Seminars Sorted by Series

Isoperimetric problems are ubiquitous in mathematics. We shall discuss some proved and some conjectural ones in symplectic geometry, together with applications to other areas of mathematics.

All finite graphs satisfy the two properties mentioned in the title. I will explain what I mean by this, and speculate on generalizations and interconnections.

We'll take a glance at the world of mathematics as viewed through the Univalent Foundations of Voevodsky. In it, "set" and "proposition" are defined in terms of something more fundamental: "type". The formal language fulfills the mathematicians'...

A basic open problem in symplectic topology is to classify Lagrangian tori in a given symplectic manifold. In recent years, ideas from mirror symmetry have led to the realization that even the simplest symplectic manifolds (eg. vector spaces or...

I will give an informal introduction to the positive Grassmannian, including its cell decomposition and its connection to cluster algebras.

I will review some theorems and conjectures about the structure of random permutations which arise in statistical mechanics. Conjectures about the cycle structure are related Bose-Einstein condensation and to universality of Wigner-Dyson statistics...

I propose to discuss classical geometric group theory, and its potential extension to homeomorphisms groups suggested recently by Kathryn Mann and Christian Rosendal.

We will describe a new construction of filtration on categories. Applications to classical questions in geometry and group theory will be discussed.

Considering some special examples as algebras of quivers I will give an informal introduction to a field of geometric realizations of noncommutative and derived varieties.

Geometers since Morse are interested in Morse Theory on the free loop space $LM$ of a Riemannian manifold $M$, because the critical points of the energy function on $LM$ are the closed geodesics on $M$. I will discuss an observed symmetry of the...

String topology, invented by Chas and Sullivan in their eponymous 1999 paper, can be viewed as a systematic study of the structure of spaces of free loops and strings on manifolds with emphasis on two basic operations: concatenation and splitting. I...

I will first give an outline of Lin-Friis-Rordam’s proof of the fact that almost commuting matrices are close to commuting matrices uniformly in the dimension. The proof is short and beautiful, but it involves an infinite-dimensional argument which...

In arithmetic geometry, there are lots of examples of natural density zero sets of primes raised from the geometry of elliptic curves or more generally abelian varieties. One may ask whether such thin set is finite or not. For example, given any two...

The Weil pairing is a bilinear form associated to an algebraic curve. I will tell you about it and why it is interesting to cryptographers. Then I'll talk about my (completely unsuccessful) attempts to make an interesting trilinear analogue.

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.