operads, and an application of these operads to p-adic stable homotopy theory. Cochains on spaces yield examples of algebras over E_\infty operads, and work of May shows that the operations present in the homology of E_\infty algebras in this case yield exactly the Steenrod operations. Moreover, a result of Mandell says that, endowed with this algebraic structure, the cochains provide algebraic models of p-adic homotopy types. We show that cochains on a spectrum yield algebras over our stable operads. We calculate the homology such algebras, showing that, in the case of a point, we get a certain completion of the generalized Steenrod algebra. Finally, we show that, endowed with this algebraic structure, the spectral cochains provide algebraic models for p-adic stable homotopy types.

• • Jan 72020

  Katharine Adamyk (UC Boulder)

  A classification of Q_0-local A(1)-modules

  This talk will present a classification theorem for modules over A(1), a subalgebra of the mod-2 Steenrod algebra. Applications of the classification theorem to lifting A(1)-modules to modules over the Steenrod algebra will be discussed, as well as applications to the computation of certain localized Adams spectral sequences.

Fall 2019

• • Dec 102019

  Candace Bethea (University of South Carolina)

  Riemann-Hurwitz with Wild Ramification

  In 2018 Marc Levine gave an arithmetic Riemann-Hurwitz formula for smooth projective varieties of the same dimension using tools from A^1 enumerative topology. This result requires a technical assumption on the residue fields. In joint work with Jesse Kass and Kirsten Wickelgren, we show by computation that the formula still holds with less restrictive assumptions for curves. In this talk I will explicitly show the result for a specific example suggested by Shuji Saito as a test case for the main theorem as well as outline the proof of the main theorem.

• • Dec 32019

  Christy Hazel (University of Oregon)

  Equivariant fundamental classes in RO(C2)-graded cohomology

  Let C2 denote the cyclic group of order two. Given a manifold with a C2-action, we can consider its equivariant Bredon RO(C2)-graded cohomology. In this talk, we give an overview of RO(C2)-graded cohomology in constant Z/2 coefficients, and then explain how a version of the Thom isomorphism theorem in this setting can be used to develop a theory of fundamental classes for equivariant submanifolds. We illustrate how these classes can be used to understand the cohomology of any C2-surface in constant Z/2 coefficients, including the ring structure.

• • Nov 262019

  Tim Campion (Notre Dame)

  Duality in homotopy theory

  We explore some implications of a fact hiding in plain sight: Namely, the n-sphere has the remarkable property that the “swap” map σ: Sn ∧ Sn → Sn ∧ Sn can be “untwisted”: it is homotopic to (-1)n ∧ 1. This simple fact remains true in equivariant and motivic contexts. One consequence is a structural fact about symmetric monoidal ∞-categories with finite colimits and duals for objects: it turns out that any such category splits as the product of three canonical subcategories (for instance, one of these subcategories is characterized by being stable). As another consequence, we show that for any finite abelian group G, the symmetric monoidal ∞-category of genuine finite G-spectra is obtained from finite G-spaces by stabilizing and freely adjoining duals for objects. This universal property vindicates one motivation sometimes given for studying genuine G-spectra: namely that genuine G-spectra (unlike naive G-spectra or Borel G-spectra) have a good theory of Spanier-Whitehead duality. We take steps toward a similar universal property for nonabelian groups and also in motivic homotopy theory.

• • Nov 192019

  Yu Zhang (Ohio State)

  Topological Quillen localization of structured ring spectra

  Homotopy groups and stable homotopy groups of spaces are central invariants in algebraic topology. Stable homotopy groups are comparatively easier to work with, at the expense of losing certain information. However, if we are working with nice spaces, nothing will be lost by working stably: A map between nilpotent spaces induces homotopy groups isomorphisms if and only if it induces stable homotopy groups isomorphisms. Structured ring spectra are spectra with certain algebraic structure encoded by the action of an operad O. For such O-algebras, the analog of stable homotopy groups are played by Topological Quillen (TQ) homology groups. In this talk, we will draw the analogy between topological spaces and O-algebras, discuss the TQ-localization of O-algebras, and show the TQ-Whitehead theorem for homotopy pro-nilpotent O-algebras. Part of the work in this talk is joint with John E. Harper.

• • Nov 122019

  Jun Hou Fung (Harvard)

  Strict units of commutative ring spectra

  Just as an ordinary commutative ring has a multiplicative group of units, a E_\infty-ring spectrum R also has a spectrum of units gl_1 R, which plays an important role for example in orientation theory and twisted cohomology theories. However, these spectra are typically very large, and to understand twists by Eilenberg-Mac Lane spaces or to isolate those units that come from geometry, it sometimes suffices to study the space of \emph{strict units} of R. Previously, Hopkins and Lurie have computed the strict units of Morava E-theories, but much remains unknown about them in general. In this talk, I will introduce these strict units and illustrate various methods for computing them, and sketch how these calculations relate to other interesting questions in homotopy theory.

• • Nov 52019

  Hood Chatham (MIT)

  An Orientation Map for Higher Real E-theory

  The real K-theory spectrum KO is ``almost complex oriented''. Here are a collection of properties that demonstrate this: (1) KO is the C2 fixed points of a complex oriented cohomology theory KU. (2) Complex oriented cohomology theories have trivial Hurewicz image, whereas KO has a small Hurewicz image -- it detects η and η2. (3) Complex oriented cohomology theories receive a ring map from MU. KO receives no ring map from MU but it receives one from MSU. (4) If E is a complex orientable cohomology theory, every complex vector bundle V is E-orientable. Not every complex vector bundle V is KO-orientable, but V⊕ V and V⊗ 2 are. Higher real E theory EO is an odd primary analogue of KO. At p=3, EO is closely related to TMF. EO is defined as the Cp fixed points of a complex oriented cohomology theory, and it has a small but nontrivial Hurewicz image, so it satisfies analogues of properties (1) and (2). I prove that it also satisfies analogues of properties (3) and (4). In particular, I produce a unital orientation map from a Thom spectrum to EO and prove that for any complex vector bundle V the bundles pV and V⊗ p are complex oriented.

• • Oct 222019

  Ningchuan Zhang (UIUC)

  Analogs of Dirichlet L-functions in chromatic homotopy theory

  The relation between Eisenstein series and the J-homomorphism is an important topic in chromatic homotopy theory at height 1. Both sides are related to the special values of the Riemann ζ-function. This relation is most clearly understood in the context of elliptic cohomology and topological modular forms. Number theorists have studied the twistings of the Riemann ζ-functions and Eisenstein series by Dirichlet characters. Motivated by the Dirichlet equivariance of these twisted Eisenstein series, we introduce the Dirichlet J-spectra in this talk. The homotopy groups of these Dirichlet J-spectra are related to the special values of the Dirichlet L-functions, and thus to congruences of the twisted Eisenstein series. If time allows, we will also explain the connection between Dirichlet J-spectra and the twisted Eisenstein series by generalizing Katz's algebro-geometric explanation of congruences of the (untwisted) Eisenstein series.

• • Oct 082019

  Zhouli Xu (MIT)

  The intersection form of spin 4-manifolds and Pin(2)-equivariant Mahowald invariants

  A fundamental problem in 4-dimensional topology is the following geography question: "which simply connected topological 4-manifolds admit a smooth structure?" After the celebrated work of Kirby-Siebenmann, Freedman, and Donaldson, the last uncharted territory of this geography question is the "11/8-Conjecture''. This conjecture, proposed by Matsumoto, states that for any smooth spin 4-manifold, the ratio of its second-Betti number and signature is least 11/8. Furuta proved the ''10/8+2''-Theorem by studying the existence of certain Pin(2)-equivariant stable maps between representation spheres. In this talk, we will present a complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald invariants of powers of certain Euler classes in the RO(Pin(2))-graded equivariant stable homotopy groups of spheres. In particular, we improve Furuta's result into a ''10/8+4''-Theorem. Furthermore, we show that within the current existing framework, this is the limit. For the proof, we use the technique of cell-diagrams, known results on the stable homotopy groups of spheres, and the j-based Atiyah-Hirzebruch spectral sequence. This is joint work with Michael Hopkins, Jianfeng Lin and XiaoLin Danny Shi.

Spring 2019

• • Jun 042019

  Anna Marie Bohmann (Vanderbilt)

  A multiplicative comparison of Segal and Waldhausen K-theory

  In influential work of the 70s and 80s, Segal and Waldhausen each construct a version of K-theory that produces spectra from certain types of categories. These constructions agree, in the sense that appropriately equivalent categories yield weakly equivalent spectra. In the 2000s, work of Elmendorf-Mandell and Blumberg-Mandell produced more structured versions of Segal and Waldhausen K-theory, respectively. These versions are "multiplicative," in the sense that appropriate notions of pairings of categories yield multiplication-type structure on their resulting spectra. In this talk, I will discuss joint work with Osorno in which we show that these constructions agree as multiplicative versions of K-theory. Consequently, we get comparisons of rings spectra built from these two constructions. Furthermore, the same result also allows for comparisons of related constructions of spectrally-enriched categories.

• • May 282019

  Jonathan Rubin (UCLA)

  Combinatorial and geometric N_∞ operads

  Let G be a finite group. A N_∞ G-operad is an equivariant generalization of an E_∞ operad. Such operads govern the natural algebraic structure on spectra over incomplete universes and on localizations of genuine commutative ring G-spectra. Since Blumberg and Hill’s pioneering work, it has been known that the homotopy theory of N_∞ operads is essentially algebraic. They proved that the homotopy type of a N_∞ operad is completely determined by a single combinatorial invariant, and subsequent work has revealed that the homotopy theory of N_∞ operads may be modeled entirely with discrete operads in the category of G-sets. On the other hand, there are natural classes of geometrically defined N_∞ operads, which generalize the classical linear isometries and infinite little discs operads. Such operads encode real representation-theoretic properties of G, in addition to purely algebraic data. In this talk, I will explain how to reduce the homotopy theory of N_∞ operads to combinatorics, and then I will discuss how the peculiarities of equivariant linear isometry and infinite little discs operads are encoded in algebra.

• • May 212019

  Teena Gerhardt (Michigan State University)

  (Topological) Hochschild homology for Green functors

  Hochschild homology of a ring has a topological analogue for ring spectra, topological Hochschild homology (THH), which plays an essential role in the trace method approach to algebraic K-theory. For a C_n-equivariant ring spectrum, one can define C_n-relative THH. This leads to the question: What is the algebraic analogue of C_n-relative THH? In this talk, I will define twisted Hochschild homology for Green functors, which allows us to describe this algebraic analogue. This also leads to a theory of Witt vectors for Green functors, as well as an algebraic analogue of TR-theory. This is joint work with Andrew Blumberg, Mike Hill, and Tyler Lawson.

• • May 142019

  Angélica Osorno (Reed)

  The combinatorics of the categorical equivariant Barratt-Eccles operad

  The Barratt-Eccles operad is an E_∞ operad given simply by taking the sequence of spaces EΣ_n. It can be constructed by first taking an operad P in categories and then taking its classifying space. Since algebras over P are precisely permutative categories, this operad is a key ingredient in the operadic infinite loop space machine that takes permutative categories to spectra. The categorical equivariant Barratt-Eccles operad P_G defined by Guillou-May-Merling plays a similar role in equivariant infinite loop space theory. In this talk I will give some results about the combinatorics of P_G. I will show that it is not finitely generated when |G| > 1, but that there is a finitely generated suboperad Q_G that is appropriately equivalent to P_G. In the cases of G=C_2 and C_3, I will moreover give an explicit finite presentation of Q_G. This finite presentation allows us to describe algebras over Q_G in a compact way. This is joint work with K. Bangs, S. Binegar, Y. Kim, K. Ormsby, D. Tamas-Parris and L. Xu.

• • May 072019

  Kate Ponto (University of Kentucky)

  Fixed point invariants as additive invariants

  The additivity of the Euler characteristic is one of its essential properties. Additivity holds and plays a similar role for the generalizations to the Lefschetz number and Reidemeister trace. For the Euler characteristic, additivity is captured by the observation that the Euler characteristic descends to a map out of algebraic K-theory. The corresponding statement for the Lefschetz number and Reidemister trace would realize a vision of Klein, McCarthy, and Williams and could provide a new approach to invariants for periodic points. I’ll describe some first steps toward this goal that connect the cyclotomic trace with the bicategorical trace. This is joint work with Jonathan Campbell.

• • Apr 162019

  TBA

  TBA

  Namboodiri: 12th, 15th, and 16th

  TBA

• • Apr 092019

  Jeremy Miller (Purdue)

  High dimensional cohomology of congruence subgroups

  The level p congruence subgroup of SL_n(Z) is defined to be the subgroup of matrices congruent to the identity matrix mod p. These groups have finite cohomological dimension. In the 1970s, Lee and Szczarba gave a conjectural description of the top dimensional cohomology groups of these congruence subgroups. In joint work in progress with Patzt and Putman, we show that this conjecture is false for p>5. In particular, these congruence subgroups have extra cohomology classes in their top degree cohomology coming from the first homology group of the associated compactified modular curve. I will also discuss joint work with Patzt and Nagpal on a stability pattern in the high dimensional cohomology of congruence subgroups.

Winter 2019

• • Mar 122019

  Nicolle Gonzales (USC)

  Categorical Bernstein Operators and the Boson-Fermion correspondence.

• • Feb 262019

  Paul Goerss (Northwestern)

  Dualizing modules in stable homotopy theory

  Let G be a very nice p-adic analytic group; I have in mind examples such as Gl_n(Z_p) or the Morava stabilizer group. The category of continuous G-modules has a very elegant theory of duality reflecting Poincaré duality for G. We would very much like to extend this to stable homotopy theory where, in various contexts, it would help explain some deep structure we have seen so far only through computations. It is easy enough to define the dualizing objects, but then we are left with understanding them. It turns out that if we are only interested in finite subgroups of G (which would be a serious start) we can get away with classical computations with characteristic classes. This is an on-going project with Agnès Beaudry, Mike Hopkins, and Vesna Stojanoska.

• • Feb 122019

  Jens Jakob Kjaer (Notre Dame)

  Unstable v₁-periodic Homotopy Groups through Goodwillie Calculus

  It is a classical result that the rational homotopy groups, π_*(X)⊗\mathbbQ, as a Lie-algebra can be computed in terms of indecomposable elements of the rational cochains on X. The closest we can get to a similar statement for general homotopy groups is the Goodwillie spectral sequence, which computes the homotopy group of a space from its spectral Lie algebra". Unfortunately both input and differentials are hard to get at. We therefore simplify the homotopy groups by taking the unstable v_h-periodic homotopy groups, v_h⁻¹π_*( ) (note h=0 recovers rational homotopy groups). For h=1 we are able to compute the K-theory based v₁-periodic Goodwillie spectral sequence in terms of derived indecomposables. This allows us to compute v₁⁻¹π_*SU(d) in a very different way from the original computation by Davis.

• • Feb 052019

  Yuri Sulyma (UT)

  Sign-graded Topological Restriction Homology

  Topological Hochschild homology (THH) is a genuine S¹ spectrum introduced in order to approximate algebraic K-theory. Its fixed points, known as topological Restriction homology (TR), are given by (a periodicization of) the de Rham-Witt complex of arithmetic geometry; the equivariant structure is reflected in the F, V, R, and d operators on the de Rham-Witt complex. However, equivariant stable homotopy theory provides yet further structure, which has hitherto been little exploited and remains arithmetically mysterious: we can consider homotopy groups graded by real representations of G, rather than just by \mathbbZ. These have been calculated for G=S¹ by Gerhardt and Angeltveit. This subsumes the case where G is cyclic of odd order; however, 2-primary cyclic groups have representations which are not restricted from an S¹-representation, namely the sign representations. We will present computations of these sign-graded TR groups, and briefly discuss applications to the slice filtration.

• • Jan 222019

  Ian Coley (UCLA)

  The K-theory of derivators

• • Jan 082019

  XiaoLin (Danny) Shi (Harvard)

  The slice spectral sequence of a C₄-equivariant height 4 Lubin-Tate theory

  We will talk about the slice spectral sequence of a C₄-equivariant spectrum. This spectrum is a variant of the detection spectrum that Hill-Hopkins-Ravenel used in their proof of the Kervaire invariant problem. After periodization and K(4)-localization, this spectrum is equivalent to a height-4 Lubin-Tate theory E₄ with C₄-action induced from the Goerss-Hopkins-Miller theorem. In particular, our computation shows that E₄^hC₁2 is 384-periodic. This is joint work with Mike Hill, Guozhen Wang, and Zhouli Xu.

Fall 2018

• • Dec 042018

  Nick Gurski (Case Western Reserve University)

  Recognizing presheaf categories, homotopically

  A beautiful theorem of Kelly classifies when a functor F:A → X identifies X as presheaves on A, up to equivalence: when F is full, faithful, and its essential image is a strong generator of small projectives. I will discuss a version of this theorem for V-enriched model categories which produces Quillen equivalences between a V-model category X and the projective model structure on V-enriched presheaves on A using homotopical analogues of the notions of strong generator and small projective. This gives a proof of the Reversed Straightening-Unstraightening Theorem of Stevenson avoiding any use of localization. This is joint work with Kim Nguyen and Daniel Schaeppi.

• • Nov 272018

  Guchuan Li (Northwestern)

  The Gross-Hopkins duals of higher real K-theory at prime 2

  The Hopkins-Mahowald higher real K-theory spectra E_n^G are generalizations of real K-theory; they are ring spectra which give some insight into higher chromatic levels while also being computable. This will be a talk based on joint work with Drew Heard and XiaoLin Danny Shi, in which we compute that higher real K-theory spectra with group G=C₂ at prime 2 and height n are Gross-Hopkins self duals with a shift 4+n. This will allow us to detect exotic invertible K(n)-local spectra.

• • Nov 202018

  Tom Bachmann (MIT)

  Power operations in normed motivic spectra

  In joint work with M. Hoyois, we established (the beginnings of) a theory of "normed motivic spectra." These are motivic spectra with some extra structure, enhancing the standard notion of a motivic E_oo-ring spectrum (this is similar to the notion of G-commutative ring spectra in equivariant stable homotopy theory). It was clear from the beginning that the homotopy groups of such normed spectra afford interesting *power operations*. In ongoing joint work with E. Elmanto and J. Heller, we attempt to establish a theory of these operations and exploit them calculationally. I will report on this, and more specifically on our proof of a weak motivic analog of the following classical result of Würgler: any (homotopy) ring spectrum with 2=0 is generalized Eilenberg-MacLane.

• • Nov 132018

  Irving Dai (Princeton University)

  Involutive Floer Homology and Applications to the Homology Cobordism Group

  In this talk, we discuss some recent applications of involutive Heegaard Floer homology (defined by Hendricks and Manolescu) to the homology cobordism group. We establish some non-torsion results and show that the homology cobordism group admits an infinite-rank summand. This is joint work with Jennifer Hom, Matthew Stoffregen, and Linh Truong.

• • Nov 062018

  James Quigley (Notre Dame)

  The parametrized Tate construction

  The Tate construction is a powerful tool in classical homotopy theory. I will begin by reviewing the Tate construction and surveying some of its applications. I will then describe an enhancement of the Tate construction to genuine equivariant homotopy theory called the parametrized Tate construction and discuss some of its applications, including C_2-equivariant versions of Lin's Theorem and the Mahowald invariant, blueshift for Real Johnson-Wilson spectra (joint work with Guchuan Li and Vitaly Lorman), and trace methods for Real algebraic K-theory (work-in-progress with Jay Shah).

• • Oct 302018

  Ayelet Lindenstrauss (Indiana University)

  On the K-theory of division algebras over local fields

  We calculate K_*(D; Z_p) where D is a central division algebra over a complete discrete valuation field K whose residue field has characteristic p. Suslin and Yufryakov have shown that if l is a prime other than p, K_*(D, Z_l) is isomorphic to K_*(K, Z_l) for all *>0 via a reduced trace map. We use a reduced trace map approach as well, comparing to K_*(K, Z_p) which has been calculated by Hesselholt and Madsen. The analogy to matrix algebras shows what a reduced trace map should be like, but it is not a natural construction nor indeed something that can always be done. Our calculation uses trace methods and Hochschild-type invariants for a maximal order A in D. Any central division algebra of degree d (that is, dimension d^2) over a field K as above contains a maximal unramified field extension L of K, which will be of dimension d. If we call K's ring of integers S and L's ring of integers T, then A tensored over S with T is close enough to being a matrix algebra that it has a reduced trace from its invariants to those of T. Comparison to that easier ring along with the action of the Galois group of L over K lets us define a reduced trace from the Hochschild-type invariants of A to those of S. (Joint with Lars Hesselholt and Michael Larsen.)

• • Oct 232018

  Allen Yuan (MIT)

  The Frobenius in higher algebra

  In classical algebra, the Frobenius provides a natural endomorphism of every ring in which p=0; this determines an action of the circle on the category of F_p algebras. In higher algebra, one can do away with the characteristic p assumption and define a Frobenius map for every E-infinity ring spectrum. In this talk, I will explain the analog of the circle action. The construction of the action features equivariant homotopy theory and a calculation of a non-group-complete variant of the K theory of F_p.

• • Oct 232018

  Tony Feng (Stanford University)

  The Artin-Tate pairing on the Brauer group of a surface

  Joint NT seminar: 2-3:20pm in Ryerson 358

  There is a canonical pairing on the Brauer group of a surface over a finite field, and an old conjecture of Tate predicts that this pairing is alternating. I will present a resolution to Tate’s conjecture, whose key ingredient is a surprising connection to Steenrod operations.

• • Oct 162018

  Inna Zakharevich (Cornell University)

  Deriving motivic measures

  A motivic measure with values in an abelian group A is a function μ: {varieties}→ A which is additive, in the sense that for any closed embedding Y\hookrightarrow X we have μ(X) = μ(Y)+μ(X\Y). Many such measures, such as point counting, Euler characteristics, or the local zeta function actually take values in a K₀-group. In this talk we will give a description of how to lift such measures to maps between spectra and show how to use these to find nontrivi elements in higher K-groups of varieties.

  Pretalk abstract: I will give a short introduction to algebraic K-theory, focusing mostly on exact categories and the Q-construction.

• • Oct 092018

  Dan Berwick-Evans (UIUC)

  A geometric model for complex analytic equivariant elliptic cohomology

  Elliptic cohomology is a natural big brother to ordinary cohomology and K-theory. In contrast to the geometric objects that provide representatives for cohomology and K-theory classes (which lead to many applications), as yet there is no such geometric description of elliptic cohomology. This talk will explain a step forward, in joint work with Arnav Tripathy, for the case of equivariant elliptic cohomology over the complex numbers. The geometric objects of interest are inspired by supersymmetric gauge theory. No prior knowledge of either elliptic cohomology or gauge theories will be assumed.