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

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.

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.

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.

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. > > > > > > > >

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.

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.

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

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

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.

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).

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).

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.

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.

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**

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.

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.

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.

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.

No more meetings of the Geometric Langlands seminar this quarter.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 $W$.

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.

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.

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.

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.

No more meetings of the Geometric Langlands seminar this quarter.

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!)

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

No more meetings of the Langlands seminar this quarter.

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.

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.

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.

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.

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.

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

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.

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.

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

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.

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.

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**

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.

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.

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).

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**

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

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".]

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.

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.

No more meetings of the seminar this quarter.

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.

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.

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.

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.

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.

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

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.

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

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

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.

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.

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.

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.

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

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.

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

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.

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.

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

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".

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.

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).

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.