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# Algebraic Geometry and Representation Theory Seminar

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.

The generalized Jacobian Jac_m(C ') of a smooth hyperelliptic curve C' associated with a module m is an algebraic group that can be described by using lines bundle of the curve C' or by using a symmetric product of the curve C' provided with a law of composition. This second definition of the Jacobian Jac_m(C') is directly related to the fibres of a Mumford system. To be precise it is a subset of the compactified Jac_m(C') which is related to the fibres. This presentation will help us to demystify the relationship of these two mathematical objects.

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.

In this talk, I present an analogue of the Hardy-Littlewood conjecture on the asymptotic distribution of prime constellations in the setting of short intervals in function fields of smooth projective curves over finite fields.

I will discuss the definition of a "short interval" on a curve as an additive translation of the space of global sections of a sufficiently positive divisor E by a suitable rational function f, and show how this definition generalizes the definition of a short interval in the polynomial setting.

I will give a sketch of the proof which includes a computation of a certain Galois group, and a counting argument, namely, Chebotarev density type theorem.

This is a joint work with Tyler Foster.

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.

We recall the notion of a hall algebra associated to a category, and explain how this construction can be done in a way that naturally includes a higher algebra structure, motivated by work of Toen and Dyckerhoff-Kapranov. We will then explain how this leads to new insights about the bi-algebra structure and related concepts.

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.

I shall review the framework of algebraic families of Harish-Chandra modules, introduced recently, by Bernstein, Higson, and the speaker. Then, I shall describe three of their applications.

The first is contraction of representations of Lie groups. Contractions are certain deformations of representations with applications in mathematical physics.

The second is the Mackey bijection, this is a (partially conjectural) bijection between the admissible dual of a real reductive group and the admissible dual of its Cartan motion group.

The third is the hidden symmetry of the hydrogen atom as an algebraic family of Harish-Chandra modules.

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.

In this talk I will describe a family of integral representations for the standard twisted L-function of a cuspidal representation of the exceptional group of type G_2. This integral representations. These integral representations are unusual in the sense that they unfold with a non-unique model. A priori this integral is not Eulerian but using remarkable machinery proposed by I. Piatetski-Shapiro and S. Rallis we prove that in fact the integral does factor. In the course of the plocal unramified calculation we use another non-standard method, approximations of generating functions. I will then describe a few applications of these integral representations to the study of the analytic behaviour of the this L-function and to various functorial lifts associated with the group G_2.

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.

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.

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]**

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.

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.

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.

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 $Gr_{2,n+3}(\mathbb{C})$.

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)

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.

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.

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.

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.

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.

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).

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).

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.

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.

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.

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*.

Let *G* be a connected reductive algebraic group defined over a field *k* of characteristic not 2, an involution of *G* defined over *k*, *H* a *k*-open subgroup of the fixed point group of and G_{k} (resp. H_{k}) the set of *k*-rational points of *G* (resp. *H*). The homogeneous space X_{k}:=G_{k}/H_{k} 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 G_{k} related to X_{k}. 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.

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.

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.

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.