Geometric Langlands Seminar

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



  • Date: Mon, 24 Sep 2012 13:28:47

  • 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.
    
    
    


  • Date: Sun, 30 Sep 2012 19:52:56

  • 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.
    
    
    
    
    


  • Date: Thu, 4 Oct 2012 18:24:15

  • 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.
    
    
    
    
    
    
    
    
    


  • Date: Mon, 8 Oct 2012 18:55:52

  • 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.
    >
    >
    >
    >
    >
    >
    >
    >
    
    


  • Date: Thu, 11 Oct 2012 18:50:18

  • 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.
    
    


  • Date: Wed, 17 Oct 2012 18:22:36

  • 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.
    
    
    
    


  • Date: Thu, 18 Oct 2012 18:51:27

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


  • Date: Mon, 22 Oct 2012 18:59:47

  • 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
    
    
    
    
    
    
    
    
    
    
    


  • Date: Mon, 29 Oct 2012 13:40:37

  • 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.
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    


  • Date: Sun, 4 Nov 2012 15:33:46

  • 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).
    
    
    
    
    
    


  • Date: Wed, 7 Nov 2012 17:14:06

  • 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).
    
    
    
    
    
    
    
    
    
    


  • Date: Thu, 8 Nov 2012 18:13:16

  • 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.
    
    
    
    
    
    
    
    


  • Date: Mon, 12 Nov 2012 18:21:29

  • 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.
    
    
    
    
    
    
    
    
    
    


  • Date: Thu, 15 Nov 2012 18:30:51

  • 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
    Description: Adobe PDF document



  • Date: Thu, 22 Nov 2012 18:29:05

  • 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.
    
    
    
    
    
    
    
    
    
    


  • Date: Tue, 27 Nov 2012 09:55:45

  • 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.
    
    
    
    
    
    
    
    
    
    
    
    


  • Date: Thu, 29 Nov 2012 18:49:57

  • 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.
    
    
    
    
    
    
    
    
    
    
    
    


  • Date: Wed, 5 Dec 2012 18:15:41

  • 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.
    
    
    
    
    
    
    
    
    
    
    
    
    
    


  • Date: Thu, 6 Dec 2012 18:52:34

  • No more meetings of the Geometric Langlands seminar this quarter.
    
    
    


  • Date: Mon, 7 Jan 2013 08:58:22

  • 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.
    
    
    
    


  • Date: Thu, 10 Jan 2013 20:03:48

  • 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.
    
    


  • Date: Tue, 15 Jan 2013 09:45:53

  • 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).
    
    
    
    
    
    
    
    
    


  • Date: Thu, 17 Jan 2013 19:25:28

  • 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.
    
    
    
    


  • Date: Mon, 21 Jan 2013 18:41:49

  • 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.
    
    
    
    
    
    
    
    


  • Date: Fri, 25 Jan 2013 11:38:40

  • 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.
    
    
    
    
    
    


  • Date: Fri, 25 Jan 2013 20:18:42

  • 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.
    
    
    
    
    
    
    
    


  • Date: Tue, 29 Jan 2013 09:39:43

  • 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.
    
    
    
    


  • Date: Fri, 1 Feb 2013 16:38:21

  • 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.
    
    
    
    


  • Date: Mon, 4 Feb 2013 18:39:18

  • 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.
    
    
    
    
    
    
    
    


  • Date: Thu, 7 Feb 2013 18:37:04

  • 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.
    
    
    
    


  • Date: Mon, 11 Feb 2013 19:16:40

  • 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.
    
    
    
    
    
    


  • Date: Thu, 14 Feb 2013 18:34:51

  • 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 $W$.
    
    
    
    


  • Date: Tue, 19 Feb 2013 18:03:52

  • 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.
    
    


  • Date: Thu, 21 Feb 2013 19:27:53

  • 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.
    
    


  • Date: Tue, 26 Feb 2013 18:36:41

  • 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.
    
    
    
    
    
    


  • Date: Thu, 28 Feb 2013 19:22:43

  • 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.
    
    
    
    
    


  • Date: Mon, 4 Mar 2013 20:09:13

  • No more meetings of the Geometric Langlands seminar this quarter.
    
    


  • Date: Mon, 1 Apr 2013 08:27:22

  • 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!)
    
    
    
    
    
    
    
    
    
    
    
    


  • Date: Thu, 4 Apr 2013 18:55:13

  • 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.
    
    
    


  • Date: Sun, 7 Apr 2013 12:42:54

  • 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.
    
    
    
    
    


  • Date: Mon, 8 Apr 2013 18:59:20

  • 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.
    
    
    
    
    


  • Date: Thu, 11 Apr 2013 20:01:20

  • 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.
    
    
    
    
    
    
    


  • Date: Mon, 15 Apr 2013 14:01:21

  • 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.
    
    
    
    
    
    
    
    
    


  • Date: Mon, 15 Apr 2013 20:23:38

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


  • Date: Thu, 18 Apr 2013 19:19:35

  • 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.
    
    
    
    
    


  • Date: Wed, 24 Apr 2013 17:02:58

  • 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.
    
    
    
    
    
    
    
    
    
    
    


  • Date: Fri, 26 Apr 2013 18:36:03

  • 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.
    
    


  • Date: Mon, 29 Apr 2013 17:16:14

  • 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.
    
    
    
    
    
    
    
    


  • Date: Wed, 1 May 2013 18:25:18

  • 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.
    
    
    
    
    
    
    
    
    
    


  • Date: Thu, 2 May 2013 18:57:07

  • 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.
    
    
    
    
    
    
    
    
    
    
    
    


  • Date: Mon, 6 May 2013 18:00:01

  • 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.
    
    
    
    
    
    


  • Date: Thu, 9 May 2013 18:59:24

  • 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.
    
    


  • Date: Mon, 13 May 2013 19:16:37

  • No more meetings of the Langlands seminar this quarter.
    
    


  • Date: Mon, 23 Sep 2013 09:51:18

  • 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.
    
    
    
    
    
    
    
    
    


  • Date: Thu, 3 Oct 2013 17:06:38

  • 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.
    
    
    


  • Date: Tue, 8 Oct 2013 08:57:14

  • 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.
    
    
    
    
    


  • Date: Tue, 8 Oct 2013 19:35:33

  • 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.
    
    
    
    
    
    
    
    
    
    
    


  • Date: Thu, 10 Oct 2013 19:23:23

  • 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.
    
    
    
    
    
    
    
    
    
    
    
    
    
    


  • Date: Mon, 14 Oct 2013 19:33:57

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


  • Date: Thu, 17 Oct 2013 20:21:54

  • 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.
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    


  • Date: Sun, 20 Oct 2013 21:54:07

  • 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.
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    


  • Date: Mon, 21 Oct 2013 19:47:55

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


  • Date: Thu, 24 Oct 2013 17:07:19

  • 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.
    
    
    
    
    
    
    
    
    


  • Date: Mon, 28 Oct 2013 18:27:50

  • 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.
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    


  • Date: Fri, 1 Nov 2013 11:43:54

  • 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
    Description: Adobe PDF document



  • Date: Tue, 5 Nov 2013 19:23:09

  • 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.
    
    
    


  • Date: Fri, 8 Nov 2013 12:21:42

  • 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.
    
    
    
    


  • Date: Tue, 12 Nov 2013 08:12:36

  • 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).
    
    


  • Date: Thu, 14 Nov 2013 08:10:48

  • 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
    Description: Adobe PDF document



  • Date: Thu, 14 Nov 2013 20:02:20

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


  • Date: Tue, 19 Nov 2013 10:34:20

  • 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".]
    
    
    
    


  • Date: Thu, 21 Nov 2013 18:24:50

  • 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.
    
    
    
    
    
    
    


  • Date: Sun, 24 Nov 2013 18:33:44

  • 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.
    
    
    
    
    
    
    
    
    


  • Date: Mon, 25 Nov 2013 20:46:52

  • No more meetings of the seminar this quarter.
    
    


  • Date: Thu, 2 Jan 2014 07:55:54

  • 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.
    
    
    
    


  • Date: Wed, 8 Jan 2014 18:03:34

  • 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.
    
    
    
    
    
    


  • Date: Thu, 9 Jan 2014 18:58:13

  • 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.
    
    
    
    
    
    
    
    


  • Date: Tue, 14 Jan 2014 08:49:13

  • 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.
    
    


  • Date: Thu, 16 Jan 2014 19:51:11

  • 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.
    
    
    
    


  • Date: Mon, 20 Jan 2014 18:57:11

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


  • Date: Thu, 23 Jan 2014 18:31:01

  • 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.
    
    
    
    
    
    
    
    


  • Date: Mon, 27 Jan 2014 19:22:59

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


  • Date: Thu, 30 Jan 2014 18:45:37

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


  • Date: Mon, 3 Feb 2014 18:39:08

  • 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.
    
    
    
    
    
    
    
    
    
    
    


  • Date: Fri, 7 Feb 2014 14:35:44

  • 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.
    
    
    
    
    
    
    
    
    
    
    


  • Date: Tue, 11 Feb 2014 10:01:31

  • 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.
    
    
    
    
    
    
    
    
    
    
    
    
    


  • Date: Fri, 14 Feb 2014 09:27:00

  • 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.
    
    
    
    


  • Date: Tue, 18 Feb 2014 08:41:00

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


  • Date: Thu, 20 Feb 2014 21:07:45

  • 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.
    
    
    
    
    
    
    
    
    
    
    


  • Date: Mon, 24 Feb 2014 17:54:47

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


  • Date: Thu, 27 Feb 2014 18:56:22

  • 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.
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    


  • Date: Thu, 6 Mar 2014 10:51:55

  • 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.
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    


  • Date: Fri, 7 Mar 2014 08:56:17

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


  • Date: Sat, 29 Mar 2014 14:17:06

  • 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".
    
    
    
    
    
    
    
    
    
    


  • Date: Thu, 3 Apr 2014 16:56:11

  • 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.
    
    
    
    
    
    
    
    
    
    
    
    
    
    


  • Date: Mon, 7 Apr 2014 18:35:51

  • 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).
    
    
    
    
    


  • Date: Mon, 14 Apr 2014 17:01:40

  • 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.
    
    
    
    
    
    
    
    
    


  • Date: Thu, 17 Apr 2014 18:47:14

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


  • Date: Tue, 22 Apr 2014 08:53:21

  • 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.
    
    
    
    


  • Date: Fri, 25 Apr 2014 08:46:11

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


  • Date: Thu, 1 May 2014 17:25:35

  • 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.
    
    
    
    
    


  • Date: Tue, 6 May 2014 08:13:43

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


  • Date: Mon, 12 May 2014 20:14:55

  • 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.
    
    
    
    


  • Date: Thu, 15 May 2014 18:44:15

  • 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.
    
    
    
    
    
    


  • Date: Mon, 19 May 2014 18:43:30

  • 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/
    
    
    
    


  • Date: Thu, 25 Sep 2014 18:38:37

  • 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).
    
    
    
    


  • Date: Thu, 2 Oct 2014 17:08:37

  • 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.
    
    
    
    
    
    
    


  • Date: Mon, 6 Oct 2014 17:11:28

  • 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.
    
    
    
    
    
    
    
    
    
    
    


  • Date: Thu, 9 Oct 2014 18:44:12

  • 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.
    
    
    


  • Date: Mon, 13 Oct 2014 18:42:23

  • 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
    Description: Adobe PDF document



  • Date: Fri, 17 Oct 2014 15:43:10

  • 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).
    
    
    
    
    


  • Date: Tue, 21 Oct 2014 09:14:12

  • 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).
    
    


  • Date: Thu, 23 Oct 2014 19:18:31

  • 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.
    
    
    
    


  • Date: Mon, 27 Oct 2014 18:48:39

  • 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.
    
    
    
    
    


  • Date: Thu, 30 Oct 2014 17:20:41

  • 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.
    
    
    
    
    
    
    
    
    


  • Date: Mon, 3 Nov 2014 18:40:31

  • 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).
    
    
    
    
    
    
    
    
    
    
    


  • Date: Thu, 6 Nov 2014 19:22:19

  • 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.
    
    
    
    
    
    
    
    
    
    
    
    
    


  • Date: Mon, 10 Nov 2014 19:42:15

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


  • Date: Tue, 11 Nov 2014 16:26:26

  • 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
    Description: Adobe PDF document



  • Date: Thu, 13 Nov 2014 19:29:18

  • 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
    Description: Adobe PDF document



  • Date: Thu, 20 Nov 2014 16:41:46

  • 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.
    
    
    
    


  • Date: Mon, 24 Nov 2014 19:20:44

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


  • Date: Thu, 27 Nov 2014 11:23:45

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


  • Date: Mon, 1 Dec 2014 18:51:44

  • No more meetings of the seminar this quarter.