Seminars Sorted by Series

Semisimple Lie groups seem to be very rigid objects. In arithmetic situations the fact that a semisimple group may degenerate into a non-semisimple one in a family is well known. In geometric situations this phenomenon has not been applied that much...

I will report on a joint work with Ronny Hadani (UT Austin) and Amit Singer (Princeton). We give an algorithmic solution to the two problems that appear in the title. The input is the numerical data (roughly, random plane sections of the molecule)...

It has been recently observed that function spaces associated to a symplectic manifold exhibit unexpected properties and surprising structures, giving rise to new tools and intuition in symplectic topology. In the talk I shall discuss these...

The talk describes how pseudoholomorphic curve based techniques can be used to understand better the dynamics of autonomous Hamiltonian systems of two degrees of freedom restricted to a compact non-degenerate energy surface.

First I will introduce the notion of Gelfand pair and its connection to invariant functions and distributions on the group, and give some examples. We will start from finite groups and compact groups and then go on to reductive groups over local...

Following the brilliant insight of Riemann, that a good understanding of the distribution of prime numbers is equivalent to a good understanding of the location of zeros of pertinent L-functions, analytic number theory has traditionally centered on...

We establish a necessary and sufficient condition for an action of a lattice by homeomorphisms of the circle to extend continuously to the ambient locally compact group. This condition is expressed in terms of the real bounded Euler class of this...

I'll give a brief survey of Heegaard Floer homology, one of the latest and most powerful descendants of the Gauge theories of the 80's and 90's, and I'll discuss established and conjectured connections between Heegaard Floer homology and Khovanov...

The theory of quantum homology, which originally arose from physics, is currently generating a great deal of interest, due in part to its striking predictions regarding enumerative algebraic geometry. In this talk we will introduce the quantum...

I'll discuss connections between the distribution of zeros and values of $L$-functions, such as the Riemann zeta function, and of characteristic polynomials of matrices from the classical compact groups. Very little background will be assumed and...

In an autonomous Hamiltonian dynamical system the dynamics evolves on an energy hypersurface by preservation of energy. Thus the energy hypersurface is foliated by the flow. This is no longer true if the system is perturbed. It is a challenging...

In this talk, we will give explicit examples of Langlands correspondence for reductive groups over the rational function field $F=k(t)$ . Fixing appropriate local ramifications, it is sometimes possible to write down explicit Hecke-eigenforms using...

Discrete problems have a habit of being beautiful but difficult. This can be true even of discrete problems whose continuous analogues are easy. For example: computing the surface area of a sphere of radius N^{1/2} in k-dimensional Euclidean space...

In this talk, I will explain my joint work with Junecue Suh on when and why the cohomology of Shimura varieties (with nontrivial integral coefficients) has no torsion, based on certain vanishing theorems we have proved recently. (All conditions...

I will present a recent joint work with Paul Bourgade (Paris) about the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices and matrices from the Gaussian...

I will introduce l-adic representations and what it means for them to be automorphic, talk about potential automorphy as an alternative to automorphy, explain what can currently be proved (but not how) and discuss what seem to me the important open...

Given a Hamiltonian on $T^n\times R^n$, we shall explain how the sequence of suitably rescaled (i.e. homogenized) Hamiltonians, converges, for a suitably defined symplectic metric. We shall then explain some applications, in particular to symplectic...

The systolic inequality says that if we take any metric on an n-dimensional torus with volume 1, then we can find a non-contractible curve in the torus with length at most C(n). A remarkable feature of the inequality is how general it is: it holds...

This will be an introduction to special value formulas for L-functions and especially the uses of modular forms in establishing some of them -- beginning with the values of the Riemann zeta function at negative integers and hopefully arriving at...

I will introduce Shimura varieties and discuss the role they play in the conjectural relashionship between Galois representations and automorphic forms. I will explain what is meant by a geometric realization of Langlands correspondences, and how...

The "hard discs" model of matter has been studied intensely in statistical mechanics and theoretical chemistry for decades. From computer simulations it appears that there is a solid--liquid phase transition once the relative area of the discs is...

In this expository talk, I will outline a plausible story of how the study of congruences between modular forms of Serre and Swinnerton-Dyer, which was inspired by Ramanujan's celebrated congruences for his tau-function, led to the formulation of...

This talk will be a biased survey of recent work on various properties of elements of infinite groups, which can be shown to hold with high probability once the elements are sampled from a large enough subset of the group (examples of groups: linear...

The local Langlands conjecture (LLC) seeks to parametrize irreducible smooth representations of a p-adic group G in terms of Weil-Deligne parameters. Bernstein's theory describes the category of smooth representations of G in terms of points on a...

Many remarkable questions about prime numbers have natural analogues in the context of elliptic curves. Among them, Artin's primitive root conjecture, the twin prime conjecture, and the Schinzel hypothesis have inspired a broad family of conjectures...

Topological spaces given by either (1) complements of coordinate planes in Euclidean space or (2) spaces of non-overlapping hard-disks in a fixed disk have several features in common. The main results, in joint work with many people, give...

In this talk we will discuss recent progresses meant as a contribution to the GLS-project, the second generation proof of the Classification of Finite Simple Groups (jointly with R. Lyons, R. Solomon, Ch. Parker).

I will explain the main notions of the microlocal theory of sheaves: the microsupport and its behaviour with respect to the operations, with emphasis on the Morse lemma for sheaves. Then, inspired by the recent work of Tamarkin but with really...

Green and Tao [GT] used the existence of a dense subset indistinguishable from theprimes under certain tests from a certain class to prove the existence of arbitrarily longprime arithmetic progressions. Tao and Ziegler [TZ] showedsome general...

The basic ingredients of Darwinian evolution, selection and mutation, are very well described by simple mathematical models. In 1973, John Maynard Smith linked game theory with evolutionary processes through the concept of evolutionarily stable...

I will survey the development of modern infinite cardinal arithmetic, focusing mainly on S. Shelah's algebraic pcf theory, which was developed in the 1990s to provide upper bounds in infinite cardinal arithmetic and turned out to have applications...

I will introduce two basic problems in random geometry. A self-avoiding walk is a sequence of steps in a d-dimensional lattice with no self-intersections. If branching is allowed, it is called a branched polymer. Using supersymmetry, one can map...

https://publications.ias.edu/rpl/paper/452

I will start with a review the basic notions of Hamiltonian/symplectic vector field and of Hamiltonian/symplectic group action, and the classical structure theorems of Kostant, Atiyah, Guillemin-Sternberg and Delzant on Hamiltonian torus actions...

Let f and g be nonlinear polynomials (in one variable) over the complex numbers. I will show that, if there exist complex numbers a and b for which the orbits {a, f(a), f(f(a)), ...} and {b, g(b), g(g(b)), ...} have infinite intersection, then f and...

The modern theory of dynamical systems, as well as symplectic geometry, have their origin with Poincare as one field with integrated Ideas. Since then these fields developed quite independently. Given the progress in these fields one can make a good...

A Lagrangian correspondence is a Lagrangian submanifold in the product of two symplectic manifolds. This generalizes the notion of a symplectomorphism and was introduced by Weinstein in an attempt to build a symplectic category. In joint work with...

It is becoming more and more clear that many of the most exciting structures of our world can be described as large networks. The internet is perhaps the foremost example, modeled by different networks (the physical internet, a network of devices...