Organizers: A. Aizenbud, M. Gorelik, D. Gourevitch

  Fall 2015-- Summer 2016

Title: The Joy of Symbol-Crunching

Abstract. 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.

Title: Fusion product structure of Demazure modules

Abstract. 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.

Title: Catalan numbers and labelled graphs (joint with A. Joseph).

Abstract. 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).

Title: From Quantum Groups to Noncommutative Geometry

Abstract. 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.

Title: The fusion products of representations of current algebras.

Abstract. 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.

Title: On p- support of an algebraic D-module.

Abstract. 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.

Title: Gel'fand-Kirillov Dimension of Algebras: Prime Spectra, Gradations and Radicals.

Abstract. 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.

Title: An introduction to derived algebraic and analytic geometry.

Abstract. 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).

Title: Hecke-Hopf algebras

Abstract. : 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.

Title: Simple Lie conformal algebras.

Abstract. 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.

Title: Truncated shifted Yangians and Nakajima monomial crystals.

Abstract. 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 g^L-action (Langlands dual Lie algebra) on representation categories of truncated shifted Yangians.

Title: Non-commutative Iwasawa algebras.

Abstract. 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.

Title: Generalized RSK

Abstract. 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.

Title: On rings stable under derivations.

Abstract. 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 i s 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.

Title: Real Galois cohomology of semisimple groups

Abstract. 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.

Title: Degenerations of Calabi-Yau manifolds and non-Archimedean analytic spaces.

Title: Low dimensional representations of finite classical groups.

Abstract. 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).

Title: Symmetric tensor categories in characteristic p.

Title: Derived Categories of Bimodules.

Title: Integrability of p-adic matrix coefficients.

Abstract. 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.

Title: Mod-p representations of p-adic metaplectic groups.

Abstract. 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.

Title: Two remarkable properties of the canonical S-graphs and the Kashiwara crystal .

Title: p-extensions of local fields with Galois groups of nilpotent class

Abstract. 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

Title: Recent applications of classical theorems on D-modules

Title: An elementary proof of Olshanskii's theorem on subgroups of a free group and its applications

Abstract: 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.

Title: Adapted pairs for maximal parabolic subalgebras and polynomiality of invariants.

Abstract: 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.

Title: Differential algebraic groups and their applications.

Abstract: 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.

Title: New tensor categories related to orthogonal and symplectic groups and the strange supergroup P(infinity)

Abstract: 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.

Title: Singular Gelfand-Tsetlin modules

Title: Primitive ideals in U(sl(infinity))

Title: Ordered tensor categories of representations of Mackey Lie algebras

Title: Supersingular representations and the mod p Langlands

Abstract: 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.

Title: Basis of root spaces for Generalized Kac-Moody algebras

Abstract: For a Generalized Kac-Moody algebra G, we consider the root spaces whose roots are supported only on the imaginary simple roots of G. In this talk, I will explain how to construct basis for these root spaces. It is joint work with G. Arun Kumar, Deniz Kus and S. Viswanath.

Title: A minimax theorem for trails

Title: Representations of reductive groups distinguished by symmetric subgroups.

Abstract: 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.

Title: The intricate Maze of Graph Complexes.

Title: The singular transfer for the Jacquet-Rallis trace formula.

Title: The Capelli problem for gl(m|n) and the spectrum of invariant differential operators.

Abstract: 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.

Title: Introduction to cluster algebras

Abstract: 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}(C).

Title: Introduction to cluster algebras (continuation)

Abstract: 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}(C).