Geometric Langlands Seminar

This is an archive of email messages concerning the Geometric Langlands Seminar for 09-10.



  • Date: Mon, 28 Sep 2009 13:35:57

  • No seminar on Thursday October 1 and Monday October 5.
    
    The seminar will meet on October 8.
    
    
    October 8, 4:30 p.m, room E 206.
    
    Dima Arinkin (University of North Carolina).
    Autoduality for Jacobians of singular curves.
    
                      Abstract
    
    Let C be a (smooth projective algebraic) curve, in other words, a Riemann
    surface. It is well known that the Jacobian J of C is a self-dual complex
    torus, that is, J is identified with the space of topologically trivial
    line bundles on J.
    
    Suppose now that C is singular. The Jacobian J of C parametrizes
    topologically trivial line bundles on C; it is smooth, but no longer
    compact. By considering torsion-free sheaves instead of line bundles, one
    obtains a natural singular compactification J' of J.
    
    The subject of this talk is line bundles on J' and their cohomology. The
    main result is the following `autoduality':
    If C has planar singularities, J is identified with a space of line
    bundles on J'.
    I also plan to discuss the Fourier-Mukai transform arising from the
    autoduality.
    
    My next talk will cover the stronger `compactified' autoduality statement
    (identifying J' and with a space of torsion-free sheaves on itself), which
    requires additional restrictions on C.
    
    The compactified Jacobians play a role in the geometric Langlands
    correspondence (for GL(n)), where they appear as fibers of the Hitchin
    fibration. However, the talk relies on classical methods of algebraic
    geometry, and should be accessible to wide audience.
    
    


  • Date: Mon, 5 Oct 2009 17:30:50

  • Thursday (October 8) , 4:30 p.m, room E 206.
    
    Dima Arinkin (University of North Carolina).
    Autoduality for Jacobians of singular curves.
    
                      Abstract
    
    Let C be a (smooth projective algebraic) curve, in other words, a Riemann 
    surface. It is well known that the Jacobian J of C is a self-dual complex 
    torus, that is, J is identified with the space of topologically trivial 
    line bundles on J.
    
    Suppose now that C is singular. The Jacobian J of C parametrizes
    topologically trivial line bundles on C; it is smooth, but no longer 
    compact. By considering torsion-free sheaves instead of line bundles, one 
    obtains a natural singular compactification J' of J.
    
    The subject of this talk is line bundles on J' and their cohomology. The 
    main result is the following `autoduality':
    If C has planar singularities, J is identified with a space of line 
    bundles on J'.
    I also plan to discuss the Fourier-Mukai transform arising from the 
    autoduality.
    
    My next talk will cover the stronger `compactified' autoduality statement 
    (identifying J' and with a space of torsion-free sheaves on itself), which
    requires additional restrictions on C.
    
    The compactified Jacobians play a role in the geometric Langlands 
    correspondence (for GL(n)), where they appear as fibers of the Hitchin 
    fibration. However, the talk relies on classical methods of algebraic 
    geometry, and should be accessible to wide audience.
    
    
    
    
    
    
    


  • Date: Wed, 7 Oct 2009 15:37:53

  • Thursday (October 8) , 4:30 p.m, room E 206.
    
    Dima Arinkin (University of North Carolina).
    Autoduality for Jacobians of singular curves.
    
                      Abstract
    
    Let C be a (smooth projective algebraic) curve, in other words, a Riemann
    surface. It is well known that the Jacobian J of C is a self-dual complex
    torus, that is, J is identified with the space of topologically trivial
    line bundles on J.
    
    Suppose now that C is singular. The Jacobian J of C parametrizes
    topologically trivial line bundles on C; it is smooth, but no longer
    compact. By considering torsion-free sheaves instead of line bundles, one
    obtains a natural singular compactification J' of J.
    
    The subject of this talk is line bundles on J' and their cohomology. The
    main result is the following `autoduality':
    If C has planar singularities, J is identified with a space of line
    bundles on J'.
    I also plan to discuss the Fourier-Mukai transform arising from the
    autoduality.
    
    My next talk will cover the stronger `compactified' autoduality statement
    (identifying J' and with a space of torsion-free sheaves on itself), which
    requires additional restrictions on C.
    
    The compactified Jacobians play a role in the geometric Langlands
    correspondence (for GL(n)), where they appear as fibers of the Hitchin
    fibration. However, the talk relies on classical methods of algebraic
    geometry, and should be accessible to wide audience.
    
    
    
    
    
    
    


  • Date: Fri, 9 Oct 2009 15:14:49

  • There will be no meeting of the Langlands seminar on Monday.
    
    Dima Arinkin will continue his talk on
       Autoduality for Jacobians of singular curves
    on WEDNESDAY (Oct 14) at 4 p.m. in E 203
    and then on Thursday (October 15) at 4:30 p.m in room E 206.
    
    (Note that the algebraic geometry will not meet this Wednesday.)
    
    I strongly recommend the students to really understand Dima's first talk.
    I think this is doable.
    
    You can download Dima Arinkin's article from
    http://arxiv.org/abs/0705.0190
    
    As far as I understand, the relevant parts of Mumford's book on
    Abelian varieties are:
     Chapter II, Section 8
     and Chapter III, Section 13.
    
    
    
    


  • Date: Mon, 12 Oct 2009 21:12:24

  • The next seminar is on
    WEDNESDAY (Oct 14) at 4 p.m. in E 203
    (please notice the unusual time and place).
    
    Dima Arinkin will continue his talk on
       autoduality for Jacobians of singular curves
    
    
    
    


  • Date: Wed, 14 Oct 2009 17:29:39

  • Tomorrow (Thursday) at 4:30 p.m in room E 206
    Dima Arinkin will continue his talk on
       Autoduality for Jacobians of singular curves.
    
    
    
    
    
    


  • Date: Fri, 16 Oct 2009 12:14:17

  • Because of the Albert lecture,
    the Langlands seminar will not meet on Monday, October 19.
    
    
    
    
    


  • Date: Mon, 19 Oct 2009 17:49:17

  • No seminar on Thursday, October 22.
    
    Next Monday (October 26) David Nadler (NWU) will start his series of talks
     joint with John Francis.
    
    
    Title of the series: Drinfeld centers and derived algebraic geometry.
    
    Abstract: In this series of talks, we would like to describe the
    background and results of our joint paper with David Ben-Zvi
    "Integral transforms and Drinfeld centers in derived algebraic geometry",
    http://arxiv.org/abs/0805.0157
    
    No preliminary knowledge of derived algebraic geometry is assumed.
    
    Our broad goal is to understand some of the basic homotopical algebra of
    the tensor category of quasicoherent sheaves on a stack. More
    specifically, we would like to understand how geometric operations on
    stacks (fiber products, loop spaces,...) interact with algebraic
    operations on categories of sheaves (tensor products, Hochschild
    homology,...).
    
    Our plan roughly breaks into three parts:
    
    1. The first talk will be an example-oriented overview of the phenomena to
    be studied. We will informally introduce the main objects and try to build
    up enough intuition so the audience will have a good sense
    as to what should be true.
    
    2. In the second part, we will explain some of the general constructions
    which we will later apply to categories of sheaves. We will follow Lurie's
    foundations on algebra in the context of oo-categories. Our focus will be
    on the notion of center, and in particular the examples of Drinfeld and 
    E_n-centers of monoidal oo-categories.
    
    3. In the third part, we will specialize to derived algebraic geometry. Here
    is a sample of the kind of theorem we will discuss:
    
    Theorem: For X a perfect stack (a notion which in characteristic zero
    includes all of the commonly occurring examples in geometric
    representation theory), the Drinfeld center of the monoidal category
    QCoh(X) of quasicoherent sheaves on X is equivalent to the category
    QCoh(LX) of quasicoherent sheaves on the loop space LX = Map(S^1, X).
    
    
    
    
    


  • Date: Thu, 22 Oct 2009 18:52:00

  • Monday (October 26), 4:30 p.m, room E 206.
    David Nadler (NWU). Drinfeld centers and derived algebraic geometry. I.
    
                         Abstract
    
    In this series of talks John Francis and I would like to describe the
    background and results of our joint paper with David Ben-Zvi
    "Integral transforms and Drinfeld centers in derived algebraic geometry",
    http://arxiv.org/abs/0805.0157
    
    No preliminary knowledge of derived algebraic geometry is assumed.
    
    Our broad goal is to understand some of the basic homotopical algebra of
    the tensor category of quasicoherent sheaves on a stack. More
    specifically, we would like to understand how geometric operations on
    stacks (fiber products, loop spaces,...) interact with algebraic
    operations on categories of sheaves (tensor products, Hochschild
    homology,...).
    
    Our plan roughly breaks into three parts:
    
    1. The first talk will be an example-oriented overview of the phenomena to
    be studied. We will informally introduce the main objects and try to build
    up enough intuition so the audience will have a good sense
    as to what should be true.
    
    2. In the second part, we will explain some of the general constructions
    which we will later apply to categories of sheaves. We will follow Lurie's
    foundations on algebra in the context of oo-categories. Our focus will be
    on the notion of center, and in particular the examples of Drinfeld and 
    E_n-centers of monoidal oo-categories.
    
    3. In the third part, we will specialize to derived algebraic geometry.
    Here is a sample of the kind of theorem we will discuss:
    
    Theorem: For X a perfect stack (a notion which in characteristic zero
    includes all of the commonly occurring examples in geometric
    representation theory), the Drinfeld center of the monoidal category
    QCoh(X) of quasicoherent sheaves on X is equivalent to the category
    QCoh(LX) of quasicoherent sheaves on the loop space LX = Map(S^1, X).
    
    
    
    
    
    
    
    
    
    
    


  • Date: Tue, 27 Oct 2009 10:03:16

  • No seminar on Thursday.
    
    The next talk on "Drinfeld centers and derived algebraic geometry" will be
    given on Monday (November 2) by John Francis (NWU).
    
    
    
    
    


  • Date: Sat, 31 Oct 2009 15:06:30

  • Monday (November 2), 4:30 p.m, room E 206.
    John Francis (NWU). Drinfeld centers and derived algebraic geometry. II.
    
    >                      Abstract
    >
    > In this series of talks John Francis and I would like to describe the
    > background and results of our joint paper with David Ben-Zvi
    > "Integral transforms and Drinfeld centers in derived algebraic geometry",
    > http://arxiv.org/abs/0805.0157
    >
    > No preliminary knowledge of derived algebraic geometry is assumed.
    >
    > Our broad goal is to understand some of the basic homotopical algebra of
    > the tensor category of quasicoherent sheaves on a stack. More
    > specifically, we would like to understand how geometric operations on
    > stacks (fiber products, loop spaces,...) interact with algebraic
    > operations on categories of sheaves (tensor products, Hochschild
    > homology,...).
    >
    > Our plan roughly breaks into three parts:
    >
    > 1. The first talk will be an example-oriented overview of the phenomena to
    > be studied. We will informally introduce the main objects and try to build
    > up enough intuition so the audience will have a good sense
    > as to what should be true.
    >
    > 2. In the second part, we will explain some of the general constructions
    > which we will later apply to categories of sheaves. We will follow Lurie's
    > foundations on algebra in the context of oo-categories. Our focus will be
    > on the notion of center, and in particular the examples of Drinfeld and
    > E_n-centers of monoidal oo-categories.
    >
    > 3. In the third part, we will specialize to derived algebraic geometry.
    > Here is a sample of the kind of theorem we will discuss:
    >
    > Theorem: For X a perfect stack (a notion which in characteristic zero
    > includes all of the commonly occurring examples in geometric
    > representation theory), the Drinfeld center of the monoidal category
    > QCoh(X) of quasicoherent sheaves on X is equivalent to the category
    > QCoh(LX) of quasicoherent sheaves on the loop space LX = Map(S^1, X).
    
    


  • Date: Sun, 1 Nov 2009 18:21:54

  • We do not meet tomorrow (Monday) because the speaker is sick.
    Presumably we meet on Thursday (November 5).
    
    > John Francis (NWU). Drinfeld centers and derived algebraic geometry. II.
    >
    >>                      Abstract
    >>
    >> In this series of talks John Francis and I would like to describe the
    >> background and results of our joint paper with David Ben-Zvi
    >> "Integral transforms and Drinfeld centers in derived algebraic
    >> geometry",
    >> http://arxiv.org/abs/0805.0157
    >>
    >> No preliminary knowledge of derived algebraic geometry is assumed.
    >>
    >> Our broad goal is to understand some of the basic homotopical algebra of
    >> the tensor category of quasicoherent sheaves on a stack. More
    >> specifically, we would like to understand how geometric operations on
    >> stacks (fiber products, loop spaces,...) interact with algebraic
    >> operations on categories of sheaves (tensor products, Hochschild
    >> homology,...).
    >>
    >> Our plan roughly breaks into three parts:
    >>
    >> 1. The first talk will be an example-oriented overview of the phenomena
    >> to
    >> be studied. We will informally introduce the main objects and try to
    >> build
    >> up enough intuition so the audience will have a good sense
    >> as to what should be true.
    >>
    >> 2. In the second part, we will explain some of the general constructions
    >> which we will later apply to categories of sheaves. We will follow
    >> Lurie's
    >> foundations on algebra in the context of oo-categories. Our focus will
    >> be
    >> on the notion of center, and in particular the examples of Drinfeld and
    >> E_n-centers of monoidal oo-categories.
    >>
    >> 3. In the third part, we will specialize to derived algebraic geometry.
    >> Here is a sample of the kind of theorem we will discuss:
    >>
    >> Theorem: For X a perfect stack (a notion which in characteristic zero
    >> includes all of the commonly occurring examples in geometric
    >> representation theory), the Drinfeld center of the monoidal category
    >> QCoh(X) of quasicoherent sheaves on X is equivalent to the category
    >> QCoh(LX) of quasicoherent sheaves on the loop space LX = Map(S^1, X).
    >
    
    


  • Date: Tue, 3 Nov 2009 19:49:36

  • Thursday (November 5), 4:30 p.m, room E 206.
    John Francis (NWU) will give the second talk in the series
    "Drinfeld centers and derived algebraic geometry".
    As far as I understand, it will be independent from the first one
    (which was given last Monday by David Nadler).
    
                         Abstract
    
    This talk will give an exposition of E_n centers, where E_n is the operad
    of configuration spaces of n-disks. The main case of interest will be the
    E_n or Drinfeld (for n=1) center of a monoidal
    oo-category. We will preface this topic with the necessary review of the
    theory of oo-categories (definition, limits and colimits,
    stability, presentability, operads and operadic monoidal structures) and,
    in particular, a description of oo-categories of functors between
    oo-categories.
    
    
    
    
    
    
    


  • Date: Thu, 5 Nov 2009 19:15:46

  • Monday (November 9), 4:30 p.m, room E 206.
    David Ben-Zvi (University of Texas at Austin).
    Drinfeld centers and derived algebraic geometry. III.
    
                         Abstract
    
    In this talk we will discuss the interaction between geometric operations on
    schemes and stacks and algebraic operations on their categories of sheaves
    in the context of derived algebraic geometry. We will introduce derived
    schemes and stacks and their oo-categories of quasicoherent sheaves. The
    latter can be considered as plain, monoidal, E_n or symmetric monoidal
    oo-categories, leading to a collection of natural algebraic operations to
    perform on them. We will demonstrate how the results of these
    operations can be described themselves as oo-categories of quasicoherent
    sheaves. As an application we discuss (time permitting) the representation
    theory of quasicoherent Hecke categories.
    
    The prerequisites are the basic notions of oo-categories, as presented in
    John Francis' talk, or a willingness to work through formal analogies,
    such as those presented in David Nadler's talk.
    
    
    
    
    
    


  • Date: Mon, 9 Nov 2009 19:07:43

  • Thursday (November 12), 4:30 p.m, room E 206.
    Ivan Losev (MIT) Irreducible finite-dimensional
    representations for finite W-algebras.
    
                         Abstract
    (Finite) W-algebras are certain associative
    algebras arising in Lie theory. They are constructed
    from a pair of a semisimple Lie algebra and its
    nilpotent element. They originate from Kostant's
    paper "On Whittaker vectors and representation theory",
    where the case of a principal nilpotent element
    was considered. During the last decade they attracted
    some interest from representation theorists (Brundan,
    Ginzburg, Goodwin, Kleshchev, Premet, the speaker
    and others) mainly due to their relation to representation
    theory of universal enveloping algebras, both
    in zero and positive characteristics. In my talk
    I will describe known results and conjectures on the
    classification of finite dimensional irreducible
    modules for W-algebras.
    
    References: arXiv:0707.3108, 0807.1023, 0812.1584.
    
    
    
    


  • Date: Tue, 10 Nov 2009 15:56:37

  • Thursday (November 12), 4:30 p.m, room E 206.
    Ivan Losev (MIT) Irreducible finite-dimensional
    representations for finite W-algebras.
    
                         Abstract
    (Finite) W-algebras are certain associative
    algebras arising in Lie theory. They are constructed
    from a pair of a semisimple Lie algebra and its
    nilpotent element. They originate from Kostant's
    paper "On Whittaker vectors and representation theory",
    where the case of a principal nilpotent element
    was considered. During the last decade they attracted
    some interest from representation theorists (Brundan,
    Ginzburg, Goodwin, Kleshchev, Premet, the speaker
    and others) mainly due to their relation to representation
    theory of universal enveloping algebras, both
    in zero and positive characteristics. In my talk
    I will describe known results and conjectures on the
    classification of finite dimensional irreducible
    modules for W-algebras.
    
    References: arXiv:0707.3108, 0807.1023, 0812.1584.
    
    
    
    
    
    
    
    


  • Date: Thu, 12 Nov 2009 19:14:23

  • No seminar on Monday (November 16).
    
    Roman Bezrukavnikov will speak on Thursday (November 19).
    
    
    
    
    
    


  • Date: Mon, 16 Nov 2009 16:59:09

  • Thursday (November 19), 4:30 p.m, room E 206.
    
    Roman Bezrukavnikov (MIT).
    Character D-modules as center of the categorical Hecke algebra.
    
                         Abstract
    
    Lusztig introduced character sheaves on a reductive algebraic
    group as a tool for understanding characters of finite Chevalley groups.
    In a joint work with Finkelberg and Ostrik we show how to view the 
    corresponding objects in the D-module setting as objects of the
    categorical center of the categorical Hecke algebra and rederive Lusztig's
    
    classification results through this approach.
    Some of our results are related to (though different from) results of 
    Ben-Zvi, Francis and Nadler.
    
    
    
    


  • Date: Thu, 19 Nov 2009 19:15:00

  • No more Langlands seminar this quarter.
    
    


  • Date: Fri, 1 Jan 2010 19:35:07

  • Happy New Year!
    
    Here is a tentative program for the winter quarter.
    
    First, Sasha Beilinson will give several talks on motive theory.
    The title of his series is "Finite-dimensionality of motives and some
    problems on algebraic cycles  (after  Kimura, O'Sullivan, and others)".
    As far as I know, Sasha will start from scratch and discuss some
    fundamental open problems, whose formulation is very easy to understand.
    He will begin either on January 7 or more likely, on January 11.
    
    Yakov Varshavsky will give several talks  on his generalization of
    Fujiwara's theorem. Presumably he will begin on January 25.
    
    In February Dmitry Kaledin will give a series of talks on his proof of
    noncommutative Hodge-to-de Rham degeneration using a noncommutative
    version of the Deligne-Illusie method. Presumably he will begin on
    February 8.
    
    
    
    
    
    
    
    
    


  • Date: Tue, 5 Jan 2010 12:59:29

  • No seminar on Thursday (January 7).
    
    On January 11 (Monday) Sasha Beilinson will give the first talk in his
    series on motives. His talks will be introductory;
    no initial knowledge about motives is required.
    
    Title: "Finite-dimensionality of motives and some
    problems on algebraic cycles  (after  Kimura, O'Sullivan, and others)".
    
              Abstract
    
    The theory of motives predicts a number of properties of
    the groups of algebraic cycles. One needs them in order to be able
    to use motives as effectively as usual homology groups.
    I will mostly discuss the finite-dimensionality property of
    motives, conjectured by Kimura and O'Sullivan, and relateted
    problems about algebraic cycles.
    
    The talks are introductory; no initial knowledge about
    motives is required. One of my goals is to explain
    HOW LITTLE WE KNOW about algebraic cycles.
    
    
    
    
    


  • Date: Thu, 7 Jan 2010 19:16:51

  • Monday (January 11), 4:30 p.m, room E 206.
    
    Alexander Beilinson.
    Title: "Finite-dimensionality of motives and some
    problems on algebraic cycles  (after  Kimura, O'Sullivan, and others)".
    
              Abstract
    
    The theory of motives predicts a number of properties of
    the groups of algebraic cycles. One needs them in order to be able to use
    motives as effectively as usual homology groups.
    I will mostly discuss the finite-dimensionality property of
    motives, conjectured by Kimura and O'Sullivan, and relateted
    problems about algebraic cycles.
    
    The talks are introductory; no initial knowledge about
    motives is required. One of my goals is to explain
    HOW LITTLE WE KNOW about algebraic cycles.
    
    
    
    
    
    
    
    
    
    
    


  • Date: Mon, 11 Jan 2010 10:01:47

  • This is just to remind that Beilinson begins his series of talks
    today and that this series can be easily understood by beginners.
    
    > Monday (January 11), 4:30 p.m, room E 206.
    >
    > Alexander Beilinson.
    > Title: "Finite-dimensionality of motives and some
    > problems on algebraic cycles  (after  Kimura, O'Sullivan, and others)".
    >
    >           Abstract
    >
    > The theory of motives predicts a number of properties of
    > the groups of algebraic cycles. One needs them in order to be able to use
    > motives as effectively as usual homology groups.
    > I will mostly discuss the finite-dimensionality property of
    > motives, conjectured by Kimura and O'Sullivan, and relateted
    > problems about algebraic cycles.
    >
    > The talks are introductory; no initial knowledge about
    > motives is required. One of my goals is to explain
    > HOW LITTLE WE KNOW about algebraic cycles.
    >
    >
    >
    >
    >
    >
    >
    >
    >
    >
    
    


  • Date: Mon, 11 Jan 2010 18:43:35

  • No seminar on Thursday.
    Beilinson will continue on January 18 (Monday).
    
    
    
    


  • Date: Thu, 14 Jan 2010 19:00:41

  • Monday (January 11), 4:30 p.m, room E 206.
    
    Alexander Beilinson will give his second talk on motives.
    It will be more or less independent of the first one.
    (But of course, understand the first talk is worth trying;
    in particular, this would help you understand the second one.)
    
    > Title: "Finite-dimensionality of motives and some
    > problems on algebraic cycles  (after  Kimura, O'Sullivan, and others)".
    >
    >           Abstract
    >
    > The theory of motives predicts a number of properties of
    > the groups of algebraic cycles. One needs them in order
    > to be able to use
    > motives as effectively as usual homology groups.
    > I will mostly discuss the finite-dimensionality property of
    > motives, conjectured by Kimura and O'Sullivan, and relateted
    > problems about algebraic cycles.
    >
    > The talks are introductory; no initial knowledge about
    > motives is required. One of my goals is to explain
    > HOW LITTLE WE KNOW about algebraic cycles.
    
    
    
    


  • Date: Mon, 18 Jan 2010 18:48:49

  • Thursday (January 21), 4:30 p.m, room E 206.
    
    Alexander Beilinson will give his last talk on motives.
    
    
    
    
    


  • Date: Fri, 22 Jan 2010 10:36:18

  • Beilinson's notes of his second and third talks on motives.are attached.
    
    Next talk:
    Monday (January 25), 4:30 p.m, room E 206.
    
    Yakov Varshavsky (Hebrew University)
    Lefschetz-Verdier trace formula, contracting correspondences, and
    a simple proof of Fujiwara's theorem.
    
                                  Abstract
    
    A theorem of Fujiwara (formerly Deligne's conjecture) asserts that the
    Lefschetz-Verdier trace formula has a particularly simple and explicit
    form in the case when a correspondence is defined over a finite field and
    is twisted by a sufficiently large power of Frobenius.
    
    In my talks I will indicate a simple proof of this theorem based on an
    algebro-geometric notion of a contracting correspondence and a classical
    construction of a deformation to the normal cone.
    
    The talk is based on the following articles (the first one is a research
    announcement, the second one is the full version):
    
    <http://front.math.ucdavis.edu/0505.5314>
    "A proof of a generalization of Deligne's conjecture"
    
    <http://front.math.ucdavis.edu/0505.5564>
    "Lefschetz-Verdier trace formula and a generalization
    of a theorem of Fujiwara"
    
    
    
    
    
    
    

    Attachment: Sasha_on_motives.pdf
    Description: Adobe PDF document



  • Date: Tue, 26 Jan 2010 16:14:42

  • Yakov Varshavsky will continue his talk on
    Thursday (January 28), 4:30 p.m, room E 206.
    
    > Yakov Varshavsky (Hebrew University)
    > Lefschetz-Verdier trace formula, contracting correspondences, and
    > a simple proof of Fujiwara's theorem.
    >
    >                               Abstract
    >
    > A theorem of Fujiwara (formerly Deligne's conjecture) asserts that the
    > Lefschetz-Verdier trace formula has a particularly simple and explicit
    > form in the case when a correspondence is defined over a finite field and
    > is twisted by a sufficiently large power of Frobenius.
    >
    > In my talks I will indicate a simple proof of this theorem based on an
    > algebro-geometric notion of a contracting correspondence and a classical
    > construction of a deformation to the normal cone.
    >
    > The talk is based on the following articles (the first one is a research
    > announcement, the second one is the full version):
    >
    > <http://front.math.ucdavis.edu/0505.5314>
    > "A proof of a generalization of Deligne's conjecture"
    >
    > <http://front.math.ucdavis.edu/0505.5564>
    > "Lefschetz-Verdier trace formula and a generalization
    > of a theorem of Fujiwara"
    >
    >
    >
    >
    >
    >
    >
    
    


  • Date: Thu, 28 Jan 2010 18:41:28

  • No seminar on Monday (February 1).
    
    On Thursday (February 4) Dmitry Kaledin will begin his series of talks.
    (The goal of this series is to explain his proof of
    the degeneration of the Hodge-to-de Rham spectral sequence in
    noncommutative algebraic geometry.)
    
    


  • Date: Mon, 1 Feb 2010 16:59:05

  • Thursday (February 4), 4:30 p.m, room E 206.
    
    Dmitry Kaledin (Moscow) will give the first talk in his series
    on the Hodge-to-de Rham degeneration theorem in
    noncommutative algebraic geometry. In particular, he will explain the
    setting of noncommutative algebraic geometry.
    
       Title of the talk:
    Finiteness conditions for DG algebras.
    
                       Abstract
    
    I am going to discuss some general facts about DG categories and DG
    algebras -- this is what constitutes the "non-commutative setting" of the
    theorem -- and then I am going to concentrate on a beautiful recent
    theorem of Bertrand Toen which claims that a proper and smooth DG algebra
    over a field K actually comes from a proper and smooth DG algebra over a
    finitely generated subring in K.
    
    
    
    
    
    
    


  • Date: Thu, 4 Feb 2010 20:29:33

  • 1. Here is a reference to Toen's article discussed in Kaledin's talk:
    
    MR2441136 (2009g:16017)
    Bertrand Toen, Anneaux de definition des dg-algebres propres et lisses.
    Bull. Lond. Math. Soc. 40 (2008), no. 4, 642--650.
    
    2. Kaledin's next talk:
    
    Monday (February 8), 4:30 p.m, room E 206.
    
    Title: Non-commutative Hodge-to-de Rham degeneration I.
    
                         Abstract
    
    In this lecture, I am going to explain two things:
    (1) the beautiful short proof of Hodge-to-de Rham degeneration
    in the usual commutative setting given by Deligne-Illusie, and all it
    requires (which is basically the Cartier isomorphism in char p).
    (2) non-commutative generalizations of differential forms
    and de Rham cohomology classes, and the statement of the
    Hodge-to-de Rham degeneration theorem in the non-commutative case.
    
    How (1) is related to (2) is the subject of subsequent lectures.
    
    I plan to go very slowly; in particular, I will assume no knowledge
    whatsoever about de Rham cohomology in positive characteristic.
    
    
    
    
    
    


  • Date: Tue, 9 Feb 2010 09:12:28

  • 1. Kaledin's next talk:
    
    Thursday (February 11), 4:30 p.m, room E 206.
    
    
    Title: Hodge-to-de Rham degeneration II
    
    Abstract. In this lecture, I will construct a version of Cartier
    isomorphism for an associative algebra A over a finite field k; A
    has to satisfy some assumptions analogous to the ones we had in the
    commutative case (dim X < p, X liftable to W_2(k)). Then if time
    permits, I will sketch the proof of the degeneration theorem.
    =======================================================================
    
    2. Here are the links mentioned in Kaledin's last talk.
    His Tokyo lectures are here:
    
    http://imperium.lenin.ru/~kaledin/tokyo/
    
    There are also his lectures from last year from Seoul, on a related
    subject (in particular, in lecture 1 there is sketch of finiteness results
    for DG algebras which he explained last time):
    
    http://imperium.lenin.ru/~kaledin/seoul/
    
    As for the degeneration theorem itself, the main paper is
    arXiv:math/0611623, and there is an overview in arXiv:0708.1574.