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I'll talk about a recent result, joint with Nir Avni, on uniform bounds on multiplicities of irreducible representations in functions on symmetric spaces like GL_n(\Z_p)/O_n(\Z_p).

The bound that we obtain is uniform in p.

Let G be a reductive group defined and deployed over a global function field. We are interested in the sum of multiplicities of irreducible representations containing a regular depth zero representation of G(O), where O is the ring of integral adeles, in the automorphic cuspidal spectrum. The sum is given in terms of the number of F_q-points of Hitchin moduli spaces of groups associated to G. When G=GL(n), it implies some cases of a conjecture of Deligne by Langlands correspondence. In this talk, I will mainly focus on the case of GL(n).

Seminar in zoom only:
https://weizmann.zoom.us/j/98304397425

In this talk, we consider representations of p-adic classical groups parabolically induced from
the products of shifted Speh representations and unitary representations of Arthur type of good parity.
We describe how to compute the socles (the maximal semisimple subrepresentations) of these representations algorithmically.
As a consequence, we can determine whether these representations are reducible or not.
In particular, our results produce many unitary representations.

This talk is based on my thesis supervised by P.-H. Chaudouard. The conjecture of Guo-Jacquet is a promising generalization to higher dimensions of Waldspurger’s well-known theorem on the relation between toric periods and central values of automorphic L-functions for GL(2). However, we are faced with divergent integrals when applying the relative trace formula approach. In this talk, we study an infinitesimal variant of this problem. Concretely, we establish global and local trace formulae for infinitesimal symmetric spaces of Guo-Jacquet. To compare regular semi-simple terms, we present the weighted fundamental lemma and certain identities between Fourier transforms of local weighted orbital integrals

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There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio
Trace(p(g)) / dim(p),
for an irreducible representation p of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.
Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank.
Rank suggests a new organization of representations based on the very few "small" ones. This stands in contrast to Harish-Chandra's philosophy of cusp forms, which is (since the 60s) the main organization principle, and is based on the (huge collection of) "Large" representations.

This talk will discuss the notion of rank for the group GLn over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for random walks.
This is joint work with Roger Howe (Yale and Texas A&M). The numerics for this work was carried with Steve Goldstein (Madison) and John Cannon (Sydney).

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Let n be a positive integer, F be a non-Archimedean locally compact field of odd residue characteristic p and G be an inner form of GL(2n,F). This is a group of the form GL(r,D) for a positive integer r and division F-algebra D of reduced degree d such that rd=2n. Let K be a quadratic extension of F in the algebra of matrices of size r with coefficients in D, and H be its centralizer in G. We study selfdual cuspidal representations of G and their distinction by H, that is, the existence of a nonzero H-invariant linear form on such representations, from the viewpoint of type theory. When F has characteristic 0, we characterize distinction by H for cuspidal representations of G in terms of their Langlands parameter, proving in this case a conjecture by Prasad and Takloo-Bighash.

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In this talk, we will discuss the theory of twisted automorphic descents, which is an extension of the automorphic descent of Ginzburg-Rallis-Soudry. 
The main goal is to construct cuspidate automorphic modules in the generic global Arthur packets by using Bessel-Fourier coefficients of automorphic representations. 
Moreover, we will discuss some applications and problems related Bessel-Fourier coefficients. 
This is a joint work with Dihua Jiang.

https://weizmann.zoom.us/j/98304397425

Please note that Israel has switched to winter time. The seminar will be 16:30 Israeli time.

Motivated by problems arising from the study of certain relative trace formulas, I discuss a notion of endoscopy in a relative setting. The main example is that of unitary Friedberg-Jacquet periods, which are related to special cycles in certain unitary Shimura varieties. After introducing the endoscopic symmetric spaces in this case, I will sketch the proof of the fundamental lemma.

https://weizmann.zoom.us/j/98304397425

Let G be a connected reductive group defined over a field k of characteristic 0. Recently Knop and Krötz showed that one can attach a Weyl group to any algebraic homogeneous G-variety defined over k. This Weyl group is called the little Weyl group. In this talk I will discuss a geometric construction of the little Weyl group for a real spherical space G/H. Our technique is based on a fine analysis of limits of conjugates of the subalgebra Lie(H) along one-parameter subgroups in the Grassmannian of subspaces of Lie(G).

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I will review how automorphic representations arise in the calculation of string theory scattering amplitudes. As follows from the work of Green, Miller, Vanhove, Pioline and others, the automorphic representations are associated with split real groups of a certain exceptional family. In the cases that are well understood, these representation have very small Gelfand-Kirillov dimension. Their Fourier expansion can be calculated using different methods and confirms physical expectation on the wavefront set. In work with Gourevitch, Gustafsson, Persson and Sahi, the method of Whittaker pairs was employed to systematize this analysis. I will also comment on the cases that are less well understood in physics and that appear to go beyond the standard notion of automorphic representations since the usual Z-finiteness condition is violated.

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In this talk, I will introduce the functorial descent from cuspidal automorphic representations \pi of GL7(A) with L^S(s, \pi, \wedge^3) having a pole at s=1 to the split exceptional group G2(A), using Fourier coefficients associated to two nilpotent orbits of E7. We show that one descent module is generic, and under mild assumptions on the unramified components of \pi, it is cuspidal and having \pi as a weak functorial lift of each irreducible summand. However, we show that the other descent module supports not only the non-degenerate Whittaker integral on G2(A), but also every degenerate Whittaker integral. Thus it is generic, but not cuspidal. This is a new phenomenon, compared to the theory of functorial descent for classical and GSpin groups. This work is joint with Joseph Hundley. 

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The classical theta correspondence establishes a relationship between automorphic representations on special orthogonal groups and automorphic representations on symplectic groups or their double covers. This correspondence is achieved by using as integral kernel a theta series that is constructed from the Weil representation. In this talk I will briefly survey earlier work on (local and global, classical and other) theta correspondences and then present an extension of the classical theta correspondence to higher degree metaplectic covers. The key issue here is that for higher degree covers there is no analogue of the Weil representation (or even a minimal representation), so additional ingredients are needed. Joint work with David Ginzburg.

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Let G be a reductive group over a local field F of characteristic zero, and H be a spherical subgroup. An irreducible representation of G is said to be distinguished by H if it has an H-invariant continuous linear functional. The study of distinguished representations is of much current interest, because of their relation to the Plancherel measure on G/H and to periods of automorphic forms.
While a complete classification seems to be out of reach, we established simple micro-local necessary conditions for distinction. The conditions are formulated in terms of the nilpotent orbits associated to the representation, in the spirit of the orbit method. Our results are strongest for Archimedean F. In this case, Rossmann showed that for any irreducible Casselman-Wallach representation, the Zariski closure of the wave-front set is the closure of a unique nilpotent complex orbit. We have shown that the restriction of this orbit to the complexified Lie algebra of H includes zero.
We apply this result to symmetric pairs, branching problems, and parabolic induction. We also have a twisted version for the case when π has a functional invariant with respect to an "additive" character of H. As an application of our theorem we derive necessary conditions for the existence of Rankin-Selberg, Bessel, Klyachko and Shalika models. Our results are compatible with the recent Gan-Gross-Prasad conjectures for non-generic representations. Our necessary conditions are rarely sufficient, but they are sufficient for one class of models: the Klyachko models for unitary representations of general linear groups.

This is a joint work with Eitan Sayag.

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Let G be a connected algebraic group.  
We define and study a convolution operation between algebraic morphisms into G.  We show that this operation yields morphisms with improved singularity properties, and in particular, that under reasonable assumptions one can always obtain a flat morphism with reduced fibers of rational singularities (termed an FRS morphism) after enough convolutions.

The FRS property is of high importance since (FRS) morphisms can be characterized by good asymptotic behaviour of the number of points of their fibers over finite rings of the form Z/p^kZ.
This further allows us to interpret the FRS property through probabilistic lenses.

We discuss some of the above, motivated by the special case of word maps which can be viewed as a relative
analogue in the settings of p-adic groups of Waring's problem from 1770 (see arXiv:1912.12556).

Joint work with Itay Glazer.

Zoom link: https://weizmann.zoom.us/j/98304397425

Consider the function field $F$ of a smooth curve over $\FF_q$, with $q\neq 2$.

L-functions of automorphic representations of $\GL(2)$ over $F$ are important objects for studying the arithmetic properties of the field $F$. Unfortunately, they can be defined in two different ways: one by Godement-Jacquet, and one by Jacquet-Langlands. Classically, one shows that the resulting L-functions coincide using a complicated computation.

I will present a conceptual proof that the two families coincide, by categorifying the question. This correspondence will necessitate comparing two very different sets of data, which will have significant implications for the representation theory of $\GL(2)$. In particular, we will obtain an exotic symmetric monoidal structure on the category of representations of $\GL(2)$

It turns out that an appropriate space of automorphic functions is a commutative algebra with respect to this symmetric monoidal structure. I will outline this construction, and show how it can be used to construct a category of automorphic representations.

Zoom meeting: https://weizmann.zoom.us/j/98304397425

We explain how a doubled version of the Beilinson-Bernstein localization functor can be understood using the geometry of the wonderful compactification of a group. Specifically, bimodules for the Lie algebra give rise to monodromic D-modules on the horocycle space, and to filtered D-modules on the group that respect a certain matrix coefficients filtration. These two categories of D-modules are related via an associated graded construction in a way compatible with localization, Verdier specialization, the Vinberg semigroup, and additional structures. This talk is based on joint work with David Ben-Zvi.

Zoom meeting: https://weizmann.zoom.us/j/98304397425

The Dulfo-Serganova functor  is a cohomology functor relating representation theory of Lie superalgebras of different ranks. 
This is a tensor functor preserving superdimension. 
Serganova conjectured that the image of a finite-dimensional simple module L under Duflo-Serganova functor is semisimple. Heidersdorf and Weissauer established this conjecture for gl-case and described DS(L). In my previous talk  I sketched a proof of semsimplicity for  osp-type. In this talk I will explain how to compute the mulitplicites in DS(L).
This is a joint project with Thorsten Heidersdorf.  
 

Unramified principal series representations of p-adic GL(r) and its metaplectic covers are important in the theory of automorphic forms. I will present a method of relating the Whittaker coinvariants of such a representation with representations of quantum affine gl_n. This involves using a Schur-Weyl duality result due to Chari and Pressley and it allows us to compute the dimension of the Whittaker model of every irreducible smooth representation with Iwahori fixed vectors.

If time permits I will explain a conjectured version of this result for the symplectic group Sp(2r) which involves quantum symmetric pairs.

In a joint work with Erez Lapid we constructed a new class of representations based on applying the RSK algorithm on Zelevinski's multisegments. Those constructions have the potential to be an alternative to the commonly used basis of standard representations. Intriguingly, this class also turned out to categorify a 45-year-old development in invariant theory: The Rota basis of standard bitableaux.
I will talk about this classical theme and its relation to representations of p-adic GL_n, as well the expected properties of our new class.

The question of determining if a given algebraic variety is rational is a notoriously difficult problem in algebraic geometry, and attempts to solve rationality problems have often produced powerful new techniques. A well-known open rationality problem is the determination of a criterion for when a cubic hypersurface of five-dimensional projective space is rational. After discussing the history of this problem, I will introduce the two conjectural rationality criteria that have been put forth and then discuss a package of tools I have developed with my collaborators to bring these two conjectures together. Our theory of Relative Bridgeland Stability has a number of other beautiful consequences such as a new proof of the integral Hodge Conjecture for Cubic Fourfolds and the construction of full-dimensional families of projective HyperKahler manifolds. Time permitting I'll discuss applications of the theory of relative stability conditions to problems other than cubic fourfolds.

The goal of my talk (based on joint work with Vladimir Retakh) is to introduce noncommutative analogs of Catalan numbers c_n which belong to the free Laurent polynomial algebra L_n in n generators. Our noncommutative Catalan numbers C_n admit interesting (commutative and noncommutative) specializations, one of them related to Garsia-Haiman (q,t)-versions, another -- to solving noncommutative quadratic equations. We also establish total positivity of the corresponding (noncommutative) Hankel matrices H_n and introduce two kinds of noncommutative binomial coefficients which are instrumental in computing the inverse of H_n and its positive factorizations, and other combinatorial identities involving C_n.
If time permits, I will explain the relationship of the C_n with the:

1. noncommutative Laurent Phenomenon, which was previously established for Kontsevich rank 2 recursions and all marked surfaces

2. noncommutative orthogonal polynomials, which can be viewed as noncommutative determinants of an extended matrix H_n.

The quotient of a monoidal category by its largest tensor ideal - given by the so-called negligible morphisms - is often a semisimple category.
I will introduce a generalization of the notion of negligible morphism for some monoidal categories and discuss the associated tensor ideals in the setting of Deligne categories and tilting modules for quantum groups and algebraic groups. It turns out that they are related to other notions from representation theory like modified dimensions and the a-function.

Representations of braid groups have appeared in many different areas such as topology, statistical mechanics, conformal field theory, braided tensor categories and others. In order to compare these, intrinsic characterizations of such representations are desirable. These have been known for some time for representations in connection with vector representations of classical Lie types, in terms of Hecke algebras and so-called BMW algebras. We review these and show how these results can be extended to include more cases related to exceptional Lie types. In particular, we obtain new classes of braid representations where the images of the generators satisfy a cubic equation. Time permitting, we discuss applications of these results such as Schur-Weyl type duality theorems and classification of braided tensor categories.

Root-reductive Lie algebras form a special type of reasonably well behaved infinite-dimensional Lie algebras. In this talk, we shall define a version of Bernstein-Gelfand-Gelfand categories O for root-reductive Lie algebras, which we called extended categories O and briefly discuss some properties of these categories. Let g be a rootreductive Lie algebra containing a splitting Borel subalgebra b satisfying a special additional condition called the Dynkin condition. The extended category O corresponding to g and b is denoted by O-bar.
The category O-bar can be decomposed analogously to the finite-dimensional cases into blocks. The main object of this talk is to give a construction of translation functors of O-bar. Then we shall see that some objects such as tilting modules arise by applying the translation functors to Verma modules just as in the finite-dimensional cases. Furthermore, the translation functors establish equivalences between some blocks of the category O-bar.

This will be a very introductory talk about Virasoro Lie algebra and its super-analogues: Ramond and Neveu-Schwarz Lie superalgebras.

Let F be a function field and G a connected split reductive group over F. We define a "strange" operator between different spaces of automorphic functions on G(A)/G(F), and show that this operator is natural from the viewpoint of the geometric Langlands program via the functions-sheaves dictionary. We discuss how to define this operator over a number field by relating it to pseudo-Eisenstein series and inversion of the standard intertwining operator. This operator is also connected to Deligne-Lusztig duality and cohomological duality of representations over a local field.

Poincare duality provides an isomorphism between the homology and cohomology of a compact manifold, up to a shift. For \pi-finite spaces, i.e. spaces with finitely many non-zero homotopy groups, all of which are finite, there is a similar duality only for Q-coefficients, but no such duality exists with F_p coefficients. However, as shown by Michael Hopkins and Jacob Lurie, there is a duality between the homology and cohomology of \pi-finite spaces with coefficients in some extra-ordinary cohomology theories called Morava K-theories. This property of Morava K-theory is called ambidexterity.I will explain what is ambidexterity, some of its consequences and our contribution to the subject.

This is a joint work with Lior Yanovski and Tomer Schlank.

The reducibility and structure of parabolic inductions is a basic problem in the representation theory of p-adic groups.  Of particular interest are principal series and degenerate principal series representations, that is parabolic induction of 1-dimensional representations of Levi subgroups.

In this talk, I will start by describing the functor of normalized induction and its left adjoint, the Jacquet functor, and by going through several examples in the group SL_4(Q_p) will describe an algorithm which can be used to determine reducibility of such representations.

This algorithm is the core of a joint project with Hezi Halawi, in which we study the structure of degenerate principal series of exceptional groups of type En (see https://arxiv.org/abs/1811.02974).

We define a functor from the category of representations of algebraic supergroups with reductive even part to the category of equivariant sheaves and show several applications of this construction to representation theory, in particular projectivity criterion, classification of blocks and computation of dimensions of irreducible representations.

Representation theory of non-compact real groups, such as SL(2,R), is a fundamental discipline with uses in harmonic analysis, number theory, physics, and more. This theory is analytical in nature, but in the course of the 20th century it was algebraized and geometrized (the key contributions are by Harish-Chandra for the former and by Beilinson-Bernstein for the latter). Roughly and generally speaking, algebraization strips layers from the objects of study until we are left with a bare skeleton, amenable to symbolic manipulation. Geometrization, again very roughly, reveals how algebraic objects have secret lives over spaces - thus more amenable to human intuition. In this talk, I will try to motivate and present one example - the calculation of the Casselman-Jacquet module of a principal series representation (I will explain the terms in the talk).

In the talk I will tell about a theory of cyclic elements in semisimple Lie algebras. This notion was introduced by Kostant, who associated a cyclic element with the principal nilpotent and proved that it is regular semisimple. In particular, I will tell about the classification of all nilpotents giving rise to semisimple and regular semisimple cyclic elements. The results are from my joint work with V. Kac and E. Vinberg.

I will define the Enright functor for contragredient Lie superalgebras and discuss its properties. If time permits, we may discuss a proof of Arakawa's Theorem for osp(1|2l).
This work is a joint project with V. Serganova.

Various algebraic and topological situations give rise to compatible sequences of representations of different groups, such as the symmetric groups, with stable asymptotic behavior. Representation stability is a recent approach to studying such sequences, which has proved effective for extracting important invariants. Coming from this point of view, I will introduce the associated character theory, which formally explains many of the approach's strengths (in char 0). Central examples are simultaneous characters of all symmetric groups, or of all Gl(n) over some finite field. Their mere existence gives applications to statistics of random matrices over finite fields, and raises many combinatorial questions.

TBA

In analysis, a convolution of two functions usually results in a smoother, better behaved function. Given two morphisms f,g from algebraic varieties X,Y to an algebraic group G, one can define a notion of convolution of these morphisms. Analogously to the analytic situation, this operation yields a morphism (from X x Y to G) with improved smoothness properties.

In this talk, I will define a convolution operation and discuss some of its properties. I will then present a recent result; if G is an algebraic group, X is smooth and absolutely irreducible, and f:X-->G is a dominant map, then after finitely many self convolutions of f, we obtain a morphism with the property of being flat with fibers of rational singularities (a property which we call (FRS)).

Moreover, Aizenbud and Avni showed that the (FRS) property has an equivalent analytic characterization, which leads to various applications such as counting points of schemes over finite rings, representation growth of certain compact p-adic groups and arithmetic groups of higher rank, and random walks on (algebraic families of) finite groups. We will discuss some of these applications, and maybe some of the main ideas of the proof of the above result.

Joint with Yotam Hendel.

We will introduce the (new) notion of approximability in triangulated categories and show its power.

The brief summary is that the derived category of quasicoherent sheaves on a separated, quasicompact scheme is an approximable triangulated category.
As relatively easy corollaries one can: (1) prove an old conjecture of Bondal and Van den Bergh, about strong generation in D^{perf}(X), (2) generalize an old theorem of of Rouquier about strong generation in D^b_{coh}(X). Rouquier proved the result only in equal characteristic, we can extend to mixed characteristic, and (3) generalize a representability theorem of Bondal and Van den Bergh,from proper schemes of finite type over fields to proper schemes of finite type over any noetherian rings.

After stating these results and explaining what they mean, we will (time permitting) also mention structural theorems. It turns out that approximable triangulated categories have a fair bit of intrinsic, internal structure that comes for free.

We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. By definition, the semisimplification of a tensor category is its quotient by the tensor ideal of negligible morphisms, i.e., morphisms f such that Tr(fg)=0 for any morphism g in the opposite direction. In particular, we compute the semisimplification of the category of representations of a finite group in characteristic p in terms of representations of the normalizer of its Sylow p-subgroup. This allows us to compute the semisimplification of the representation category of the symmetric group S_{n+p} in characteristic p, where n=0,...,p-1, and of the Deligne category Rep^{ab} S_t, t in N. We also compute the semisimplification of the category of representations of the Kac-De Concini quantum group of the Borel subalgebra of sl_2. Finally, we study tensor functors between Verlinde categories of semisimple algebraic groups arising from the semisimplification construction, and objects of finite type in categories of modular representations of finite groups (i.e., objects generating a fusion category in the semisimplification).
This is joint work with Victor Ostrik.

Given a p-adic group G, number theorists are interested in producing admissible representations of G: representations which have a well-defined character functional. One way to produce such representations is by "Jacquet induction" from smaller groups, whose characters can be understood inductively. The complementary space of "new" characters which are not obtained by induction (complementary with respect to a natural metric on the space of characters) is given by what is called "elliptic" characters. Given a representation V of G, the "new" input from its character is captured by the operator Ax(V), with A (the Bernstein-Deligne-Kazhdan A-operator) the projector to the elliptic component (note that this is different from the component of the character lattice valued in elliptic elements). I will talk about my ongoing work with Xuhua He on extending this operator to a trace functional Ax(V) for V a finitely-generated representation (whose Grothendieck group is well understood), which works by first constructing a virtual elliptic admissible representation from any finitely generated representation.

Let K be a commutative ring. Consider the groups GLn(K). Bernstein and Zelevinsky have studied the representations of the general linear groups in case the ring K is a nite eld. Instead of studying the representations of GLn(K) for each n separately, they have studied all the representations of all the groups GLn(K) si- multaneously. They considered on R := nR(GLn(K)) structures called parabolic (or Harish-Chandra) induction and restriction, and showed that they enrich R with a structure of a so called positive self adjoint Hopf algebra (or PSH algebra). They use this structure to reduce the study of representations of the groups GLn(K) to the following two tasks:
1. Study a special family of representations of GLn(K), called cuspidal representa- tions. These are representations which do not arise as direct summands of parabolic induction of smaller representations.
2. Study representations of the symmetric groups. These representation also has a nice combinatorial  description, using partitions.
In this talk I will discuss the study of representations of GLn(K) where K is a nite quotient of a discrete valuation ring (such as Z=pr or k[x]=xr, where k is a nite eld). One reason to study such representation is that all continuous complex representations of the groups GLn(Zp) and GLn(k[[x]]) (where Zp denotes the p-adic integers) arise from these nite quotients. I will explain why the natural generalization of the Harish-Chandra functors do not furnish a PSH algebra in this case, and how is this related to the Bruhat decomposition and Gauss elimination. In order to overcome this issue we have constructed a generalization of the Harish-Chandra functors. I will explain this generalization, describe some of the new functors properties, and explain how can they be applied to studying complex representations.
 The talk will be based on a joint work with Tyrone Crisp and Uri Onn.
 

Abstract. As is well known and easy to prove the Weyl algebras A_n over a field of characteristic zero are simple. Hence any non-zero homomorphism from A_n to A_m is an embedding and m \geq n. V. Bavula conjectured that the same is true over the fields with finite characteristic. It turned out that exactly one half of his conjecture is correct (m \geq n but there are homomorphisms which are not embeddings).
If we replace the Weyl algebra by its close relative symplectic Poisson algebra (polynomial algebra with F[x_1, ..., x_n; y_1, ..., y_n] variables and Poisson bracket given by {x_i, y_i} =1 and zero on the rest of the pairs), then independently of characteristic all homomorphisms are embeddings.

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.

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.

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.

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.

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).

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.  

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.

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).

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.

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).

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.

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.

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.

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).

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.

   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  a topological isomorphism, then there is a homotopy   of topological isomorphisms with   and  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  Q_X, and a morphism   such that . Similarly we have .

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

   We establish this in two main cases. One case is where  is a diffeomorphism that restricts to  G-isomorphisms on the reduced fibers of  and . The other case is where  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  . 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 Q_X is admissibly biholomorphic to the quotient of a G-module V.

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.