Seminars Sorted by Series

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...

This is a shameless repeat of a Math Conversations I gave about four years ago, and maybe four years before that as well, explaining 2-adic shift registers.

I will introduce a math problem from deep learning regarding the regularization effect of gradient flow dynamics for underdetermined problems.

I’ll introduce the incompressible Euler equations and talk about the solution’s behavior when the vorticity has odd symmetry.

I will survey what is known about the rationality of algebraic varieties, including recent progress and open questions. There will be a surprising connection to whiskey.

Gauge theory studies partial differential equations with a large group of local symmetries, and it is the geometric language to formulate many fundamental physical phenomena. Starting in the 1980s, mathematicians began to unravel surprising...

In the absence of a vaccine, isolation is about the only available means to control an epidemic. I would like to share with everyone some things I learned from a project I worked on a few years ago studying the consequences of delays and...

Through interactions with engineers and computer scientists over the years, including some current visitors at IAS, I have become pretty sold on the idea that machine learning is rich in questions which are interesting to pure mathematicians and...

In 1976 Dennis Sullivan gave an example of a smooth vector-field on a compact (Riemannian) 5-dimensional manifold in which all the orbits are closed but for which there is no upper bound to the length of a closed orbit. (At first this doesn't even...

In the course of this collaboration, both sides learned about the other field; to my surprise, the biologists learned to "speak" some mathematics. Also, when they saw how we approached answering their initial questions, the questions changed. And...

Discrepancy theory tells us that it is possible to partition vectors into sets so that each set looks surprisingly similar to every other. By "surprisingly similar" we mean much more similar than a random partition. Randomized Controlled Trials are...

In the 60s and 70s, there was a flurry of activity concerning the question of whether or not various subgroups of homeomorphism groups of manifolds are simple, with beautiful contributions by Fathi, Kirby, Mather, Thurston, and many others. A...

In 1978, Charles Conley classified all continuous dynamical systems. His theorem, dubbed the "fundamental theorem of dynamical systems" states that the orbits of any continuous map on a compact metric space fall into two classes: gradient-like and...

A great achievement of physics in the second half of the twentieth century has been the prediction of conformal symmetry of the scaling limit of critical statistical physics systems. Around the turn of the millenium, the mathematical understanding...

A growing number of mathematicians are having fun explaining mathematics to computers using proof assistant softwares. This process is called formalization. For instance, together with Kevin Buzzard and Johan Commelin, I recently formalized enough...

In this talk I will first recall three classical theorems in the theory of finite dimensional Hamiltonian systems, then I will use the periodic nonlinear Schrodinger equation as an example of an infinite dimensional Hamiltonian system and I will...

There is a rich interplay between the fields of knot theory and 3- and 4-manifold topology. In this talk, I will describe a weak notion of equivalence for knots called concordance, and highlight some historical and recent connections between knot...

The reversibility paradox is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics. A first result reconciling the fundamental microscopic physical processes (with time reversal symmetry) and...

We review a "Weyl law" in embedded contact homology, relating periods of orbits of the Reeb vector field on a contact three-manifold to volume. (This was also mentioned in the talk by Dan Cristofaro-Gardiner.) We explain a clever argument by Kei...

In 2016, Ellenberg and Gijswijt made a breakthrough on the famous cap-set problem, which asks about the maximum size of a subset of \mathbb{F}_3^n not containing a three-term arithmetic progression. Ellenberg and Gijswijt proved that any such set...

This is a joint work with Piermarco Cannarsa and Wei Cheng. Most of the lecture is about the distance function to a closed subset in Euclidean subset, at the level of a beginning graduate student. If $A$ is a closed subset of the Euclidean space $...

Cryptography and Machine Learning have shared a curious history: a scientific success for one often provided an example of an impossible task for the other. Today, the goals of the two fields are aligned. Cryptographic models and tools can and...

I will discuss a little about the context and solution of the rectangular peg problem: for every smooth Jordan curve and rectangle in the Euclidean plane, one can place four points on the curve at the vertices of a rectangle similar to the one given...

The Mumford-Shah functional has been introduced by Mumford and Shah in 1989 as a variational model for image reconstruction. Since then, it has been widely studied both from a theoretical and an applied point of view. In this talk we will focus on...

In 1979, Kaufman constructed a remarkable surjective Lipschitz map from a cube to a square whose derivative has rank $1$ almost everywhere. In this talk, we will present some higher-dimensional generalizations of Kaufman's construction that lead to...

It turns out that certain additive patterns in the integers are very hard to get rid of. An instance of this is captured in a conjecture of Erdős, which states that as long as a set of natural numbers is 'somewhat dense' -- namely the sum of the...

Artificial intelligence or "deep learning" is becoming ubiquitous in new fields of mathematical applications stemming from the internet economy. This has led to the creation of powerful new tools. We would like to explore how these techniques can be...

Several of the most important problems in combinatorial number theory ask for the size of the largest subset of some abelian group or interval of integers lacking points in a fixed arithmetic configuration. One example of such a question is, "What...

Let $C$ be an algebraic curve over the rational numbers, that is, a 1-dimensional complex manifold that is defined by polynomial equations with rational coefficients. A celebrated result of Faltings implies that all algebraic points on $C$ come in...

Hyperbolic polynomials are a multivariate generalization of real-rooted polynomials that originated in the study of partial differential equations and have since found applications in many other fields, including operator theory, optimization, and...

In high dimensions, what does it look like when we take the intersection of a set of random half-spaces with either the sphere or the Hamming cube? This is one phrasing of the so-called perceptron problem, whose study originated with a toy model of...