Seminars Sorted by Series

In spin systems, the existence of a spectral gap has far-reaching consequences. "Frustration-free" spin systems form a subclass that is special enough to make the spectral gap problem amenable and, at the same time, broad enough to be physically...

There is a very short proof that a graph is 3-colorable: you simply give the coloring - it is linear in the size of the graph. How long a proof is needed that a given graph is *not* 3-colorable? The best we know is exponential in the size of the...

In this (short and informal) talk I will present the three fundamental factors that determine the quality of a statistical machine learning algorithm. I will then depict a classic strategy for handling these factors, which is relatively well...

Unfortunately counting prime numbers is hard. Fortunately, we can cheat by counting 'approximate prime numbers' which is much easier. Moreover, this allows us to say something about the primes themselves, and works in situations which seem well...

Constructive vs Pure Existence proofs have been a topic of intense debate in foundations of mathematics. Constructive proofs are nice as they demonstrate the existence of a mathematical object by describing an algorithm for building it. In computer...

The topology of the zero set and nesting properties of a random homogeneous real polynomial of large degree has a universal behavior depending only on the dimension. We discuss this and an apparent relation to super-critical percolation in...

For a formal group law G the group of automorphisms Aut(G) acts on the space of deformations Def(G). The invariants of this action miraculously recover an object of huge interest to algebraic topologists, and this connection led to much progress in...

The classical Liouville theorem claims that any positive harmonic function in $R^n$ is a constant function. Nadirashvili conjectured that any non-constant harmonic function in $R^3$ has a zero set of infinite area. The conjecture is true and we will...

We will describe several situations in number theory and geometry in which one recovers a sought-after structure by first constructing a “random” approximation to it.

The Univalent Foundations of Voevodsky offer not only a formal language for use in computer verification of proofs, but also a foundation of mathematics alternative to set theory, in which propositions and their proofs are mathematical objects, and...

Our object of interest will be tuples matrices over a field.

I will explain how different views of this object by diverse fields of mathematics give rise to important questions in these areas, which turn out to be surprisingly tightly connected...

Harmonic measure of a portion of the boundary is the probability that a Brownian traveler starting inside the domain exits through this portion of the boundary. It is also a simplest building block of any harmonic function in a domain. Some...

What do you do with a person who behaves in the worst possible way at every point in time? Well, I don’t know. But if you ask instead about an operator that picks out the worst possible behavior of a function, we sometimes know how to control it. We...

In addition to offering a formal system for doing ordinary (or "set-level") mathematics, Vladimir Voevodsky’s Univalent Foundations also suggest a new way of studying homotopy theory, called "synthetic homotopy theory".

I will show how synthetic...

The end of the previous century saw radical changes to three-dimensional topology, which arose from two completely different approaches. One breakthrough came from Thurston's introduction of hyperbolic geometry into the field. The second one came...

Hyperbolic 3-manifolds with arithmetic fundamental group exhibit many remarkable number theoretic properties. Is it possible that such manifolds live over finite fields (whatever that means)? In this talk I will give some evidence for this...

I will consider a very simple open problem in the theory of ODEs and give a brief overview on what is known about it. The problem is also an excuse to talk about a widely open subject in modern PDEs.

We will recall the central limit theorem for random numbers, and then discuss the general principle of universality and what it might mean specifically in an analog of the central limit theorem for random groups.

The multivariate Tutte polynomial, known to physicists as the Potts-model partition function, can be defined for any finite graph. The function has a hidden convexity property that implies several nontrivial results concerning the combinatorics of...

An atom is made of a positively charged nucleus and negatively charged electrons, interacting with each other via Coulomb forces. In this talk, I will review what is known, from a mathematical perspective, about this paradigmatic model, with a...

The isoperimetric inequality says that balls have the smallest perimeter among all sets of a fixed volume in Euclidean space. We give an elegant analytic proof of this fact.

The uncertainty principle says that a function and its Fourier transform can not be well-localized simultaneously. We will first discuss a version of this statement for a collection of functions forming a basis for $L^2$ space. Then we will connect...

In 1984, Robert Lazarsfeld solved an old conjecture of Remmert and Van de Ven, which stated that there are no non-trivial complex manifolds that can be covered by a projective space. His result was a consequence of Shigefumi Mori's breakthrough...

In 2004 Jean Bourgain proved, with Netz Katz and Terry Tao, the "sum-product theorem in finite fields". He referred to this result (and proof technique) as a "goose which lays golden eggs". Indeed, in subsequent years, he has published a couple of...

Understanding the mechanics of crumpling, i.e. of isotropically compressing thin elastic sheets, is a challenging problem of theoretical and applied interest. We will present an interesting conjecture on the order of magnitude of the elastic energy...

The PCP Theorem states that any mathematical proof can be encoded in a way that allows verifying it probabilistically while reading only a small number of bits of the (new) proof. This result has several applications in Theoretical Computer Science...

Given a domain, the harmonic measure is a measure that relates any boundary function to its harmonic extension; it is also the hitting probability of the boundary for a Brownian motion moving inside the domain. We will talk about the relationship...

The Ax–Grothendieck theorem from the 1960s says that an injective polynomial $f \colon \mathbb{C}^n \rightarrow \mathbb{C}^n$ is also surjective. It is one of the first examples of the powerful technique in algebraic geometry of using finite fields...

Solutions to some differential equations are related to geometric structures on the underlying manifold. For instance certain hypergeometric equations are related to the uniformization of Riemann surfaces. I will start by recalling some classical...

The study of the planar-circular-restricted 3-body problem led to Poincaré's "last geometric theorem", nowadays known as the Poincaré-Birkhoff theorem. It is a fixed point theorem for certain area-preserving annulus homeomorphisms. Birkhoff's proof...

The idea of corrugation goes back to Whitney, who proved that homotopy classes of immersed curves in the plane are classified by their rotation number. Generalizing this result, Smale and Hirsch proved that the space of immersions of a manifold X...

This is a light survey of the origins of contact topology and its applications to dynamics. We will use anecdotes and images to illustrate ideas.

I will construct a family of curves in the square that illustrates the interplay between hyperbolic dynamics and pathology.

An unlikely couple devised one of the first spread spectrum communication systems. Today these systems use sophisticated mathematics and are ubiquitous. This is a verbatim repeat (by popular demand) of a talk I gave about 6 years ago.

When designing the first computer built at IAS, von Neumann rejected floating-point arithmetic as neither necessary nor convenient. In 1997 William Kahan at Berkeley, who designed the famously accurate algorithms on Hewlett-Packard calculators, said...

The Ax-Grothendieck theorem from the 1960s says that an injective polynomial $f : \mathbb C^n \to \mathbb C^n$ is also surjective. It is one of the first examples of the powerful technique in algebraic geometry of using finite fields to prove...

Percolation is a simple model for the movement of liquid through a porous medium or the spread of a forest fire or an epidemic: the edges of some graph are declared open or closed depending on independent coin tosses, and then connected open...

I will describe the construction and applications of Khovanov homology, a combinatorially defined invariant for knots that categorifies the Jones polynomial.

Optimal transport has been used to have new insights on a variety of mathematical questions, ranging from functional inequalities to economics. We will discuss some of the unexpected uses of optimal transport, as a simple proof of the isoperimetric...

Given $N$ distinct points on the plane, what's the minimal number, $g(N)$, of distinct distances between them? Erdős conjectured in 1946 that $g(N)\geq O(N/(log N)^{1/2})$. In 2010, Guth and Katz showed that $g(N)\geq O(N/log N)$ using the...

A main theme in extremal combinatorics is about asking when the random construction is close to optimal. A famous conjecture of Erd\H{o}s-Simonovits and Sidorenko states that if $H$ is a bipartite graph, then the random graph with edge density $p$...

To understand human memory one needs to understand both the ability to acquire vast amounts of information and at the same time the limited ability to recall random material. We have recently proposed a model for recalling random unstructured...

In the 1960's, Quillen found a remarkable relationship between a certain class of cohomology theories and the theory of formal groups. This discovery has had a profound impact on the development of stable homotopy theory. In this talk, I'll give a...

In this talk, we will discuss some of the recent advances in high-dimensional robust statistics. In particular, we will focus on designing faster and simpler robust algorithms for fundamental statistical and machine learning problems.

I will talk about an approach to proving exponential mixing for some kinetic, non-diffusive stochastic processes, that have recently become popular in computational statistics community.

I will explain the source of Grothendieck philosophy of motives, and tell of applications.

The goal of the talk is to explain the statement and proof of a beautiful result due to Margulis (1967) later extended by Plante and Thurston (1972) that imposes restrictions on the growth of the fundamental group of 3-manifolds that support Anosov...