Seminars Sorted by Series

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...

In this talk I’ll consider the Schrödinger in heterogeneous media and shall discuss long-time transport properties of localized initial conditions, and more precisely the regime of asymptotic ballistic transport. To this aim I’ll introduce an...

I will describe the problem of finding the homotopy type of the space of quantum Hamiltonians in dimension d under certain constraints. The common assumptions are that the interactions have finite range and that the ground state is separated from...

For the Bargmann--Fock field on $R^d$ with $d > 2$, we prove that the critical level $l_c(d)$ of the percolation model formed by the excursion sets $\{ f \ge l \}$ is strictly positive. This implies that for every l sufficiently close to 0 (in...

Given a finite graph, the arboreal gas is the measure on forests (subgraphs without cycles) in which each edge is weighted by a parameter $\beta>0$. Equivalently this model is bond percolation conditioned to be a forest, the independent sets of the...

New kinds of vortex sheets with vorticity confined to the boundary layer are proposed and investigated in detail. Exact solutions of the steady Navier-Stokes equations for a planar vortex sheet in arbitrary background strain are found in terms of...

Given a polynomial in one variable, what is the simplest formula for the roots in terms of the coefficients? Hilbert conjectured that for polynomials of degree 6,7 and 8, any formula must involve functions of at least 2, 3 and 4 variables...

In this talk I will describe a topological approach to some problems about algebraic functions due to Klein and Hilbert. As a sample application of these methods, I will explain the solution to the following problem of Felix Klein: Let $\Phi_{g,n}$...

A hiker holding an analog altimeter (with a needle indicating altitude modulo a constant $\chi$) in one hand and a compass in the other traces an {\em altimeter-compass ray (ACR)} by walking in such a way that the altimeter and compass needles are...

The infinity Laplacian (informally, the "second derivative in the gradient direction") is a simple yet mysterious operator with many applications. "Tug of war" is a two player random turn game played as follows: SETUP: Assign each player one of two...

In 1991 a new type of transfer to the Moon was operationally demonstrated by the Japanese spacecraft, Hiten, using ballistic capture. It was designed by this speaker and James Miller. This is capture about the Moon which is automatic so that no...

In this talk I will recall the statement of the Friedlander conjecture on the cohomology of a reductive group G and give a short recollection on known results. I will present a new approach which relies on the study of the A^1 homotopy type of the...

This talk will give a more detailled account of the ``basic" results in A^1-homotopy theory which are used in our approach: Hurewicz theorem, Lower central series spectral sequence, A^1-derived functors of nonadditive functors, Eilenberg-Moore...

This talk will give a more detailled account of the ``basic" results in A^1-homotopy theory which are used in our approach: Hurewicz theorem, Lower central series spectral sequence, A^1-derived functors of nonadditive functors, Eilenberg-Moore...

In this talk I will review some of the history, goals, and results of the theory of automorphic forms and Galois representations.