This is an archive of email messages concerning the Geometric Langlands Seminar for 09-10.
No seminar on Thursday October 1 and Monday October 5.
The seminar will meet on October 8.
October 8, 4:30 p.m, room E 206.
Dima Arinkin (University of North Carolina).
Autoduality for Jacobians of singular curves.
Abstract
Let C be a (smooth projective algebraic) curve, in other words, a Riemann
surface. It is well known that the Jacobian J of C is a self-dual complex
torus, that is, J is identified with the space of topologically trivial
line bundles on J.
Suppose now that C is singular. The Jacobian J of C parametrizes
topologically trivial line bundles on C; it is smooth, but no longer
compact. By considering torsion-free sheaves instead of line bundles, one
obtains a natural singular compactification J' of J.
The subject of this talk is line bundles on J' and their cohomology. The
main result is the following `autoduality':
If C has planar singularities, J is identified with a space of line
bundles on J'.
I also plan to discuss the Fourier-Mukai transform arising from the
autoduality.
My next talk will cover the stronger `compactified' autoduality statement
(identifying J' and with a space of torsion-free sheaves on itself), which
requires additional restrictions on C.
The compactified Jacobians play a role in the geometric Langlands
correspondence (for GL(n)), where they appear as fibers of the Hitchin
fibration. However, the talk relies on classical methods of algebraic
geometry, and should be accessible to wide audience.
Thursday (October 8) , 4:30 p.m, room E 206.
Dima Arinkin (University of North Carolina).
Autoduality for Jacobians of singular curves.
Abstract
Let C be a (smooth projective algebraic) curve, in other words, a Riemann
surface. It is well known that the Jacobian J of C is a self-dual complex
torus, that is, J is identified with the space of topologically trivial
line bundles on J.
Suppose now that C is singular. The Jacobian J of C parametrizes
topologically trivial line bundles on C; it is smooth, but no longer
compact. By considering torsion-free sheaves instead of line bundles, one
obtains a natural singular compactification J' of J.
The subject of this talk is line bundles on J' and their cohomology. The
main result is the following `autoduality':
If C has planar singularities, J is identified with a space of line
bundles on J'.
I also plan to discuss the Fourier-Mukai transform arising from the
autoduality.
My next talk will cover the stronger `compactified' autoduality statement
(identifying J' and with a space of torsion-free sheaves on itself), which
requires additional restrictions on C.
The compactified Jacobians play a role in the geometric Langlands
correspondence (for GL(n)), where they appear as fibers of the Hitchin
fibration. However, the talk relies on classical methods of algebraic
geometry, and should be accessible to wide audience.
Thursday (October 8) , 4:30 p.m, room E 206.
Dima Arinkin (University of North Carolina).
Autoduality for Jacobians of singular curves.
Abstract
Let C be a (smooth projective algebraic) curve, in other words, a Riemann
surface. It is well known that the Jacobian J of C is a self-dual complex
torus, that is, J is identified with the space of topologically trivial
line bundles on J.
Suppose now that C is singular. The Jacobian J of C parametrizes
topologically trivial line bundles on C; it is smooth, but no longer
compact. By considering torsion-free sheaves instead of line bundles, one
obtains a natural singular compactification J' of J.
The subject of this talk is line bundles on J' and their cohomology. The
main result is the following `autoduality':
If C has planar singularities, J is identified with a space of line
bundles on J'.
I also plan to discuss the Fourier-Mukai transform arising from the
autoduality.
My next talk will cover the stronger `compactified' autoduality statement
(identifying J' and with a space of torsion-free sheaves on itself), which
requires additional restrictions on C.
The compactified Jacobians play a role in the geometric Langlands
correspondence (for GL(n)), where they appear as fibers of the Hitchin
fibration. However, the talk relies on classical methods of algebraic
geometry, and should be accessible to wide audience.
There will be no meeting of the Langlands seminar on Monday. Dima Arinkin will continue his talk on Autoduality for Jacobians of singular curves on WEDNESDAY (Oct 14) at 4 p.m. in E 203 and then on Thursday (October 15) at 4:30 p.m in room E 206. (Note that the algebraic geometry will not meet this Wednesday.) I strongly recommend the students to really understand Dima's first talk. I think this is doable. You can download Dima Arinkin's article from http://arxiv.org/abs/0705.0190 As far as I understand, the relevant parts of Mumford's book on Abelian varieties are: Chapter II, Section 8 and Chapter III, Section 13.
The next seminar is on WEDNESDAY (Oct 14) at 4 p.m. in E 203 (please notice the unusual time and place). Dima Arinkin will continue his talk on autoduality for Jacobians of singular curves
Tomorrow (Thursday) at 4:30 p.m in room E 206 Dima Arinkin will continue his talk on Autoduality for Jacobians of singular curves.
Because of the Albert lecture, the Langlands seminar will not meet on Monday, October 19.
No seminar on Thursday, October 22. Next Monday (October 26) David Nadler (NWU) will start his series of talks joint with John Francis. Title of the series: Drinfeld centers and derived algebraic geometry. Abstract: In this series of talks, we would like to describe the background and results of our joint paper with David Ben-Zvi "Integral transforms and Drinfeld centers in derived algebraic geometry", http://arxiv.org/abs/0805.0157 No preliminary knowledge of derived algebraic geometry is assumed. Our broad goal is to understand some of the basic homotopical algebra of the tensor category of quasicoherent sheaves on a stack. More specifically, we would like to understand how geometric operations on stacks (fiber products, loop spaces,...) interact with algebraic operations on categories of sheaves (tensor products, Hochschild homology,...). Our plan roughly breaks into three parts: 1. The first talk will be an example-oriented overview of the phenomena to be studied. We will informally introduce the main objects and try to build up enough intuition so the audience will have a good sense as to what should be true. 2. In the second part, we will explain some of the general constructions which we will later apply to categories of sheaves. We will follow Lurie's foundations on algebra in the context of oo-categories. Our focus will be on the notion of center, and in particular the examples of Drinfeld and E_n-centers of monoidal oo-categories. 3. In the third part, we will specialize to derived algebraic geometry. Here is a sample of the kind of theorem we will discuss: Theorem: For X a perfect stack (a notion which in characteristic zero includes all of the commonly occurring examples in geometric representation theory), the Drinfeld center of the monoidal category QCoh(X) of quasicoherent sheaves on X is equivalent to the category QCoh(LX) of quasicoherent sheaves on the loop space LX = Map(S^1, X).
Monday (October 26), 4:30 p.m, room E 206.
David Nadler (NWU). Drinfeld centers and derived algebraic geometry. I.
Abstract
In this series of talks John Francis and I would like to describe the
background and results of our joint paper with David Ben-Zvi
"Integral transforms and Drinfeld centers in derived algebraic geometry",
http://arxiv.org/abs/0805.0157
No preliminary knowledge of derived algebraic geometry is assumed.
Our broad goal is to understand some of the basic homotopical algebra of
the tensor category of quasicoherent sheaves on a stack. More
specifically, we would like to understand how geometric operations on
stacks (fiber products, loop spaces,...) interact with algebraic
operations on categories of sheaves (tensor products, Hochschild
homology,...).
Our plan roughly breaks into three parts:
1. The first talk will be an example-oriented overview of the phenomena to
be studied. We will informally introduce the main objects and try to build
up enough intuition so the audience will have a good sense
as to what should be true.
2. In the second part, we will explain some of the general constructions
which we will later apply to categories of sheaves. We will follow Lurie's
foundations on algebra in the context of oo-categories. Our focus will be
on the notion of center, and in particular the examples of Drinfeld and
E_n-centers of monoidal oo-categories.
3. In the third part, we will specialize to derived algebraic geometry.
Here is a sample of the kind of theorem we will discuss:
Theorem: For X a perfect stack (a notion which in characteristic zero
includes all of the commonly occurring examples in geometric
representation theory), the Drinfeld center of the monoidal category
QCoh(X) of quasicoherent sheaves on X is equivalent to the category
QCoh(LX) of quasicoherent sheaves on the loop space LX = Map(S^1, X).
No seminar on Thursday. The next talk on "Drinfeld centers and derived algebraic geometry" will be given on Monday (November 2) by John Francis (NWU).
Monday (November 2), 4:30 p.m, room E 206. John Francis (NWU). Drinfeld centers and derived algebraic geometry. II. > Abstract > > In this series of talks John Francis and I would like to describe the > background and results of our joint paper with David Ben-Zvi > "Integral transforms and Drinfeld centers in derived algebraic geometry", > http://arxiv.org/abs/0805.0157 > > No preliminary knowledge of derived algebraic geometry is assumed. > > Our broad goal is to understand some of the basic homotopical algebra of > the tensor category of quasicoherent sheaves on a stack. More > specifically, we would like to understand how geometric operations on > stacks (fiber products, loop spaces,...) interact with algebraic > operations on categories of sheaves (tensor products, Hochschild > homology,...). > > Our plan roughly breaks into three parts: > > 1. The first talk will be an example-oriented overview of the phenomena to > be studied. We will informally introduce the main objects and try to build > up enough intuition so the audience will have a good sense > as to what should be true. > > 2. In the second part, we will explain some of the general constructions > which we will later apply to categories of sheaves. We will follow Lurie's > foundations on algebra in the context of oo-categories. Our focus will be > on the notion of center, and in particular the examples of Drinfeld and > E_n-centers of monoidal oo-categories. > > 3. In the third part, we will specialize to derived algebraic geometry. > Here is a sample of the kind of theorem we will discuss: > > Theorem: For X a perfect stack (a notion which in characteristic zero > includes all of the commonly occurring examples in geometric > representation theory), the Drinfeld center of the monoidal category > QCoh(X) of quasicoherent sheaves on X is equivalent to the category > QCoh(LX) of quasicoherent sheaves on the loop space LX = Map(S^1, X).
We do not meet tomorrow (Monday) because the speaker is sick. Presumably we meet on Thursday (November 5). > John Francis (NWU). Drinfeld centers and derived algebraic geometry. II. > >> Abstract >> >> In this series of talks John Francis and I would like to describe the >> background and results of our joint paper with David Ben-Zvi >> "Integral transforms and Drinfeld centers in derived algebraic >> geometry", >> http://arxiv.org/abs/0805.0157 >> >> No preliminary knowledge of derived algebraic geometry is assumed. >> >> Our broad goal is to understand some of the basic homotopical algebra of >> the tensor category of quasicoherent sheaves on a stack. More >> specifically, we would like to understand how geometric operations on >> stacks (fiber products, loop spaces,...) interact with algebraic >> operations on categories of sheaves (tensor products, Hochschild >> homology,...). >> >> Our plan roughly breaks into three parts: >> >> 1. The first talk will be an example-oriented overview of the phenomena >> to >> be studied. We will informally introduce the main objects and try to >> build >> up enough intuition so the audience will have a good sense >> as to what should be true. >> >> 2. In the second part, we will explain some of the general constructions >> which we will later apply to categories of sheaves. We will follow >> Lurie's >> foundations on algebra in the context of oo-categories. Our focus will >> be >> on the notion of center, and in particular the examples of Drinfeld and >> E_n-centers of monoidal oo-categories. >> >> 3. In the third part, we will specialize to derived algebraic geometry. >> Here is a sample of the kind of theorem we will discuss: >> >> Theorem: For X a perfect stack (a notion which in characteristic zero >> includes all of the commonly occurring examples in geometric >> representation theory), the Drinfeld center of the monoidal category >> QCoh(X) of quasicoherent sheaves on X is equivalent to the category >> QCoh(LX) of quasicoherent sheaves on the loop space LX = Map(S^1, X). >
Thursday (November 5), 4:30 p.m, room E 206.
John Francis (NWU) will give the second talk in the series
"Drinfeld centers and derived algebraic geometry".
As far as I understand, it will be independent from the first one
(which was given last Monday by David Nadler).
Abstract
This talk will give an exposition of E_n centers, where E_n is the operad
of configuration spaces of n-disks. The main case of interest will be the
E_n or Drinfeld (for n=1) center of a monoidal
oo-category. We will preface this topic with the necessary review of the
theory of oo-categories (definition, limits and colimits,
stability, presentability, operads and operadic monoidal structures) and,
in particular, a description of oo-categories of functors between
oo-categories.
Monday (November 9), 4:30 p.m, room E 206.
David Ben-Zvi (University of Texas at Austin).
Drinfeld centers and derived algebraic geometry. III.
Abstract
In this talk we will discuss the interaction between geometric operations on
schemes and stacks and algebraic operations on their categories of sheaves
in the context of derived algebraic geometry. We will introduce derived
schemes and stacks and their oo-categories of quasicoherent sheaves. The
latter can be considered as plain, monoidal, E_n or symmetric monoidal
oo-categories, leading to a collection of natural algebraic operations to
perform on them. We will demonstrate how the results of these
operations can be described themselves as oo-categories of quasicoherent
sheaves. As an application we discuss (time permitting) the representation
theory of quasicoherent Hecke categories.
The prerequisites are the basic notions of oo-categories, as presented in
John Francis' talk, or a willingness to work through formal analogies,
such as those presented in David Nadler's talk.
Thursday (November 12), 4:30 p.m, room E 206.
Ivan Losev (MIT) Irreducible finite-dimensional
representations for finite W-algebras.
Abstract
(Finite) W-algebras are certain associative
algebras arising in Lie theory. They are constructed
from a pair of a semisimple Lie algebra and its
nilpotent element. They originate from Kostant's
paper "On Whittaker vectors and representation theory",
where the case of a principal nilpotent element
was considered. During the last decade they attracted
some interest from representation theorists (Brundan,
Ginzburg, Goodwin, Kleshchev, Premet, the speaker
and others) mainly due to their relation to representation
theory of universal enveloping algebras, both
in zero and positive characteristics. In my talk
I will describe known results and conjectures on the
classification of finite dimensional irreducible
modules for W-algebras.
References: arXiv:0707.3108, 0807.1023, 0812.1584.
Thursday (November 12), 4:30 p.m, room E 206.
Ivan Losev (MIT) Irreducible finite-dimensional
representations for finite W-algebras.
Abstract
(Finite) W-algebras are certain associative
algebras arising in Lie theory. They are constructed
from a pair of a semisimple Lie algebra and its
nilpotent element. They originate from Kostant's
paper "On Whittaker vectors and representation theory",
where the case of a principal nilpotent element
was considered. During the last decade they attracted
some interest from representation theorists (Brundan,
Ginzburg, Goodwin, Kleshchev, Premet, the speaker
and others) mainly due to their relation to representation
theory of universal enveloping algebras, both
in zero and positive characteristics. In my talk
I will describe known results and conjectures on the
classification of finite dimensional irreducible
modules for W-algebras.
References: arXiv:0707.3108, 0807.1023, 0812.1584.
No seminar on Monday (November 16). Roman Bezrukavnikov will speak on Thursday (November 19).
Thursday (November 19), 4:30 p.m, room E 206.
Roman Bezrukavnikov (MIT).
Character D-modules as center of the categorical Hecke algebra.
Abstract
Lusztig introduced character sheaves on a reductive algebraic
group as a tool for understanding characters of finite Chevalley groups.
In a joint work with Finkelberg and Ostrik we show how to view the
corresponding objects in the D-module setting as objects of the
categorical center of the categorical Hecke algebra and rederive Lusztig's
classification results through this approach.
Some of our results are related to (though different from) results of
Ben-Zvi, Francis and Nadler.
No more Langlands seminar this quarter.