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Algebraic Geometry and Representation Theory Seminar

TuesdayNov 28, 201711:15
Algebraic Geometry and Representation Theory SeminarRoom 155
Speaker:Yuanqing Cai Title:Weyl group multiple Dirichlet seriesAbstract:opens in new windowin html    pdfopens in new window

Weyl group multiple Dirichlet series are Dirichlet series in r complex variables which initially converge on a cone in C^r, possess analytic continuation to a meromorphic function on the whole complex space, and satisfy functional equations whose action on C^r is isomorphic to the Weyl group of a reduced root system. I will review different constructions of such series and discuss the relations between them.

WednesdayNov 22, 201711:15
Algebraic Geometry and Representation Theory SeminarRoom 155
Speaker:Alexander Elashvili Title:About Index of Lie AlgebrasAbstract:opens in new windowin html    pdfopens in new window
In my talk I plan to give overview of results about of index of biparaboic subalgebras of classical Lie algebras and formulate conjecture about asymptotic biheviar of lieandric numbers.
TuesdayNov 21, 201711:15
Algebraic Geometry and Representation Theory SeminarRoom 155
Speaker:Raf Cluckers Title:Uniform p-adic integration and applications Abstract:opens in new windowin html    pdfopens in new window

As a concrete variant of motivic integration, I will discuss uniform p-adic integration and constructive aspects of results involved. Uniformity is in the p-adic fields, and, for large primes p, in the fields F_p((t)) and all their finite field extensions. Using real-valued Haar measures on such fields, one can study integrals, Fourier transforms, etc. We follow a line of research that Jan Denef started in the eighties, with in particular the use of model theory to study various questions related to p-adic integration. A form of uniform p-adic quantifier elimination is used. Using the notion of definable functions, one builds constructively a class of complex-valued functions which one can integrate (w.r.t. any of the variables) without leaving the class. One can also take Fourier transforms in the class. Recent applications in the Langlands program are based on Transfer Principles for uniform p-adic integrals, which allow one to get results for F_p((t)) from results for Q_p, once p is large, and vice versa. These Transfer Principles are obtained via the study of general kinds of loci, some of them being zero loci. More recently, these loci are playing a role in the uniform study of p-adic wave front sets for (uniformly definable) p-adic distributions, a tool often used in real analysis.
This talk contains various joint works with Gordon, Hales, Halupczok, Loeser, and Raibaut, and may mention some work in progress with Aizenbud about WF-holonomicity of these distributions, in relation to a question raized by Aizenbud and Drinfeld.

WednesdayNov 15, 201714:15
Algebraic Geometry and Representation Theory SeminarRoom 155
Speaker:Liran ShaulTitle:Injective modules in higher algebraAbstract:opens in new windowin html    pdfopens in new windowNOTE UNUSUAL DAY AND TIME
The notion of an Injective module is one of the most fundamental notions in homological algebra over rings. In this talk, we explain how to generalize this notion to higher algebra. The Bass-Papp theorem states that a ring is left noetherian if and only if an arbitrary direct sum of left injective modules is injective. We will explain a version of this result in higher algebra, which will lead us to the notion of a left noetherian derived ring. In the final part of the talk, we will specialize to commutative noetherian rings in higher algebra, show that the Matlis structure theorem of injective modules generalize to this setting, and explain how to deduce from it a version of Grothendieck's local duality theorem over commutative noetherian local DG rings.
TuesdayOct 31, 201711:15
Algebraic Geometry and Representation Theory SeminarRoom 155
Speaker:Walter Gubler Title:The non-Archimedean Monge-Ampere problemAbstract:opens in new windowin html    pdfopens in new window

Calabi conjectured that the complex Monge-Ampere equation on compact Kaehler manifolds has a unique solution.
This was solved by Yau in 1978. In this talk, we present a non-archimedean version on projective Berkovich spaces.
In joint work with Burgos, Jell, Kunnemann and Martin, we improve a result of  Boucksom, Favre and Jonsson in the equicharacteristic 0 case. We give also a result in positive equicharacteristic using test ideals.

WednesdaySep 06, 201711:15
Algebraic Geometry and Representation Theory SeminarRoom 1
Speaker:Thorsten HeidersdorfTitle:Reductive groups attached to representations of the general linear supergroup GL(m|n)Abstract:opens in new windowin html    pdfopens in new window

Let Rep(GL(m|n)) denote the category of finite-dimensional algebraic representations of the supergroup Gl(m|n). Nowadays the abelian structure (Ext^1 between irreducibles, block description,...) is well understood. Kazhdan-Lusztig theory gives an algorithmic solution for the character problem, and in special cases even explicit character formulas. However we understand the monoidal structure hardly at all (e.g. the decomposition of tensor products into the indecomposable constituents). I will talk about the problem of decomposing tensor products "up to superdimension 0", i.e. about the structure of Rep(GL(m|n))/N where N is the ideal of indecomposable representations of superdimension 0.

TuesdayJul 11, 201711:15
Algebraic Geometry and Representation Theory SeminarRoom 108
Speaker:Jasmine Fittouhi Title:The uncovering of fibers’ Mumford systemAbstract:opens in new windowin html    pdfopens in new window
This talk is dedicate to the description of the fibers resulting from the Mumford system of degree g>0. Each fiber is linked to a hyperelliptic curve; we will focus our description more specifically to the ones linked to singular hyperelliptic curves. These fibers are arranged hierarchically by stratification which allows us to provide a geometrical as well as an algebraic understanding of fibers that result in an isomorphism between a fiber and a part of a commutative algebraic group associated to its singular hyperelliptic curves in other words the generalized Jacobian.
TuesdayJun 20, 201711:15
Algebraic Geometry and Representation Theory SeminarRoom 141
Speaker:Luc IllusieTitle:Revisiting vanishing cycles and duality in étale cohomologyAbstract:opens in new windowin html    pdfopens in new windowNOTE UNUSUAL ROOM
Abstract: In the early 1980's Gabber proved compatibility of nearby cycles with duality and Beilinson compatibility of vanishing cycles with duality. I will explain new insights and results on this topic, due to Beilinson, Gabber, and Zheng.
TuesdayJun 06, 201711:15
Algebraic Geometry and Representation Theory SeminarRoom 108
Speaker:Klaus KunnemannTitle:Positivity properties of metrics in non-archimedean geometryAbstract:opens in new windowin html    pdfopens in new window
We describe the Calabi-Yau problem on complex manifolds and its analog in non-archimedean geometry. We discuss positivity properties of metrics on line bundles over non-archimedean analytic spaces and applications to the solution of the non-archimedean Calabi-Yau problem in the equicharacteristic zero case.
TuesdayMay 30, 201711:15
Algebraic Geometry and Representation Theory Seminar
Speaker:Siddhartha Sahi Title:Multivariate Hypergeometric functions with a parameterAbstract:opens in new windowin html    pdfopens in new windowDe Picciotto Building, Room 25

The theory of hypergeometric functions with matrix argument was developed in the 1950s by S. Bochener for Hermitian matrices, and by C. Herz for symmetric matrices. This theory admits a common generalization to the setting of symmetric cones, which is discussed in the book by Faraut-Koranyi. It also has applications to the study of non-central distributions in statistics and to the theory of random matrices.

In the 1980s, I.G. Macdonald introduced a one parameter family of multivariate hypergeometric functions, which, for special values of the parameter, are the *radial* parts of the matrix hypergeometric functions. He also formulated a number of natural conjectures about these functions, which in the matrix case can be proved by appropriate integral formulas. However this technique is unavailable in the general setting and as a result these conjectures have remained open.


In recent work with G. Olafsson we have solved most of these conjectures, using ideas from the theory of Cherednik algebras and Jack polynomials. Among other results we obtain sharp estimates for the exponential kernel that allow us to establish a rigorous theory of the Fourier and Laplace transforms, and we prove an explicit formula for the Laplace transform of a Jack polynomial conjectured by Macdonald. This opens the door for several future developments in the associated harmonic analysis, some of which we also treat. This includes (1) the Paley-Wiener theorem, (2) Laplace transform identities for hypergeometric functions, and (3) the "so-called" Ramanujan master theorem.

 

Note the unusual room [De Picciotto Building, Room 25]

TuesdayMay 23, 201711:15
Algebraic Geometry and Representation Theory SeminarRoom 141
Speaker:Yakov Varshavsky Title:On the depth r Bernstein projector.Abstract:opens in new windowin html    pdfopens in new windowPLEASE NOTE THE UNUSUAL ROOM

This is a joint work with Bezrukavnikov and Kazhdan. The goal of my talk is to give an explicit formula for the Bernstein projector to representations of depth $\leq r$. As a consequence, we show that the depth zero Bernstein projector is supported on topologically unipotent elements and it is equal to the restriction of the character of the Steinberg representation. As another application, we deduce that the depth $r$ Bernstein projector is stable. Moreover, for integral depths our proof is purely local.

TuesdayApr 25, 201711:15
Algebraic Geometry and Representation Theory SeminarRoom 108
Speaker:Crystal Hoyt Title:A new category of sl(infinity)-modules related to Lie superalgebras Abstract:opens in new windowin html    pdfopens in new window
The (reduced) Grothendieck group of the category of finite-dimensional representations of the Lie superalgebra gl(m|n) is an sl(infinity)-module with the action defined via translation functors, as shown by Brundan and Stroppel. This module is indecomposable and integrable, but does not lie in the tensor category, in other words, it is not a subquotient of the tensor algebra generated by finitely many copies of the natural and conatural sl(infinity)-modules. In this talk, we will introduce a new category of sl(infinity)-modules in which this module is injective, and describe the socle filtration of this module. Joint with: I. Penkov, V. Serganova
TuesdayApr 18, 201711:15
Algebraic Geometry and Representation Theory SeminarRoom 108
Speaker:Mikhail IgnatyevTitle:Coadjoint orbits, Kostant–Kumar polynomials and tangent cones to Schubert varietiesAbstract:opens in new windowin html    pdfopens in new window
TBA
TuesdayFeb 21, 201711:15
Algebraic Geometry and Representation Theory SeminarRoom 155
Speaker:Stephen LichtenbaumTitle:A conjectured cohomological description of special values of zeta-functions.Abstract:opens in new windowin html    pdfopens in new window

Let X be a regular scheme, projective and flat over Spec Z. We give a conjectural formula in terms of motivic cohomology, singular cohomology and de Rham cohomology  for the special value of the zeta-function of X at any rational integer. We will explain how this reduces to the standard formula for the residue of the Dedekind zeta-function at s = 1. 

FridayFeb 03, 201710:30
Algebraic Geometry and Representation Theory SeminarRoom 155
Speaker:Vadim Schechtman Title:Fourier transformation and hyperplane arrangementsAbstract:opens in new windowin html    pdfopens in new windowPLEASE NOTE UNUSUAL DAY
Linear algebra problems related to the Fourier transformation of perverse sheaves smooth along a hyperplane arrangement in an affine space, together with some examples coming from the representation theory will be discussed.
TuesdayJan 31, 201711:15
Algebraic Geometry and Representation Theory SeminarRoom 155
Speaker:Boris Tsygan Title:What do algebras form? (Revisited)Abstract:opens in new windowin html    pdfopens in new window
We will start with the observation that assocciative algebras form a two-category with a trace functor where one-morphisms are bimodules, two-morphisms are bimodule homomorphisms, and the trace of an (A,A) bimodule M is M/[M,A]. We then explain in what sense the derived version of the above is true, I.e. what happens when one replaces bimodule homomorrphisms and the trace by their derived functors that are Hochschild (com)homology. We will explain how the beginnings of noncommutative differential calculus can bee deduced from the above. This is a continuation of a series of works of MacClure and Smith, Tamarkin, Lurie, and others, and a joint work with Rebecca Wei.
TuesdayJan 24, 201711:15
Algebraic Geometry and Representation Theory SeminarRoom 155
Speaker:Shamgar Gurevich Title:“Size" of a representation of a finite group controls the size of its character valuesAbstract:opens in new windowin html    pdfopens in new window

Many problems about finite groups (e.g., convergence of random walks, properties of word maps, spectrum of Cayley graphs, etc.) can be approached in terms of sums of group characters. More precisely, what intervenes in such sums are the character ratios: 
X_r(g) / dim(r),       g in G, 
where r is an irreducible representation of G, and X_r is its character. This leads to the quest for good estimates on the character ratios.
In this talk I will introduce a precise notion of "size" for representations of finite classical groups and show that it tends to put together those with character ratios of the same order of magnitude.
As an application I will show how one might generalize to classical groups the following result of Diaconis-Shahshahani (for k=2) and Berestycki -Schramm -Zeitouni (for general k): The mixing time for the random walk on the group G=S_n using the cycles of length k is (1/k) n log(n).
The talk should be accessible for beginning graduate students, and is part from our joint project with Roger Howe (Yale and Texas A&M).

TuesdayJan 10, 201711:15
Algebraic Geometry and Representation Theory SeminarRoom 155
Speaker:Jianrong LiTitle:Finite-dimensional representations of quantum affine algebrasAbstract:opens in new windowin html    pdfopens in new window
I will talk about finite dimensional representations of quantum affine algebras. The main topics are Chari and Pressley's classification of finite-dimensional simple modules over quantum affine algebras, Frenkel and Reshetikhin's theory of q-characters of finite dimensional modules, Frenkel-Mukhin algorithm to compute q-characters, T-systems, Hernandez-Leclerc's conjecture about the cluster algebra structure on the ring of a subcategory of the category of all finite dimensional representations of a quantum affine algebra. I will also talk about how to obtain a class of simple modules called minimal affinizations of types A, B using mutations (joint work with Bing Duan, Yanfeng Luo, Qianqian Zhang).
TuesdayJan 03, 201711:15
Algebraic Geometry and Representation Theory SeminarRoom 155
Speaker:Elena Gal Title:A geometric approach to Hall algebrasAbstract:opens in new windowin html    pdfopens in new windowNOTE CHANGE IN DATE TO JAN.03 2017, room 155
The Hall algebra associated to a category can be constructed using the Waldhausen S-construction. We will give a systematic recipe for this and show how one can use it to construct higher associativity data. We will discuss a natural extension of this construction providing a bi-algebraic structure for Hall algebra. As a result we obtain a more transparent proof of Green's theorem about the bi-algebra structure on the Hall algebra.
TuesdayDec 27, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 155
Speaker:Vera SerganovaTitle:P (n) via categorification of Temperley- Lieb algebra and Sp(infinity)Abstract:opens in new windowin html    pdfopens in new window
TuesdayDec 20, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 155
Speaker:Leonid Makar-LimanovTitle:On a bizarre geometric property of a counterexample to the Jacobian conjectureAbstract:opens in new windowin html    pdfopens in new window

If f, g are two polynomials in C[x,y] such that J(f,g)=1, but C[f,g] does not coincide with C[x,y], then the mapping  given by these polynomials ( (x,y) maps to (f(x,y), g(x,y)) ) has a rather unexpected property which will be discussed in the talk.  

TuesdayDec 13, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 155
Speaker:Arkady Berenstein Title:Canonical bases in quantum Schubert cellsAbstract:opens in new windowin html    pdfopens in new window
The goal of my talk (based on a recent joint paper with Jacob Greenstein) is to provide an elementary construction of the canonical basis B(w) in each quantum Schubert cell U_q(w) and to establish its invariance under Lusztig's symmetries. In particular, I will explain how to directly construct the upper global basis B^up, will show that B(w) is contained in B^up, and that a large part of the latter is preserved by the (modified) Lusztig's symmetries.
TuesdayDec 06, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 155
Speaker:Crystal Hoyt Title:The Duflo-Serganova functor and character rings of Lie superalgebrasAbstract:opens in new windowin html    pdfopens in new window
TuesdayNov 29, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 155
Speaker:Dmitry GourevitchTitle:Whittaker supports of representations of reductive groupsAbstract:opens in new windowin html    pdfopens in new window
TuesdayNov 22, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 155
Speaker:Michael Chmutov Title:An affine version of Robinson-Schensted Correspondence for Kazhdan-Lusztig theoryAbstract:opens in new windowin html    pdfopens in new window
In his study of Kazhdan-Lusztig cells in affine type A, Shi has introduced an affine analog of Robinson-Schensted Correspondence. We generalize the Matrix-Ball Construction of Viennot and Fulton to give a more combinatorial realization of Shi's algorithm. As a byproduct, we also give a way to realize the affine correspondence via the usual Robinson-Schensted bumping algorithm. Next, inspired by Honeywill, we extend the algorithm to a bijection between the extended affine symmetric group and collection of triples (P, Q, r) where P and Q are tabloids and r is a dominant weight.
TuesdayNov 01, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 155
Speaker:Anthony JosephTitle:S-graphs, trails and identities in Demazure modulesAbstract:opens in new windowin html    pdfopens in new window
WednesdaySep 21, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 1
Speaker:Jian-Rong Li Title:Introduction to cluster algebras (continuation)Abstract:opens in new windowin html    pdfopens in new windowcorrect date 21/09/2016
Cluster algebras are a class of commutative rings introduced by Fomin and Zelevinsky in 2000. I will give an introductory talk about cluster algebras. The main examples are the cluster algebra of type A2, the coordinate ring of $SL_4/N$, and the homogeneous coordinate ring of the Grassmannian $Gr_{2,n+3}(\mathbb{C})$.
WednesdaySep 14, 201611:00
Algebraic Geometry and Representation Theory SeminarRoom 1
Speaker:Jian-Rong Li Title:Introduction to cluster algebrasAbstract:opens in new windowin html    pdfopens in new window

Cluster algebras are a class of commutative rings introduced by Fomin and Zelevinsky in 2000. I will give an introductory talk about cluster algebras. The main examples are the cluster algebra of type A2, the coordinate ring of $SL_4/N$, and the homogeneous coordinate ring of the Grassmannian $Gr_{2,n+3}(\mathbb{C})$.  

WednesdayAug 03, 201610:30
Algebraic Geometry and Representation Theory SeminarRoom 1
Speaker:Siddhartha SahiTitle:The Capelli problem for gl(m|n) and the spectrum of invariant differential operatorsAbstract:opens in new windowin html    pdfopens in new window
The "generalized" Capelli operators form a linear basis for the ring of invariant differential operators on symmetric cones, such as GL/O and GL/Sp. The Harish-Chandra images of these operators are specializations of certain polynomials defined by speaker and studied together with F. Knop. These "Knop-Sahi" polynomials are inhomogeneous polynomials characterized by simple vanishing conditions; moreover their top homogeneous components are Jack polynomials, which in turn are common generalizations of spherical polynomials on symmetric cones. In the talk I will describe joint work with Hadi Salmasian that extends these results to the setting of the symmetric super-cones GL/OSp and (GLxGL)/GL.
WednesdayJun 29, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 1
Speaker:Michal ZydorTitle:The singular transfer for the Jacquet-Rallis trace formulaAbstract:opens in new windowin html    pdfopens in new window

The Jacquet-Rallis relative trace formula was introduced as a tool towards solving the global conjectures of Gan-Gross-Prasad for unitary groups. I will present some recent progress in developing the full formula.
I will show how to extend the transfer of regular orbital integrals to singular geometric terms using a mix of local and global methods.
(Joint with Pierre-Henri Chaudouard)

WednesdayJun 22, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 1
Speaker:Vasily Dolgushev Title:The intricate Maze of Graph ComplexesAbstract:opens in new windowin html    pdfopens in new window
ThursdayJun 16, 201614:00
Algebraic Geometry and Representation Theory SeminarRoom 1
Speaker:Itay GlazerTitle:Representations of reductive groups distinguished by symmetric subgroupsAbstract:opens in new windowin html    pdfopens in new window

We will discuss representation theory of a symmetric pair (G,H), where G is a complex reductive group, and H is a real form of G. The main objects of study are the G-representations with a non trivial H-invariant functional, called the H-distinguished representations of G.


I will give a necessary condition for a G-representation to be H-distinguished and show that the multiplicity of such representations is less or equal to the number of double cosets B\G/H, where B is a Borel subgroup of G.

WednesdayJun 15, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 1
Speaker:Anthony JosephTitle:A minimax theorem for trailsAbstract:opens in new windowin html    pdfopens in new window
WednesdayJun 08, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 1
Speaker:Yotam Hendel Title:Supersingular representations and the mod p LanglandsAbstract:opens in new windowin html    pdfopens in new window

Let F/Q_p be a finite extension, supersingular representations are the irreducible mod p representations of GL_n(F) which do not appear as a subquotient of a principal series representation, and similarly to the complex case, they are the building blocks of the representation theory of GL_n(F). Historically, they were first discovered by L. Barthel and R. Livne some twenty years ago and they are still not understood even for n=2.

For F=Q_p, the supersingular representations of GL_2(F) have been classified by C. Breuil, and a local mod p Langlands correspondence was established between them and certain mod p Galois representations.

When one tries to generalize this connection and move to a non-trivial extension of Q_p, Breuil's method fails; The supersingular representations in that case have complicated structure and instead of two as in the case F=Q_p we get infinitely many such representations, when there are essentially only finitely many on the Galois side.

In this talk we give an exposition of the subject and explore, using what survives from Breuil's methods, the universal modules whose quotients contain all the supersingular representations in the difficult case where F is a non-trivial extension of Q_p.

ThursdayMay 26, 201614:00
Algebraic Geometry and Representation Theory SeminarRoom 1
Speaker:Ivan Penkov Title:Ordered tensor categories of representations of Mackey Lie algebrasAbstract:opens in new windowin html    pdfopens in new window
WednesdayMay 25, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 155
Speaker:Ivan PenkovTitle:Primitive ideals in U(sl(infinity))Abstract:opens in new windowin html    pdfopens in new window
WednesdayMay 18, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 1
Speaker:Dimitar Granthcharov Title:Singular Gelfand-Tsetlin modulesAbstract:opens in new windowin html    pdfopens in new window
ThursdayMay 05, 201614:00
Algebraic Geometry and Representation Theory SeminarRoom 1
Speaker:Vera Serganova Title: New tensor categories related to orthogonal and symplectic groups and the strange supergroup P(infinity)Abstract:opens in new windowin html    pdfopens in new window
We study a symmetric monoidal category of tensor representations of the ind group O(infinity). This category is Koszul and its Koszul dual is the category of tensor representations of the strange supergroup P(infinity). This can be used to compute Ext groups between simple objects in both categories. The above categories are missing the duality functor. It is possible to extend these categories to certain rigid tensor categories satisfying a nice universality property. In the case of O(infinity) such extension depends on a parameter t and is closely related to the Deligne’s category Rep O(t). When t is integer, this new category is a highest weight category and the action of translation functors in this category is related to the representation of gl(infinity) in the Fock space.
WednesdayMay 04, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 1
Speaker:Andrey Minchenko Title:Differential algebraic groups and their applicationsAbstract:opens in new windowin html    pdfopens in new window
At the most basic level, differential algebraic geometry studies solution spaces of systems of differential polynomial equations. If a matrix group is defined by a set of such equations, one arrives at the notion of a linear differential algebraic group, introduced by P. Cassidy. These groups naturally appear as Galois groups of linear differential equations with parameters. Studying linear differential algebraic groups and their representations is important for applications to finding dependencies among solutions of differential and difference equations (e.g. transcendence properties of special functions). This study makes extensive use of the representation theory of Lie algebras. Remarkably, via their Lie algebras, differential algebraic groups are related to Lie conformal algebras, defined by V. Kac. We will discuss these and other aspects of differential algebraic groups, as well as related open problems.
WednesdayApr 20, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 1
Speaker:Prof. Florence Fauquant-Millet Title:Adapted pairs for maximal parabolic subalgebras and polynomiality of invariantsAbstract:opens in new windowin html    pdfopens in new window
In this talk we will see how adapted pairs - introduced by A. Joseph about ten years ago, the analogue of principal s-triples for non reductive Lie algebras - may be used to prove the polynomiality of some algebras of invariants associated to a maximal parabolic subalgebra.
WednesdayApr 13, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 1
Speaker:Mark Shusterman Title:An elementary proof of Olshanskii's theorem on subgroups of a free group and its applicationsAbstract:opens in new windowin html    pdfopens in new windowplease note change in room

I will present an elementary proof of the following theorem of Alexander Olshanskii:

Let F be a free group and let A,B be finitely generated subgroups of infinite index in F. Then there exists an infinite index subgroup C of F which contains both A and a finite index subgroup of B.

The proof is carried out by introducing a 'profinite' measure on the discrete group F, and is valid also for some groups which are not free.Some applications of this result will be discussed:


1. Group Theory - Construction of locally finite faithful actions of countable groups.

2. Number Theory - Discontinuity of intersections for large algebraic extensions of local fields.

3. Ergodic Theory - Establishing cost 1 for groups boundedly generated by subgroups of infinite index and finite cost.

WednesdayApr 06, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 108
Speaker:Dmitry GourevitchTitle:Recent applications of classical theorems on D-modulesAbstract:opens in new windowin html    pdfopens in new window
WednesdayMar 30, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Victor Abrashkin Title:p-extensions of local fields with Galois groups of nilpotent class <pAbstract:opens in new windowin html    pdfopens in new windowmoved into room 155

Let K be a complete discrete valuation field with finite residue field of characteristic p>0. Let G  be the absolute Galois group of K and for a natural M, let  G(M) be the maximal quotient of G of nilpotent class <p and period p^M. Then G(M) can be identified  with a group obtained from a Lie Z/p^M-algebra L via (truncated) Campbell-Hausdorff composition law. Under this identification the ramification subgroups in upper numbering G(M)^(v)correspond to ideals L^(v) of L. It will be explained an  explicit construction of L and the ideals L^(v). The case of fields K of characteristic p was obtained by the author in 1990's (recently refined), the case of fields K of mixed characteristic requires the assumption that K contains a primitive p^M-th root of unity (for the case M=1 cf. Number Theory Archive).

WednesdayMar 23, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Anthony JosephTitle:Two remarkable properties of the canonical S-graphs and the Kashiwara crystal Abstract:opens in new windowin html    pdfopens in new window
WednesdayMar 02, 201611:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Laura Peskin Title:Mod-p representations of p-adic metaplectic groupsAbstract:opens in new windowin html    pdfopens in new window
I will discuss a classification of the mod-p representations (i.e., of representations with coefficients in an algebraic closure of F_p) of the metaplectic double cover of a p-adic symplectic group. I'll review techniques from the mod-p representation theory of p-adic reductive groups, and explain how to modify them in order to classify representations of covering groups. This is joint work with Karol Koziol.
WednesdayJan 27, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Max GurevichTitle:Integrability of p-adic matrix coefficientsAbstract:opens in new windowin html    pdfopens in new window
Many works in relative p-adic harmonic analysis aim to describe which representations of a reductive group G can be embedded inside the space of smooth functions on a homogeneous space G/H. A related question is whether such an embedding can be realized in a canonical form such as an H-integral over a matrix coefficient. In a joint work with Omer Offen we treated the symmetric case, i.e., when H is the fixed point group of an involution. As part of the answer we provide a precise criterion for such integrability, which reduces in the group case to Casselman’s known square-integrability criterion.
WednesdayJan 20, 201611:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Amnon Yekutieli Title:Derived Categories of BimodulesAbstract:opens in new windowin html    pdfopens in new window

Homological algebra plays a major role in noncommutative ring theory. One important homological construct related to a noncommutative ring A is the dualizing complex, which is a special kind of complex of A-bimodules. When A is a ring containing a central field K, this concept is well-understood now. However, little is known about dualizing complexes when the ring A does not contain a central field (I shall refer to this as the noncommutative arithmetic setting). The main technical issue is finding the correct derived category of A-bimodules.
In this talk I will propose a promising definition of the derived category of A-bimodules in the noncommutative arithmetic setting. Here A is a (possibly) noncommutative ring, central over a commutative base ring K (e.g. K = Z). The idea is to resolve A: we choose a DG (differential graded) ring A', central and flat over K, with a DG ring quasi-isomorphism A' -> A. Such resolutions exist. The enveloping DG ring A'^{en} is the tensor product over K of A' and its opposite. Our candidate for the "derived category of A-bimodules" is the category D(A'^{en}), the derived category of DG A'^{en}-modules. A recent theorem says that the category D(A'^{en}) is independent of the resolution A', up to a canonical equivalence. This justifies our definition.
Working within D(A'^{en}), it is not hard to define dualizing complexes over A, and to prove all their expected properties (like when K is a field). We can also talk about rigid dualizing complexes in the noncommutative arithmetic setting.
What is noticeably missing is a result about existence of rigid dualizing complexes. When the K is a field, Van den Bergh had discovered a powerful existence result for rigid dualizing complexes. We are now trying to extend Van den Bergh's method to the noncommutative arithmetic setting. This is work in progress, joint with Rishi Vyas.
In this talk I will explain, in broad strokes, what are DG rings, DG modules, and the associated derived categories and derived functors. Also, I will try to go into the details of a few results and examples, to give the flavor of this material.

WednesdayJan 13, 201611:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Pavel EtingofTitle:Symmetric tensor categories in characteristic pAbstract:opens in new windowin html    pdfopens in new window
WednesdayJan 06, 201611:15
Algebraic Geometry and Representation Theory SeminarRoom 208
Speaker:Shamgar GurevitchTitle:Low Dimensional Representations of Finite Classical GroupsAbstract:opens in new windowin html    pdfopens in new windowPLEASE NOTE UNUSUAL ROOM

Finite group theorists have established many formulas that express interesting properties of a finite group in terms of sums of characters of the group. An obstacle to applying these formulas is lack of control over the dimensions of representations of the group. In particular, the representations of small dimension tend to contribute the largest terms to these sums, so a systematic knowledge of these small representations could lead to proofs of some of these facts. This talk will discuss a new method for systematically constructing the small representations of finite classical groups. I will explain the method with concrete examples and applications. 


This is part from a joint project with Roger Howe (Yale).

MondayJan 04, 201611:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Mattias Jonsson Title:Degenerations of Calabi-Yau manifolds and non-Archimedean analytic spacesAbstract:opens in new windowin html    pdfopens in new window
Various considerations, from mirror symmetry and elsewhere, have lead people to consider 1-parameter degenerating families of Calabi-Yau manifolds, parameterized by the punctured unit disc. A conjecture by Kontsevich-Soibelman and Gross-Wilson describe what the limiting metric space should be, under suitable hypotheses. I will present joint work with Sebastien Boucksom, in which we show a measure theoretic version of this conjecture. The precise result involves a partial compactification of the family, obtained by adding a non-Archimedean analytic space, in the sense of Berkovich, as the central fiber.
WednesdayDec 30, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Mikhail Borovoi Title:Real Galois cohomology of semisimple groupsAbstract:opens in new windowin html    pdfopens in new window
In a 2-page note of 1969, Victor Kac described automorphisms of finite order of simple Lie algebras over the field of complex numbers C. He used certain diagrams that were later called Kac diagrams. In this talk, based on a joint work with Dmitry Timashev, I will explain the method of Kac diagrams for calculating the Galois cohomology set H^1(R,G) for a connected semisimple algebraic group G over the field of real numbers R. I will use real forms of groups of type E_7 as examples. No prior knowledge of Galois cohomology, Kac diagrams, or groups of type E_7 will be assumed.
WednesdayDec 23, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Lenny Makar-Limanov Title:On rings stable under derivationsAbstract:opens in new windowin html    pdfopens in new window
Let z be an algebraic function of n variables and A(z) the algebra generated by all variables and all partial derivatives of z (of all orders). If z is a polynomial then A(z) is just a polynomial algebra, but when z is not a polynomial then it is not clear what is the structure of this algebra. I'll report on known cases and formulate a conjecture.
WednesdayDec 16, 201512:30
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Arkady Berenstein Title:Generalized RSKAbstract:opens in new windowin html    pdfopens in new window

The goal of my talk (based on joint work with Dima Grigoriev, Anatol Kirillov, and Gleb Koshevoy) is to generalize the celebrated Robinson-Schensted-Knuth (RSK) bijection between the set of matrices with nonnegative integer entries, and the set of the planar partitions.

Namely, for any pair of injective valuations on an integral domain we construct a canonical bijection K, which we call the generalized RSK, between the images of the valuations, i.e., between certain ordered abelian monoids.

Given a semisimple or Kac-Moody group, for each reduced word ii=(i_1,...,i_m) for a Weyl group element we produce a pair of injective valuations on C[x_1,...,x_m] and argue that the corresponding bijection K=K_ii, which maps the lattice points of the positive octant onto the lattice points of a convex polyhedral cone in R^m, is the most natural generalization of the classical RSK and, moreover, K_ii can be viewed as a bijection between Lusztig and Kashiwara parametrizations of the dual canonical basis in the corresponding quantum Schubert cell.

Generalized RSKs are abundant in "nature", for instance, any pair of polynomial maps phi,psi:C^m-->C^m with dense images determines a pair of  injective valuations on C[x_1,...,x_n] and thus defines a generalized RSK bijection K_{phi,psi} between two sub-monoids of Z_+^m.

When phi and psi are birational isomorphisms, we expect that K_{phi,psi} has a geometric "mirror image", i.e., that there is a rational function f on C^m whose poles complement the image of phi and psi so that the tropicalization of the composition psi^{-1}phi along f equals to K_{phi,psi}. We refer to such a geometric data as a (generalized) geometric RSK, and view f as a "super-potential". This fully applies to each ii-RSK situation, and we find a super-potential f=f_ii which helps to compute K_ii.

While each K_ii has a "crystal" flavor, its geometric (and mirror) counterpart f_ii emerges from the cluster twist of the relevant double Bruhat cell studied by Andrei Zelevinsky, David Kazhdan, and myself.

WednesdayDec 09, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Konstantin Ardakov Title:Non-commutative Iwasawa algebrasAbstract:opens in new windowin html    pdfopens in new window
Non-commutative Iwasawa algebras are completed group rings of compact p-adic Lie groups with mod-p, or p-adic integer, coefficients. They can also be viewed as rings of continuous p-adic distributions on the group in question. These algebras have found applications in several areas of number theory, including non-commutative Iwasawa theory and the p-adic local Langlands correspondence, but they also provide interesting examples of non-commutative Noetherian rings which are similar in certain respects to universal enveloping algebras of finite dimensional Lie algebras. After giving the basic definitions and some examples, I will advertise some open questions on the algebraic structure of these Iwasawa algebras.
WednesdayDec 02, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Oded Yacobi Title:Truncated shifted Yangians and Nakajima monomial crystalsAbstract:opens in new windowin html    pdfopens in new window
In geometric representation theory slices to Schubert varieties in the affine Grassmannian are affine varieties which arise naturally via the Satake correspondence. This talk centers on algebras called truncated shifted Yangians, which are quantizations of these slices. In particular we will describe the highest weight theory of these algebras using Nakajima's monomial crystal. This leads to conjectures about categorical ' -action (Langlands dual Lie algebra) on representation categories of truncated shifted Yangians.
WednesdayNov 25, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Andrey MinchenkoTitle:Simple Lie conformal algebrasAbstract:opens in new windowin html    pdfopens in new window

The notion of a Lie conformal algebra (LCA) comes from physics, and is related to the operator product expansion. An LCA is a module over a ring of differential operators with constant coefficients, and with a bracket which may be seen as a deformation of a Lie bracket. LCA are related to linearly compact differential Lie algebras via the so-called annihilation functor. Using this observation and the Cartan's classification of linearly compact simple Lie algebras, Bakalov, D'Andrea and Kac classified finite simple LCA in 2000. 

I will define the notion of LCA over a ring R of differential operators with not necessarily constant coefficients, extending the known one for R=K[x]. I will explain why it is natural to study such an object and will suggest an approach for the classification of finite simple LCA over arbitrary differential fields.

MondayNov 23, 201514:30
Algebraic Geometry and Representation Theory SeminarRoom 108
Speaker:Arkady Berenstein Title:Hecke-Hopf algebrasAbstract:opens in new windowin html    pdfopens in new window

It is well-known that Hecke algebras H_q(W) do not have interesting Hopf algebra structures because, first, the only available one would emerge only via an extremely complicated isomorphism with the group algebra of W and, second, this would  make H_q(W) into yet another cocommutative Hopf algebra.

The goal of my talk (based on joint work with D. Kazhdan) is to extend each Hecke algebra H_q(W) to a non-cocommutative Hopf algebra (we call it Hecke-Hopf algebra of W) that contains H_q(W) as a coideal.

Our Hecke-Hopf algebras have a number of applications: they generalize Bernstein presentation of Hecke algebras, provide new solutions of quantum Yang-Baxter equation and a large category of endo-functors of H_q(W)-Mod, and suggest further generalizations of Hecke algebras.

WednesdayNov 18, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Oren Ben-Bassat Title:Introduction to derived algebraic and analytic geometry Abstract:opens in new windowin html    pdfopens in new window

I will present a 'categorical' way of doing analytic geometry in which analytic geometry is seen as a precise analogue of algebraic geometry. Our approach works for both complex analytic geometry and p-adic analytic geometry in a uniform way. I will focus on the idea of an 'open set' as used in these various areas of math and how it is characterised categorically. In order to do this, we need to study algebras and their modules in the category of Banach spaces.  The categorical characterization that we need uses homological algebra in these 'quasi-abelian' categories which is work of Schneiders and Prosmans.  In fact, we work with the larger category of  Ind-Banach spaces for reasons I will explain. This gives us a way to establish foundations of  analytic geometry and to compare with the standard notions such as the theory of affinoid algebras, Grosse-Klonne's theory of dagger algebras (over-convergent functions), the theory of Stein domains and others.  I will explain how this extends to a formulation of derived analytic geometry following the relative algebraic geometry approach of Toen, Vaquie and Vezzosi.

This is joint work with Federico Bambozzi (Regensburg) and Kobi Kremnizer (Oxford).

WednesdayNov 11, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Be'eri GreenfeldTitle:Gel'fand-Kirillov Dimension of Algebras: Prime Spectra, Gradations and RadicalsAbstract:opens in new windowin html    pdfopens in new window

We study properties of affine algebras with small Gel'fand-Kirillov dimension, from the points of view of the prime spectrum, gradations and radical theory.

As an application, we are able to prove that Z-graded algebras with quadratic growth, and graded domains with cubic growth have finite (and efficiently bounded) classical Krull dimension; this is motivated by Artin's conjectured geometric classification of non-commutative projective surfaces, and by opposite examples in the non-graded case.

As another application, we prove a graded version of a dichotomy question raised by Braun and Small, between primitive algebras (namely, algebras admitting faithful irreducible representations) and algebras satisfying polynomial identities.

If time permits, we discuss approximations of the well-studied Koethe problem and in particular prove a stability result for certain radicals under suitable growth conditions.

We finally propose further questions and possible directions, which already stimulated new constructions of monomial algebras.

This talk is partially based on a joint work with A. Leroy, A. Smoktunowicz and M. Ziembowski.

MondayNov 09, 201515:00
Algebraic Geometry and Representation Theory SeminarRoom 208
Speaker:Thomas BitounTitle:On p- support of an algebraic D-moduleAbstract:opens in new windowin html    pdfopens in new windowplease note unusual day, time, room
The p-support is a characteristic p variety attached to an algebraic D-module, for p large enough. It lives in the (Frobenius-twisted) cotangent space. We will discuss how it can be seen as a refined characteristic variety/singular support of the D-module. Further key words: Azumaya algebra, p-curvature.
WednesdayNov 04, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Venkatesh Title:The fusion products of representations of current algebrasAbstract:opens in new windowin html    pdfopens in new window
The current algebra G[t] associated to a simple Lie algebra G is the Lie algebra of polynomial maps from complex plane to G. It is naturally graded with the grading defined by the degree of the polynomials. The fusion product, of Feigin and Loktev, is a graded G[t]-module, which is a refinement of the tensor product of finite dimensional cyclic G[t]-modules. More precisely, one starts with the tensor product of finite dimensional cyclic G[t]-modules, each localized at distinct points. It is again a cyclic G[t]-module generated by the tensor products of cyclic vectors. The graded module associated with the resulting cyclic module is defined to be the fusion product. Feigin and Loktev conjectured that the fusion product as a graded space is independent of the localization parameters for sufficiently well behaved modules. In this talk, we will see that this conjecture is true in most of the special cases.
WednesdayOct 28, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Dimitri Gurevich Title:From Quantum Groups to Noncommutative GeometryAbstract:opens in new windowin html    pdfopens in new window
Since creation theory of Quantum Groups numerous attempts to elaborate an appropriate differential calculus were undertaken. Recently, a new type of Noncommutative Geometry has been obtained on this way. Namely, we have succeeded in introducing the notions of partial derivatives on the enveloping algebras U(gl(m)) and constructing the corresponding de Rham complexes. All objects arising in our approach are deformations of their classical counterparts. In my talk I plan to introduce some basic notions of the theory of Quantum Groups and to exhibit possible applications of this type Noncommutative Geometry to quantization of certain dynamical models.
WednesdayOct 21, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Polyxeni LamprouTitle:Catalan Numbers and Labelled GraphsAbstract:opens in new windowin html    pdfopens in new window

The Catalan numbers form a sequence of integers C_t. A collection of sets H_t with |H_t|= C_t for all t is called a Catalan set. Many examples of Catalan sets are known; the triangulations of the (t+2)-gon, the Dyck paths from (0,0) to (0, 2t) and the nilpotent ideals in the Borel subalgebra of sl_t to name but a few. In my talk I will present a new example of a Catalan set, which has a remarkable property: for all t, H_t decomposes into a (non-disjoint) union of C_{t-1} distinct subsets each of cardinality 2^{t-1}. Moreover, one may define certain interesting labelled graphs for H_t and obtain the above decomposition in a natural way. The subgraphs corresponding to the aforementioned subsets are labelled hypercubes with some edges missing. The motivation of this work was the study of the additive structure of the Kashiwara crystal B(infty).

WednesdayOct 14, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:R. VenkateshTitle:Fusion product structure of Demazure modulesAbstract:opens in new windowin html    pdfopens in new window

In this talk, we study Demazure modules which occur in a level l irreducible integrable representation of an untwisted affine Lie algebra. We also assume that they are stable under the action of the standard maximal parabolic subalgebra of the affine Lie algebra. We prove that such a module is isomorphic to the fusion product of "prime" Demazure modules, where the prime factors are indexed by dominant integral weights which are either a multiple of l or take value less than l on all simple coroots. Our proof depends on a technical result which we prove in all the classical cases and G_2.  We do not need any assumption on the underlying simple Lie algebra when the last "prime" factor is too small. This is joint work with Vyjayanthi Chari, Peri Shereen and Jeffrey Wand.

WednesdayOct 07, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 1
Speaker:Doron ZeilbergerTitle:The Joy of Symbol-CrunchingAbstract:opens in new windowin html    pdfopens in new window

19th century mathematicians (Gauss, Riemann, Markov, to name a few) spent a lot of their time doing tedious numerical computations. Sometimes they were assisted by (human) computers, but they still did a lot themselves. All this became unnecessary with the advent of computers, who made number-crunching million times faster (and more reliable).

20th- and 21st- century mathematicians spent (and still spend) a lot of their time doing tedious symbolic computations. Thanks to the more recent advent of Computer Algebra Systems (e.g. Maple, Mathematica, and the free system SAGE), much of their labor can be delegated to computers, who, of course, can go much faster, much further, and more reliably.

But humans are still needed! First, to teach the computer how to crunch symbols efficiently, but, just as importantly, to inspire them to formulate general conjectures, and methods of proof, for which humans are (still) crucial. I will mention several examples, most notably, a recent proof, by (the human) Guillaume Chapuy, of a conjecture made with the help of my computer Shalosh B. Ekhad (who rigorously proved many special cases), generalizing, to multi-permutations, Amitai Regev's  celebrated asymptotic formula for the number of permutations of length n avoiding an increasing subsequence of length d.

WednesdayJul 29, 201513:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Leonid Makar-LimanovTitle:Possibly a solution of the two dimensional JC (Jacobian Conjecture).Abstract:opens in new windowin html    pdfopens in new window

Several years ago I introduced Newton polytopes related to the potential counterexamples to the JC. This approach permitted to obtain some additional information which though interesting, was not sufficient to get a contradiction. It seems that a contradiction can be obtained by comparing Newton polytopes for the left and right side of a (somewhat mysterious) equality G_x=-y_F.

WednesdayJul 29, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Jacob Greenstein Title:Double canonical basesAbstract:opens in new windowin html    pdfopens in new window
WednesdayJul 15, 201513:30
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Jacob Greenstein Title:Koszul duality for semidirect productsAbstract:opens in new windowin html    pdfopens in new window
WednesdayJul 15, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Anthony JosephTitle:The Representation Theory of Invariant Subalgebras constructed from g AlgebrasAbstract:opens in new windowin html    pdfopens in new window
WednesdayJul 01, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Antoine DucrosTitle:Piecewise-linear and non-archimedean geometriesAbstract:opens in new windowin html    pdfopens in new window
This will be kind of a survey talk (including classical results, more recent ones, and a joint work with Amaury Thuillier which is still in progress ) about the deep links which exist between non-archimedean geometry over a valued field and piecewise linear geometry. I will mainly focus on the properties of some subsets of non-archimedean analytic spaces (in the sense of Vladimir Berkovich), called the skeleta, that inherit a canonical piecewise linear structure.
WednesdayJun 24, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Yuri ZarkhinTitle:Galois groups and splitting fields of Mori trinomialsAbstract:opens in new windowin html    pdfopens in new window
We discuss a certain class of irreducible polynomials over the rationals that was introduced by Shigefumi Mori forty years ago in his Master Thesis. We prove that the Galois group of a Mori polynomial coincides with the corresponding full symmetric groups and the splitting field is "almost" unramified over its quadratic subfield.
WednesdayJun 24, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Yuri ZarkhinTitle:Galois groups and splitting fields of Mori trinomialsAbstract:opens in new windowin html    pdfopens in new window
We discuss a certain class of irreducible polynomials over the rationals that was introduced by Shigefumi Mori forty years ago in his Master Thesis. We prove that the Galois group of a Mori polynomial coincides with the corresponding full symmetric groups and the splitting field is ``almost" unramified over its quadratic subfield.
MondayJun 15, 201515:15
Algebraic Geometry and Representation Theory SeminarRoom 1
Speaker:Gerald SchwarzTitle:Oka Principles and the Linearization ProblemAbstract:opens in new windowin html    pdfopens in new windowNote the unusual day, time and place. Note that this is the second talk from the same seminar on this date.

   Let Q be a Stein space and L a complex Lie group. Then Grauert's Oka Principle states that the canonical map of the  isomorphism classes of holomorphic principle L-bundles over Q to the isomorphism classes of topological principle L-bundles over Q is an isomorphism. In particular he showed that if P, P' are holomorphic principle L-bundles and The Actual Formula a topological isomorphism, then there is a homotopy  The Actual Formula of topological isomorphisms with The Actual Formula  and The Actual Formula a holomorphic isomorphism.

   Let X and Y be Stein G-manifolds where G is a reductive complex Lie group. Then there is a  quotient Stein space  QX, and a morphism The Actual Formula  such that The Actual Formula. Similarly we haveThe Actual Formula .

   Suppose that The Actual Formula is a G-biholomorphism. Then the induced mapping The Actual Formula  has the following property: for any The Actual Formula , The Actual Formula  is G-isomorphic to  The Actual Formula (the fibers are actually affine G-varieties). We say that  The Actual Formula is admissible. Now given an admissible The Actual Formula, assume that we have a G-equivariant homeomorphism  The Actual Formula lifting The Actual Formula. Our goal is to establish an  Oka principle, saying that The Actual Formula has a deformation The Actual Formula  with The Actual Formula  and The Actual Formula biholomorphic.

   We establish this in two main cases. One case is where The Actual Formula is a diffeomorphism that restricts to  G-isomorphisms on the reduced fibers of The Actual Formula and The Actual Formula. The other case is where The Actual Formula restricts to G-isomorphisms on the fibers and X satisfies an auxiliary condition, which usually holds. Finally, we give applications to the Holomorphic Linearization Problem. Let G act holomorphically on The Actual Formula . When is there a change of coordinates such that the action of G becomes linear? We  prove that this is true, for X satisfying the same auxiliary condition as before,  if and only if the quotient QX is admissibly biholomorphic to the quotient of a G-module V.

MondayJun 15, 201514:05
Algebraic Geometry and Representation Theory SeminarRoom 1
Speaker:Aloysius Helminck Title:Orbits of parabolic subgroups on generalized symmetric spacesAbstract:opens in new windowin html    pdfopens in new windowNote the unusual day, time and place. Note that this talk will be followed by another one.

Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, The Actual Formula an involution of G defined over k, H a k-open subgroup of the fixed point group of   The Actual Formula and Gk (resp. Hk) the set of k-rational points of G (resp. H). The  homogeneous space Xk:=Gk/Hk is a generalization of a real reductive symmetric space to arbitrary fields and is called a generalized symmetric space.

Orbits of parabolic k-subgroups on these generalized symmetric spaces occur in various situations, but are especially of importance in the study of representations of Gk related to Xk. In this talk we present a number of structural results for these parabolic k-subgroups that are of importance for the study of these generalized symmetric space and their applications.

WednesdayJun 10, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Lenny Makar-Limanov Title:A description of two-generated subalgebras of a polynomial ring in one variable and a new proof of the AMS theoremAbstract:opens in new windowin html    pdfopens in new window
The famous AMS (Abhyankar-Moh-Suzuki) theorem states that if two polynomials $f$ and $g$ in one variable with coefficients in a field $F$ generate all algebra of polynomials, i.e. any polynomial $h$ in one variable can be expressed as $h = H(f, g)$ where $H$ is a polynomial in two variables, then either the degree of $f$ divides the degree of $g$, or the degree of $g$ divides the degree of $f$, or the degree of $f$ and the degree of $g$ are divisible by the characteristic of the field $F$. There were several wrong published proofs of this theorem and there are many correct published proofs of this theorem but all of them either long or not self-contained. Recently I found a (relatively) short and self-contained proof which is not published yet and which I can explain in one-two hours.
WednesdayJun 03, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Sasha Yomdin Title:Reciprocity laws and K-theoryAbstract:opens in new windowin html    pdfopens in new window

We associate to a full flag F in an n-dimensional variety X over a field k, a "symbol map" $\mu_F :K(F_X) \to \Sigma^n K(k)$. Here, F_X is the field of rational functions on X, and K(.) is the K-theory spectrum.

We prove a "reciprocity law" for these symbols: Given a partial flag, the sum of all symbols of full flags refining it is 0. Examining this result on the level of K-groups, we derive the following known reciprocity laws: the degree of a principal divisor is zero, the Weil reciprocity law, the residue theorem, the Contou-Carrère reciprocity law (when X is a smooth complete curve) as well as the Parshin reciprocity law and the higher residue reciprocity law (when
X is higher-dimensional).

This is a joint work with Evgeny Musicantov.

WednesdayMay 27, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Avner SegalTitle:A Family of New-way Integrals for the Standard L-function of Cuspidal Representations of the Exceptional Group of Type G2. Abstract:opens in new windowin html    pdfopens in new window
In a joint work with N. Gurevich we have constructed a family of Rankin-Selberg integrals representing the standard twisted L-function of a cuspidal representation of the exceptional group of type G2. This integral representations use a degenerate Eisenstein series on the family of quasi-split forms of Spin8 associated to an induction from a character on the Heisenberg parabolic subgroup. This integral representations are unusual in the sense that they unfold with a non-unique model. A priori this integral is not factorizable but using remarkable machinery proposed by I. Piatetski-Shapiro and S. Rallis we prove that in fact the integral does factor. As the local generating function of the local L-factor was unknown to us, we used the theory of C*-algebras in order to approximate it and perform the unramified computation. If time permits, I will discuss the poles of the relevant Eisenstein series and some applications to the theory of CAP representations of G2.
WednesdayMay 20, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Dan CarmonTitle:Autocorrelations of the Moebius function over function fieldsAbstract:opens in new windowin html    pdfopens in new window
In this talk we shall discuss results on autocorrelations of the arithmetic Moebius function of polynomials over finite fields, in the limit of a large base field. Special consideration will be given to base fields of characteristic 2, where both methods and results substantially differ from those applicable in odd characteristics. The methods used are mostly elementary, with a hint of algebraic geometry.
WednesdayMay 13, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Luc Illusie Title:Around the Thom-Sebastiani theoremAbstract:opens in new windowin html    pdfopens in new window

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 $(f \oplus g)(x,y) = f(x) + g(y), x = (x_0,...,x_m), y = (y_0,...,y_n)$. 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 Künneth formula for $R\Psi$ in the framework of Deligne's theory of nearby cycles over general bases.

WednesdayApr 29, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Marko Tadic Title:Square-integrable representations of classical p-adic groups and their Jacquet modulesAbstract:opens in new windowin html    pdfopens in new window
In the talk we shall present formulas for Jacquet modules of square integrable representations of segment type, formulas for special Jacquet modules of a general square integrable representation and a new proof of the Matic’s formula for the Jacquet modules of strongly positive (square integrable) representations of classical p-adic groups.
WednesdayApr 22, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Andrey MinchenkoTitle:Conformal and differential Lie algebrasAbstract:opens in new windowin html    pdfopens in new window
WednesdayApr 15, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Efrat Bank Title:Prime polynomial values of linear functions in short intervalsAbstract:opens in new windowin html    pdfopens in new window

In this talk I will present a function field analogue of a conjecture in number theory. This conjecture is a combination of several famous conjectures, including the Hardy-Littlewood prime tuple conjecture, conjectures on the number of primes in arithmetic progressions and in short intervals, and the Goldbach conjecture. I prove an asymptotic formula for the number of simultaneous prime values of n linear functions, in the limit of a large finite field.
A key role is played by the computation of some Galois groups.

WednesdayMar 18, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Erez LapidTitle:Representation theory of inner forms of GL(n) over a local non-archimedean field - old and new resultsAbstract:opens in new windowin html    pdfopens in new window
Representation theory of inner forms of GL(n) over a local non-archimedean field - old and new results - PART TWO
WednesdayMar 18, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Erez LapidTitle:Representation theory of inner forms of GL(n) over a local non-archimedean field - old and new resultsAbstract:opens in new windowin html    pdfopens in new window
Representation theory of inner forms of GL(n) over a local non-archimedean field - old and new results - PART TWO
WednesdayMar 11, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Erez LapidTitle:Representation theory of inner forms of GL(n) over a local non-archimedean field - old and new resultsAbstract:opens in new windowin html    pdfopens in new window
Representation theory of inner forms of GL(n) over a local non-archimedean field - old and new results - PART ONE
WednesdayFeb 25, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Polyxeni LamprouTitle:The polyhedral structure of B(infinity): graphs, tableaux and Catalan setsAbstract:opens in new windowin html    pdfopens in new window
WednesdayFeb 18, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Bernhard KroetzTitle:On the tempered embedding theorem for real spherical spacesAbstract:opens in new windowin html    pdfopens in new window
WednesdayFeb 11, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Vera SerganovaTitle:gl(\infty) and Deligne categoriesAbstract:opens in new windowin html    pdfopens in new window
WednesdayFeb 04, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Max GurevichTitle:Ladder representations and Galois distinctionAbstract:opens in new windowin html    pdfopens in new window
The space GL_n(E)/GL_n(F), for a quadratic extension E/F of p-adic fields, serves as an approachable case for the study of harmonic analysis on p-adic symmetric spaces on one hand, while having ties with Asai L-functions on the other. It is long known that a GL_n(F)-distinguished representation of GL_n(E) must be contragredient to its own Galois conjugate. Conversely, a conjecture often attributed to Jacquet states that the last-mentioned condition is close to being sufficient for distinction. We show the conjecture is valid for the class of ladder representations which was recently explored by Lapid and Minguez. Along the way, we will suggest a reformulation of the conjecture which concerns standard modules in place of irreducible representations.
WednesdayJan 28, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Tsachik GelanderTitle:On the asymptotic of L_2 invariants of arithmetic groups.Abstract:opens in new windowin html    pdfopens in new window
WednesdayJan 07, 201511:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Shaul ZemelTitle:On Lattices over Valuation Rings of Arbitrary RankAbstract:opens in new windowin html    pdfopens in new window
We show how the simple property of 2-Henselianity suffices to reduce the classification of lattices over a general valuation ring in which 2 is invertible (with no restriction on the value group) to classifying quadratic spaces over the residue field. The case where 2 is not invertible is much more difficult. In this case we present the generalized Arf invariant of a unimodular rank 2 lattice, and show how in case the lattice contains a primitive vector with norm divisible by 2, a refinement of this invariant and a certain class suffice for classifying these lattices.
WednesdayDec 31, 201411:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Victor KacTitle:Non-commutative geometry and non-commutative integrable systemsAbstract:opens in new windowin html    pdfopens in new window
WednesdayNov 19, 201411:00
Algebraic Geometry and Representation Theory SeminarRoom 261
Speaker:Ary ShavivTitle:Affine generalized root systems and symmetrizable affine Kac-Moody superalgebrasAbstract:opens in new windowin html    pdfopens in new window
Correspondence between different types of Lie algebras and abstract root systems is a classical and useful tool. In the end of the 19th century E.J. Cartan and W. Killing classified real root systems and finite dimensional complex Lie algebras. They showed the correspondence between reduced root systems and these algebras. I.G. Macdonald classified affine root systems in the beginning of the 1970's. V.G. Kac later realized these systems are, in most cases, real parts of Kac-Moody algebras of affine type. V. Serganova classified generalized root systems in 1996 and showed their almost perfect correspondence to basic classical Lie superalgebras. We defined a generalization we call affine generalized root systems, and studied their correspondence to symmetrizable affine Kac-Moody superalgebras. In the talk we will define the above types of root systems, present their precise correspondences to Lie (super)algebras, and present the main points of our classification of affine generalized root systems.