This is an archive of email messages concerning the Geometric Langlands Seminar for 11-12.
As usual, the seminar will meet on Mondays and/or Thursdays in room E206 at 4:30 p.m. First meeting: October 3. Here is the program for the nearest future. Oct 3 (Monday): Dinesh Thakur (Arizona), Multizeta and related structures in Function Field Arithmetic. Oct 6 (Thursday), Oct 10 (Monday), Oct 17 (Monday), and Oct 20 (Thursday): Michael Temkin (Hebrew University), Non-archimedean analytic geometry via Berkovich spaces and birational geometry via Riemann-Zariski spaces. Oct 13 (Thursday): James Arthur (Toronto), TBA.
Monday (Oct 3), 4:30 p.m, room E 206.
Dinesh Thakur (Arizona).
Multizeta and related structures in Function Field Arithmetic.
Abstract
We will discuss arithmetic of special values of function field valued
Gamma, Zeta, and Multizeta: relations between them, relations with periods
of pure and mixed t-motives and Shtukas, Galois groups, class groups,
class modules etc. We do not understand the fundamental group or
associator-type connection, but do have some hints for the underlying
structures.
Thursday (Oct 6), 4:30 p.m, room E 206.
Michael Temkin (Hebrew University) will give the first lecture of his
minicourse on
Non-archimedean analytic geometry via Berkovich spaces and birational
geometry via Riemann-Zariski spaces.
Abstract
The aim of this minicourse is to outline basic theories of Berkovich analytic
spaces and Riemann-Zariski birational spaces, and various connections
between them. In particular, I will show how RZ spaces are used to study
analytic spaces and vice versa. The minicourse will consists of four talks
and its approximate plan is as follows:
Talk 1. Banach rings and their Berkovich spectrum, affinoid algebras and
spaces.
Talk 2. General analytic spaces and basic classes of morphisms. Relations to
other categories: algebraization and GAGA, generic fibers of formal schemes
and Raynaud's theory.
Talk 3. Classical RZ spaces and local properties of analytic spaces. Analytic
curves and stable reduction theorem.
Talk 4. Relative RZ spaces. Applications of absolute and relative RZ spaces:
stable modification, desingularization, Nagata compactification.
A good reference for the first three talks is my lecture notes at
http://www.math.huji.ac.il/~temkin/papers/Lecture_Notes_Berkovich_Analytic_Spaces.pdf
The last talk will be related to some extent to the (old fashioned) slides at
http://www.math.huji.ac.il/~temkin/lectures/RZ.pdf
Monday (Oct 10), 4:30 p.m, room E 206. Michael Temkin. Non-archimedean analytic geometry via Berkovich spaces and birational geometry via Riemann-Zariski spaces. II.
Thursday (Oct 13), 4:30 p.m, room E 206.
James Arthur (Toronto). Representations of orthogonal and symplectic groups.
Abstract
Suppose that G is a connected, quasisplit orthogonal or symplectic group
over a number field F. I shall review the statements of theorems that
classify automorphic representations of G. The proof of these theorems
rests on an extended argument that is ultimately based on the comparison
of trace formulas, specifically, the stabilization of the trace formula
for G, and the conditional stabilization of the twisted trace formula for
GL(N). I shall try to give some overview of the proof, insofar as this is
feasible in the time available. If time permits, I could also add a
couple or remarks on some of the basic implications of the
theorems, and perhaps add a few
comments on how the classification would extend to inner forms of G.
Monday (Oct 17), 4:30 p.m, room E 206. Michael Temkin. Non-archimedean analytic geometry via Berkovich spaces and birational geometry via Riemann-Zariski spaces. III.
Thursday (Oct 20), 4:30 p.m, room E 206. Michael Temkin. Non-archimedean analytic geometry via Berkovich spaces and birational geometry via Riemann-Zariski spaces. IV.
Monday (Oct 24), 4:30 p.m, room E 206. Michael Temkin. Non-archimedean analytic geometry via Berkovich spaces and birational geometry via Riemann-Zariski spaces. V.
No seminar on Thursday (Oct 27).
Monday (Oct 31), 4:30 p.m, room E 206.
Victor Ginzburg. Quantum cohomology of symplectic resolutions, its
monodromy, and applications to representation theory
Abstract
This talk is an introduction to Bezrukavnikov's lectures on
November 3 and 7. We follow closely the exposition given by Okounkov a few
weeks ago.
Symplectic resolutions include cotangent bundles to flag varieties,
Hilbert scheme of points, and quiver varieties. It turns out that
equivariant quantum cohomology of these varieties, and an associated
Dubrovin connection, are closely related to representation theory of
various interesting noncommutative algebras, such as enveloping algebras,
Yangians, and symplectic reflection algebras. Motivated by mirror symmetry
considerations, Bezrukavnikov made a conjecture that relates the derived
category of coherent sheaves on a symplectic resolution to the monodromy
of the Dubrovin connection. This conjecture is a vast generalization of,
and has applications to Lusztig's theory of canonical bases.
Thursday (Nov 3) and Monday (Nov 7), 4:30 p.m, room E 206.
Roman Bezrukavnikov (MIT). Geometry and algebra of quantized symplectic
resolutions.
Abstract
Quantization of a symplectic resolution of singularities produces
noncommutative algebras of interest in representation theory: the most
classical example is (a central reduction of) the universal enveloping of
a semi-simple Lie algebra, more modern ones come from symplectic
reflection algebras (in particular rational double affine Hecke algebras)
and more general quiver algebras. I will describe an attempt to control
numerics of their representations (over base fields of various
characteristics) by a Kazhdan-Lusztig style formalism, and link it to
Bridgeland stability conditions and equivariant quantum cohomology. The
talks are based on joint work with Anno, Mirkovic and Okounkov.
Monday (Nov 7), 4:30 p.m, room E 206.
Roman Bezrukavnikov (MIT).
Geometry and algebra of quantized symplectic resolutions. II.
Abstract
Quantization of a symplectic resolution of singularities produces
noncommutative algebras of interest in representation theory: the most
classical example is (a central reduction of) the universal enveloping of
a semi-simple Lie algebra, more modern ones come from symplectic
reflection algebras (in particular rational double affine Hecke algebras)
and more general quiver algebras. I will describe an attempt to control
numerics of their representations (over base fields of various
characteristics) by a Kazhdan-Lusztig style formalism, and link it to
Bridgeland stability conditions and equivariant quantum cohomology. The
talks are based on joint work with Anno, Mirkovic and Okounkov.
No more meetings of the seminar this quarter (as far as I can see).
We begin the winter quarter with a series of talks by Ian Le (NWU) on Deligne's article "Finitude de l'extension de Q engendr\'ee par des traces de Frobenius, en caracteristique finie", which is available here: http://math.uchicago.edu/~drinfeld/Deligne_article.pdf (In my talks last spring I used the result of this article without explaining the proof.) Ian's first talk will be on Thursday January 5. After Ian's talks, Beilinson will give a series of lectures on his work "On the crystalline period map", arXiv:1111.3316 As usual, we meet on Mondays and/or Thursdays, 4:30 p.m, room E 206. Happy New Year!
Thursday (Jan 5), 4:30 p.m, room E 206
Ian Le (NWU). Deligne's proof of the finiteness of the extension of Q
generated by traces of Frobenius in characteristic p>0.
Deligne's article is available here:
http://math.uchicago.edu/~drinfeld/Deligne_article.pdf
Notes on Deligne's proof by H.Esnault and M.Kerz are here:
http://math.uchicago.edu/~drinfeld/Esnault-Kerz2.pdf
Abstract
In his paper, Deligne shows that for a scheme Z over F_q, and an
l-adic local system F on Z that has determinant of finite order, the field
generated by the traces of Frobenius acting on the fibers of F is a finite
extension of Q. This result was used by Drinfeld in his recent paper, "On
a conjecture of Deligne."
The outline of the talk is as follows:
1) Review of the case of curves, for which the theorem follows from work
of Lafforgue on the Langlands correspondence. We will rephrase these
results in a form useful for our purposes. This will involve recalling
some important facts about the structure of l-adic sheaves from Deligne's
Weil II.
2) Quantitative version of Lafforgue's result in the case of curves. One
consequence of Lafforgue's work is that for a curve X over F_q and an
l-adic local system F on X, there exists N such that the field
generated by the traces of Frobenius at all points of X is generated by
the traces of Frobenius at the points X(F_{q^n}) for n<N. Here, we will
give bounds, in terms of the "complexity' of a curve (as
measuredby H^1), on the size of the N. The key idea in obtaining this
bound comes from showing that the traces of Frobenius at points of
X(F_{q^n}), viewed as a function on the finite set X(F_{q^n}), are "almost
orthogonal." This can be viewed as an analogue of the
orthogonality of characters in group theory.
3) Let Z be a scheme over F_q and F an l-adic local system of Z. By
constructing curves of low complexity through any point in Z, we will show
that the field generated by the traces of Frobenius acting on F is
generated by the traces of Frobenius at the points Z(F_{q^n}) for n less
than some N.
Monday (Jan 9), 4:30 p.m, room E 206
Ian Le (NWU). Deligne's proof of the finiteness of the extension of Q
generated by traces of Frobenius in characteristic p>0. II.
> Deligne's article is available here:
http://math.uchicago.edu/~drinfeld/Deligne_article.pdf
> Notes on Deligne's proof by H.Esnault and M.Kerz are here:
http://math.uchicago.edu/~drinfeld/Esnault-Kerz2.pdf
> Abstract
>
> In his paper, Deligne shows that for a scheme Z over F_q, and an l-adic
local system F on Z that has determinant of finite order, the
field
> generated by the traces of Frobenius acting on the fibers of F is a
finite
> extension of Q. This result was used by Drinfeld in his recent paper,
"On
> a conjecture of Deligne."
>
> The outline of the talk is as follows:
>
> 1) Review of the case of curves, for which the theorem follows from work
of Lafforgue on the Langlands correspondence. We will rephrase these
results in a form useful for our purposes. This will involve recalling
some important facts about the structure of l-adic sheaves from
Deligne's
> Weil II.
>
> 2) Quantitative version of Lafforgue's result in the case of curves. One
consequence of Lafforgue's work is that for a curve X over F_q and an
l-adic local system F on X, there exists N such that the field
> generated by the traces of Frobenius at all points of X is generated by
the traces of Frobenius at the points X(F_{q^n}) for n<N. Here, we will
give bounds, in terms of the "complexity' of a curve (as
> measuredby H^1), on the size of the N. The key idea in obtaining this
bound comes from showing that the traces of Frobenius at points of
X(F_{q^n}), viewed as a function on the finite set X(F_{q^n}), are
"almost
> orthogonal." This can be viewed as an analogue of the
> orthogonality of characters in group theory.
>
> 3) Let Z be a scheme over F_q and F an l-adic local system of Z. By
constructing curves of low complexity through any point in Z, we will
show
> that the field generated by the traces of Frobenius acting on F is
generated by the traces of Frobenius at the points Z(F_{q^n}) for n less
than some N.
No seminar on Thursday.
Monday (Jan 16), 4:30 p.m, room E 206.
Takeshi Saito (University of Tokyo)
A Riemann-Roch formula for l-adic sheaves on
varieties over a local field (joint work with K. Kato)
Abstract
The wild ramification of a l-adic representation of a local field
is measured by the Swan conductor. We generalize the definition
to a constructible l-adic sheaf on a variety over a local field and prove
a Riemann-Roch formula.
Monday (Jan 16), 4:30 p.m, room E 206.
Takeshi Saito (University of Tokyo)
A Riemann-Roch formula for l-adic sheaves on
varieties over a local field (joint work with K. Kato)
Abstract
The wild ramification of a l-adic representation of a local field
is measured by the Swan conductor. We generalize the definition
to a constructible l-adic sheaf on a variety over a local field and prove
a Riemann-Roch formula.
No seminar on Thursday (Jan 19). On Monday (Jan 23) Beilinson will begin his series of lectures on his work "On the crystalline period map", arXiv:1111.3316
Monday (Jan 23), 4:30 p.m, room E 206.
Alexander Beilinson will begin his series of lectures on
p-adic Hodge theory.
(As you see from his abstract, he will begin from scratch ! )
Title: Introduction to p-adic Hodge theory.
Abstract
I will discuss a recent approach to p-adic Hodge
theory (the Fontaine-Jannsen C_{st} conjecture) based on the
p-adic Poincar'e lemma.
The first talk(s) review the de Rham/log crystalline part of the story
developed in works of Kato and Hyodo-Kato from the end of 1980s.
No knowledge of the subject is assumed.
This to remind you that today Beilinson is beginning his series of
lectures on p-adic Hodge theory.
He will begin from scratch, and he says that the theory is becoming
very simple !
******
Monday (Jan 23), 4:30 p.m, room E 206.
Title: Introduction to p-adic Hodge theory.
Abstract
I will discuss a recent approach to p-adic Hodge
theory (the Fontaine-Jannsen C_{st} conjecture) based on the
p-adic Poincar'e lemma.
The first talk(s) review the de Rham/log crystalline part of the story
developed in works of Kato and Hyodo-Kato from the end of 1980s.
No knowledge of the subject is assumed.
No seminar on Thursday. Beilinson will continue on Monday (Jan 30).
Some articles mentioned in Beilinson's talk can be downloaded here: http://math.uchicago.edu/~drinfeld/p-adic_periods/Kato_log-structures.pdf http://math.uchicago.edu/~drinfeld/p-adic_periods/Bhargav_Bhatt&deJong.pdf http://math.uchicago.edu/~drinfeld/p-adic_periods/Bhatt-p-adic_derived_de_Rham.pdf http://math.uchicago.edu/~drinfeld/p-adic_periods/Brinon+Brian_Conrad.pdf Here is a book on log structures written by Ogus (Beilinson says it is good): http://math.uchicago.edu/~drinfeld/p-adic_periods/Ogus_logbook.pdf
Beilinson will continue on Monday (Jan 30), 4:30 p.m, room E 206. As I said, some articles mentioned in Beilinson's first talk and also the book on log structures written by Ogus can be downloaded here: http://math.uchicago.edu/~drinfeld/p-adic_periods/Kato_log-structures.pdf http://math.uchicago.edu/~drinfeld/p-adic_periods/Bhargav_Bhatt&deJong.pdf http://math.uchicago.edu/~drinfeld/p-adic_periods/Bhatt-p-adic_derived_de_Rham.pdf http://math.uchicago.edu/~drinfeld/p-adic_periods/Brinon+Brian_Conrad.pdf http://math.uchicago.edu/~drinfeld/p-adic_periods/Ogus_logbook.pdf
Beilinson will continue on Thursday (Feb 2), 4:30 p.m, room E 206. **** Probably to understand Beilinson, one should have a look at the following articles: http://math.uchicago.edu/~drinfeld/p-adic_periods/Kato_log-structures.pdf http://math.uchicago.edu/~drinfeld/p-adic_periods/Bhargav_Bhatt&deJong.pdf
The date of the next seminar will be announced later.
*******
The articles by Kato and Hyodo-Kato are available here:
http://math.uchicago.edu/~mmorrow/Miscpapers/
On the other hand, Fontaine's articles mentioned in Sasha's first talk are
available here:
http://math.uchicago.edu/~drinfeld/p-adic_periods/Fontaine-corps_de_periodes.pdf
http://math.uchicago.edu/~drinfeld/p-adic_periods/Fontaine-reps_semistables.pdf
Beilinson will continue on
Monday (Feb 6), 4:30 p.m, room E 206.
I didn't understand the Hyodo-Kato theory.
So I asked Sasha to explain it again from scratch.
*******
Illusie's notes mentioned by Sasha are here:
http://www.math.u-psud.fr/~illusie/Illusie-Sapporo1.pdf
http://www.math.u-psud.fr/~illusie/Illusie-Sapporo-Hyodo-Kato.pdf
The articles by Kato and Hyodo-Kato are available here:
http://math.uchicago.edu/~mmorrow/Miscpapers/
Fontaine's articles mentioned in Sasha's first talk are available here:
http://math.uchicago.edu/~drinfeld/p-adic_periods/Fontaine-corps_de_periodes.pdf
http://math.uchicago.edu/~drinfeld/p-adic_periods/Fontaine-reps_semistables.pdf
Thursday (Feb 9), 4:30 p.m, room E 206.
David Kazhdan (Hebrew University). Arithmetic varieties.
(Beilinson will continue next Monday.)
[I asked Kazhdan to speak on his old work on conjugation of
arithmetic varieties. Note that the proofs of the general results in the
theory of Shimura varieties are still based on it! ]
Abstract of Kazhdan's talk
Let G be a semisimple Lie group such that the symmetric space D=G/K admits a
G-invariant complex structure. Then for any arithmetic subgroup $\Gamma$
in G, the quotient of D by $\Gamma$ is an analytic variety which has a
canonical structure of an algebraic variety over the field of complex
numbers, C. So if $\sigma$ is any automorphism of C one can consider the
$\sigma$-conjugate variety. I will discuss the proof of the following result.
Theorem. The conjugate variety is a quotient of D by some
arithmetic subgroup $\Gamma'$ .
1. Beilinson will continue (and hopefully finish) his talk on Monday (Feb 13), 4:30 p.m, room E 206. 2. The unpublished lemma by Kazhdan mentioned in his talk is available here: http://math.uchicago.edu/~drinfeld/Kazhdan_to_McMullen_March_10_2010.pdf
No more meetings of the seminar this quarter.
The first meeting of the seminar is on April 2 (Monday).
April 2 (Monday), 4:30 p.m, room E 206
Bertrand Toen. University of Montpellier.
Introduction to derived Artin n-stacks. I.
Abstract
The purpose of this series of lectures is to provide an introduction to
the notion of derived Artin n-stacks. I will start from the very
beginning (no knowledge of derived algebraic geometry is required):
simplicial algebras, Grothendieck topologies on simplicial algebras,
stacks, geometric stacks ...
These notions will be illustrated by examples, such as the moduli of objects
in a nice enough dg-category. In a second part I will present some ideas
for a proof that derived Artin n-stacks are local for the flat topology.
This will be used in order to show the existence of stratifications by
gerbes.
Monday (April 2), 4:30 p.m, room E 206
Bertrand Toen. University of Montpellier.
Introduction to derived Artin n-stacks. I.
Abstract
The purpose of this series of lectures is to provide an introduction to
the notion of derived Artin n-stacks. I will start from the very
beginning (no knowledge of derived algebraic geometry is required):
simplicial algebras, Grothendieck topologies on simplicial algebras,
stacks, geometric stacks ...
These notions will be illustrated by examples, such as the moduli of
objects in a nice enough dg-category. In a second part I will present some
ideas for a proof that derived Artin n-stacks are local for the flat
topology. This will be used in order to show the existence of
stratifications by gerbes.
Thursday (April 5), 4:30 p.m, room E 206 Bertrand Toen. University of Montpellier. Introduction to derived Artin n-stacks. II.
Monday (April 9), 4:30 p.m, room E 206 Bertrand Toen. University of Montpellier. Introduction to derived Artin n-stacks. III.
No seminar on Thursday.
On Monday Sam Raskin will begin his series of talks.
*****************
M.Artin proved the equivalence of two definitions of algebraic stack in
his article "Versal deformations and algebraic stacks". The article is
available here:
http://math.uchicago.edu/~drinfeld/Artin_on_stacks.pdf
The proof is on p.184-186, which corresponds to p.20-22 of the PDF file.
*****************
Here is the title and abstract of the talk on
Monday (April 16), 4:30 p.m, room E 206
Sam Raskin (Harvard University)
A geometric approach to the Feigin-Frenkel theorem
Abstract
We will present a new approach to the theorem of Beilinson and Drinfeld
completely describing the derived global sections of spherical D-modules
on the affine Grassmannian at critical level as modules over the
corresponding affine Kac-Moody algebra. Unlike their proof, this approach
does not rely on Feigin-Frenkel theorem relating the critical-level center
of the affine Kac-Moody algebra attached to a semisimple Lie algebra to
the space of opers for the Langlands dual group, and gives a new proof of
this theorem via a construction of Beilinson-Drinfeld-Frenkel-Gaitsgory.
Monday (April 16), 4:30 p.m, room E 206
Sam Raskin (Harvard University)
A geometric approach to the Feigin-Frenkel theorem
Abstract
We will present a new approach to the theorem of Beilinson and Drinfeld
completely describing the derived global sections of spherical D-modules
on the affine Grassmannian at critical level as modules over the
corresponding affine Kac-Moody algebra. Unlike their proof, this approach
does not rely on Feigin-Frenkel theorem relating the critical-level center
of the affine Kac-Moody algebra attached to a semisimple Lie algebra to
the space of opers for the Langlands dual group, and gives a new proof of
this theorem via a construction of Beilinson-Drinfeld-Frenkel-Gaitsgory.
Thursday (April 19), 4:30 p.m, room E 206
Sam Raskin (Harvard University)
A geometric approach to the Feigin-Frenkel theorem. II.
Abstract
We will present a new approach to the theorem of Beilinson and Drinfeld
completely describing the derived global sections of spherical D-modules
on the affine Grassmannian at critical level as modules over the
corresponding affine Kac-Moody algebra. Unlike their proof, this approach
does not rely on Feigin-Frenkel theorem relating the critical-level center
of the affine Kac-Moody algebra attached to a semisimple Lie algebra to
the space of opers for the Langlands dual group, and gives a new proof of
this theorem via a construction of Beilinson-Drinfeld-Frenkel-Gaitsgory.
No seminar on Monday.
Next meeting:
Thursday (April 26), 4:30 p.m, room E 206
Sam Raskin (Harvard University)
A geometric approach to the Feigin-Frenkel theorem. III.
Abstract
We will present a new approach to the theorem of Beilinson and Drinfeld
completely describing the derived global sections of spherical D-modules
on the affine Grassmannian at critical level as modules over the
corresponding affine Kac-Moody algebra. Unlike their proof, this approach
does not rely on Feigin-Frenkel theorem relating the critical-level center
of the affine Kac-Moody algebra attached to a semisimple Lie algebra to
the space of opers for the Langlands dual group, and gives a new proof of
this theorem via a construction of Beilinson-Drinfeld-Frenkel-Gaitsgory.
Thursday (April 26), 4:30 p.m, room E 206
Sam Raskin (Harvard University)
A geometric approach to the Feigin-Frenkel theorem. III.
Abstract
We will present a new approach to the theorem of Beilinson and Drinfeld
completely describing the derived global sections of spherical D-modules
on the affine Grassmannian at critical level as modules over the
corresponding affine Kac-Moody algebra. Unlike their proof, this approach
does not rely on Feigin-Frenkel theorem relating the critical-level center
of the affine Kac-Moody algebra attached to a semisimple Lie algebra to
the space of opers for the Langlands dual group, and gives a new proof of
this theorem via a construction of Beilinson-Drinfeld-Frenkel-Gaitsgory.
No seminar on Monday.
On Thursday (May 3) Dima Arinkin will begin his series of talks on his
recent article with Dennis Gaitsgory, see
http://arxiv.org/abs/1201.6343
In this VERY IMPORTANT work they give a precise formulation of the
geometric Langlands conjecture (the so-called "Best Hope"). The previous
formulations (by me and by others) were either incorrect or not quite
precise, i.e., only "modulo something".
**********
Thursday (May 3), 4:30 p.m, room E 206
Dima Arinkin (University of North Carolina)
Singular support of coherent sheaves and the Langlands conjecture. I.
Abstract
In its `categorical' version, the geometric Langlands conjecture predicts
deep ties between quasi-coherent sheaves on the moduli stack of local
systems and D-modules on the moduli stack of principal bundles. It is very
natural to state the conjecture as an equivalence between the derived
categories of these objects.
However, such naive formulation turns out to be inconsistent: the two
categories do not match. In a joint work with D.Gaitsgory, we develop a
way to correct this issue. We study the notion of singular support for
(ind)-coherent sheaves on a locally complete intersection. We then suggest
that in order for the geometric Langlands conjecture to give an
equivalence of categories, quasicoherent sheaves should be replaced with
ind-coherent sheaves whose singular support is contained in a certain
natural subset.
In this series of talks, I have two goals. Firstly, I plan to present the
general theory of ind-coherent sheaves and their singular support. I will
then explain a version of the Langlands conjecture using ind-coherent
sheaves, and show that this new version is compatible with natural
constructions (such as the Eisenstein series), unlike the naive statement
using quasi-coherent sheaves.
On Thursday Dima Arinkin will begin his series of talks on his recent
article with Dennis Gaitsgory, see
http://arxiv.org/abs/1201.6343
In this VERY IMPORTANT work they give a precise formulation of the
geometric Langlands conjecture (the so-called "Best Hope"). The previous
formulations (by me and by others) were either incorrect or not quite
precise, i.e., only "modulo something".
**********
Thursday (May 3), 4:30 p.m, room E 206
Dima Arinkin (University of North Carolina)
Singular support of coherent sheaves and the Langlands conjecture. I.
Abstract
In its `categorical' version, the geometric Langlands conjecture predicts
deep ties between quasi-coherent sheaves on the moduli stack of local
systems and D-modules on the moduli stack of principal bundles. It is very
natural to state the conjecture as an equivalence between the derived
categories of these objects.
However, such naive formulation turns out to be inconsistent: the two
categories do not match. In a joint work with D.Gaitsgory, we develop a
way to correct this issue. We study the notion of singular support for
(ind)-coherent sheaves on a locally complete intersection. We then suggest
that in order for the geometric Langlands conjecture to give an
equivalence of categories, quasicoherent sheaves should be replaced with
ind-coherent sheaves whose singular support is contained in a certain
natural subset.
In this series of talks, I have two goals. Firstly, I plan to present the
general theory of ind-coherent sheaves and their singular support. I will
then explain a version of the Langlands conjecture using ind-coherent
sheaves, and show that this new version is compatible with natural
constructions (such as the Eisenstein series), unlike the naive statement
using quasi-coherent sheaves.
Monday (May 7), 4:30 p.m, room E 206
Dima Arinkin (University of North Carolina)
Singular support of coherent sheaves and the Langlands conjecture. II.
Abstract
In its `categorical' version, the geometric Langlands conjecture predicts
deep ties between quasi-coherent sheaves on the moduli stack of local
systems and D-modules on the moduli stack of principal bundles. It is very
natural to state the conjecture as an equivalence between the derived
categories of these objects.
However, such naive formulation turns out to be inconsistent: the two
categories do not match. In a joint work with D.Gaitsgory, we develop a
way to correct this issue. We study the notion of singular support for
(ind)-coherent sheaves on a locally complete intersection. We then suggest
that in order for the geometric Langlands conjecture to give an
equivalence of categories, quasicoherent sheaves should be replaced with
ind-coherent sheaves whose singular support is contained in a certain
natural subset.
In this series of talks, I have two goals. Firstly, I plan to present the
general theory of ind-coherent sheaves and their singular support. I will
then explain a version of the Langlands conjecture using ind-coherent
sheaves, and show that this new version is compatible with natural
constructions (such as the Eisenstein series), unlike the naive statement
using quasi-coherent sheaves.
Dima Arinkin will continue (and probably finish) his talk on Thursday (May 10), 4:30 p.m, room E 206 Here is information about two workshops in the enarest future: ********** A school on Algebraic Microlocal Analysis will take place at NWU from May 14 to May 26, 2012, see http://www.math.northwestern.edu/~tsygan/conf.html ********** Here is an announcement of the workshop to be held at UIC this weekend: Towards a Local Proof of the Local Langlands Correspondence University of Chicago and University of Illinois at Chicago, May 12-13, 2012 It aims to provide graduate students and young researchers with an overview of the recent developments in the field. There will be series of talks on the following subjects, emphasizing the geometric nature of the correspondence: Introduction to the Lubin-Tate spaces (by A. Caraiani). The cohomology of the Lubin-Tate tower (by M. Strauch). Moduli of p-divisible groups (by J. Weinstein). Introduction to the theory of smooth representations of p-adic groups via types and covers (by P. Kutzko). Local Langlands correspondence and zeta functions of varieties over finite fields (by M. Boyarchenko). For more information, please visit the webpage: http://math.uchicago.edu/~lxiao/workshop_site/
No seminar until Gerard Laumon's talk on May 24 (Thursday). Let me recall that there is a workshop "Towards a Local Proof of the Local Langlands Correspondence" at UIC this weekend, see http://math.uchicago.edu/~lxiao/workshop_site/ and a school on Algebraic Microlocal Analysis at NWU from May 14 to May 26, see http://www.math.northwestern.edu/~tsygan/conf.html