Seminars Sorted by Series

In my talk I will describe our joint work with David Kazhdan on the global Langlands correspondence over function fields for arbitrary split reductive groups. Our main result asserts that for every pair $(\pi,\omega)$, where $\pi$ is a cuspidal...

We construct a weak representation of the category of framed affine tangles on a disjoint union of triangulated categories $D_n$. The categories we use are that of coherent sheaves on Springer fibers over a nilpotent element of $SL(2n)$ with two...

This will be a relaxed, very elementary introduction to the central ideas and applications of derived algebraic geometry. My objective is to describe the Artin-Lurie Representability Theorem, and to give a number of interesting examples.

This is the first in a series of (probably) three talks whose goal is to understand Faltings’ paper "A p-adic Simpson correspondence". Roughly speaking, this is a correspondence between Higgs bundles over a smooth proper curve X over the field of p...

The aim of this talk is to introduce some of the technologies used in derived algebraic geometry. Beginning with an explanation of the notion of infinity-groupoid, I will develop the notion of infinity-categories. I will discuss the relationship...

Let G be a reductive algebraic group over a local field k . Hiraga, Ichino and Ikeda have recently proposed a general conjecture for the formal degree of a discrete series representation of G(k) , using special values of the adjoint L-function and...

Spectral properties of random band matrices can be expressed in terms certain statistical mechanics models. Some of these models are generalizations of the Ising model except that the spins belong the the Poincare superdisc and there is...

Spectral properties of random band matrices can be expressed in terms certain statistical mechanics models. Some of these models are generalizations of the Ising model except that the spins belong the the Poincare superdisc and there is...

We will introduce certain operators A(m) on the equivariant cohomology ring of the Laumon quasiflags spaces M_d . The character of A(m) will be equal to the generating function Z(m) of the integrals of the Chern polynomial of the tangent bundle of...

We discuss sharp regularity results for the solutions of the polyharmonic equation in an arbitrary open set. Then we introduce an appropriate notion of capacity which allows to describe the precise correlation between the smoothness of the solution...

We discuss the distribution of the extreme eigenvalues for several classes of large (N X N) Hermitian random matrices. For a class of sparse matrices, the distribution is approximately Tracy--Widom. For band matrices with band width W(N) , the...

Mathematics has contributed immensely to the development of secure cryptosystems and protocols. Yet our networks are terribly insecure, and we are constantly threatened with the prospect of imminent doom. Furthermore, even though such warnings have...

In the talk, I will describe recent attempts to understand the mysterious and beautiful geometry of nodal lines of random spherical harmonics and of random plane waves. If time permits, I will also discuss asymptotic statistical topology of other...

THIS LECTURE IS BEING POSTPONED UNTIL NEXT TERM.

Abstract: In addition to formal definitions and theorems, mathematical theories also contain clever, context-sensitive notations, usage conventions, and proof methods. To mechanize advanced mathematical results it is essential to capture these more...

I am going to talk about triangulated categories in algebra, geometry and physics and about differential-graded (DG) enhancements of triangulated categories. I will discuss such properties of DG enhancements as uniqueness and existing. It can be...

I will present a recent joint work with Ya.G. Sinai. We investigate the ``randomness" of the classical Möbius function by means of a statistical mechanical model for square-free numbers and we prove some new results, including a non-standard limit...

Let Gamma be a non-cocompact lattice on a right-angled building X. Examples of such X include products of trees, or Bourdon's building I_{p,q}, which has apartments hyperbolic planes tesselated by right-angled p-gons and all vertex links the...

We explain in this talk how Ramanujan graphs can be used to devise optimal cycle codes and review how other graph families related to a construction proposed by Margulis yield interesting families of quantum codes with logarithmic minimum distance...

(1) My lecture revolves around a question: Does the world embody beautiful ideas? That is a question that people have thought about for a long time. Pythagoras and Plato intuited that the world should embody beautiful ideas; Newton and Maxwell...

This is from joint works with D. Knopf and I. M. Sigal. In this talk I will present a new strategy in studying neckpinching of mean curvature flow. Different from previous results, we do not use backward heat kernel, entropy estimates or subsequent...

The random loop representations of Toth ('93) and Aizenman-Nachtergaele ('94) can be extended to describe certain SU(2)-invariant spin-1 Heisenberg models. Quantum spin correlations are given in terms of loop correlations. Existence of long-range...

A large class of one dimensional stochastic particle systems are predicted to share the same universal long-time/large-scale behavior. By studying certain integrable models within this (Kardar-Parisi-Zhang) universality class we access what should...

We extend the notion of lattice polarization for K3 surfaces to families in a way that gives control over the action of monodromy on the algebraic cycles, and discuss the uses of this new theory in the study of families of K3 surfaces admitting...

Let $F$ be a local field with finite residue characteristic $p$, let $C$ be an algebraically closed field of characteristic $p$, and let $\mathbf G$ be a connected reductive $F$-group. With Abe, Henniart, Herzig, we classified irreducible admissible...

I will give examples and motivations, about the local systems/Higgs bundles correspondence, the case of variations of Hodge structures and the case of irregular singularities. I hope this will help to enjoy the forthcoming lectures of T. Mochizuki...

Harmonic bundles are flat bundles equipped with a pluri-harmonic metric. They are very useful in the study of flat bundles on complex projective manifolds. Indeed, according to the fundamental theorem of Corlette, any semisimple flat bundle on a...

Harmonic bundles are flat bundles equipped with a pluri-harmonic metric. They are very useful in the study of flat bundles on complex projective manifolds. Indeed, according to the fundamental theorem of Corlette, any semisimple flat bundle on a...

Let $\mathcal G$ be a split reductive group over a $p$-adic field $F$, and $G$ the group of its $F$-points.The main insight of the local Langlands program is that to every irreducible smooth representation $(\rho, G, V )$ should correspond a...

We will explain how the circle method can be used in the setting of thin orbits, by sketching the proof (joint with Bourgain) of the asymptotic local-global principle for Apollonian circle packings. We will mention extensions of this method due to...

Markoff triples are integer solutions of the equation $x^2+y^2+z^2 = 3xyz$ which arose in Markoff's spectacular and fundamental work (1879) on diophantine approximation and has been henceforth ubiquitous in a tremendous variety of different fields...

We report on some recent work with Peter Sarnak. For integers $k$, we consider the affine cubic surfaces $V_k$ given by $M(x) = x_1^2 + x_2 + x_3^2 − x_1 x_2 x_3 = k$. Then for almost all $k$, the Hasse Principle holds, namely that $V_k(Z)$ is non...

For integer parameters $n \geq 3$, $a \geq 1$, and $k \geq 0$ the Markoff-Hurwitz equation is the diophantine equation \[ x_1^2 + x_2^2 + \cdots + x_n^2 = ax_1x_2 \cdots x_n + k.\] In this talk, we establish an asymptotic count for the number of...

The classical affine cubic surface of Markoff has a well-known interpretation as a moduli space for local systems on the once-punctured torus. We show that the analogous moduli spaces for general topological surfaces form a rich family of log Calabi...

Let M be a compact manifold with negative curvature. The Quantum Unique Ergodicity conjecture of Rudnick and Sarnak says that eigenfunctions of the Laplacian on M get equidistributed as the eigenvalue tends to infinity. A weaker version, called...