Seminars Sorted by Series

I will discuss some open questions about the relation between Stein and Weinstein structures.

The harmonic measure is an important tool, which allows one to reconstruct a harmonic function from its values on the boundary. But it also admits a very simple and beautiful probabilistic interpretation: it is the probability that the path of the...

I will introduce an extremely natural model for random hyperbolic surfaces and discuss how little we know about it in large genus.

The statement that general relativity is a deterministic theory finds its mathematical formulation in the Strong Cosmic Censorship conjecture due to Roger Penrose.  I will introduce the conjecture and report on some recent progress.

Let G be a group, and let (g,h) be a pair in G x G. Consider the group of symmetries of G x G generated by the "moves" sending (g,h) to (g,gh), (g,g^{-1}h), (g,hg), (g,hg^{-1}), (gh,h),...etc. An old question from the 50's, motivated by the study of...

Bieberbach's 1912 theory of Euclidean crystallographic groups provides a satisfying qualitative classification of flat Riemannian manifolds. In 1977 Milnor asked whether a similar picture could extend to flat affine manifolds, that is, when the...

Symmetric polynomials are often characterized as characters of modules over Lie algebras. Such characters are symmetric as they are invariant under the action of the Weyl group. In the "super case", this group generalizes to the Weyl groupoid. We...

In 2017, Miller conjectured, based on computational evidence, that for any fixed prime $p$ the density of entries in the character table of $S_n$ that are divisible by $p$ goes to $1$ as $n$ goes to infinity.  K. Soundararajan and I proved this...

https://www.nasa.gov/feature/how-dark-matter-could-be-measured-in-the-solar-system

In 1850, Liouville proved a rather surprising fact: any $C^{3}$ conformal map in a three-dimensional domain is a Möbius transformation; this is in stark contrast with the two-dimensional case, where conformal maps abound. Since then, Liouville's...

Undergraduate mathematicians are taught Hilbert's dream that theorems should be built up from a solid axiomatic base, and that the whole structure of mathematics is (or should be) a solid verifiable whole. However, this is rather far from how...

In 1939, Mahler asked whether the product of the volumes of a centrally symmetric convex body and its polar is minimized by a cube. He gave a positive answer to this question in dimension 2. In this talk I will explain how this is related to...

Suppose you have an approximate homomorphism from an Abelian group A to Hom(V, W); is it close to a genuine homomorphism ?
This question can be asked with various different notions of “close”. I will describe one that arises in the context of higher...

I've been working a lot on "random surfaces" in recent years.  These are "canonical" random fractal Riemannian manifolds (just as Brownian motion is a canonical random fractal curve) and they come up in many areas of physics and mathematics.  In a...

Information geometry studies the mathematical properties of probabilistic models. Classically, we view the parameter space of a model as a Riemannian manifold, and use tools from differential geometry to study properties of the parameterized class...

Scalar curvature geometry is characterized by remarkable extremality and rigidity properties due to minimal hypersurfaces on the one hand and harmonic spinor fields on the other. Are there hidden connections between these viewpoints? We do not know...

The mapping class group of a surface is a very important, but still mysterious group. Natural actions of the mapping class group appear on representation varieties of surface groups. In some cases, e.g. when this action preserves a metric, we know...

In dimension two, Urs Lang and Mario Bonk proved that a surface, homeomorphic to the plane, is bi-Lipschitz to the Euclidean space if its total Gauss curvature is smaller than that of the hemisphere. In this talk, I will explain what is known in...

We discuss some questions that arise when studying rational and integral points on curves, especially elliptic curves. For example, for a "random" such curve, how many rational points should it have? This will be a talk suitable for a general math...

Abstract: In 2010, Larry Guth wrote a beautiful essay "Metaphors in systolic geometry", where he poetically described several approaches to Gromov's celebrated systolic inequality. A nontrivial special case of this inequality claims that a...

In the second half of the 19th century, it was discovered that algebra and geometry had nothing to do with each other. I will discuss this fact and some consequences.

In this talk, I will discuss a possible symplectic version of Smale's conjecture on diffeomorphism groups. We will provide some evidence for it and suggest some preliminary questions about complex hyperbolic manifolds to explore. 

In theoretical computer science, we often aim to prove lower bounds and demonstrate the computational hardness of solving certain problems. However, some of these "negative" results can be directly applied to cryptography, to base the security of...

Click here for abstract details. 

As telegraph lines proliferated through Europe and North America in the 1850s, plans were drawn up for a transatlantic telegraph cable.  Extended telegraph lines were modelled by William Thomson (Lord Kelvin), who showed that a transatlantic cable...

In 1971, Martin Gardner proposed a deceptively simple problem about 'kissing cubes' in his Mathematical Games column in the Scientific American, and more than three decades later it is still unsolved. In this talk I will introduce the problem, the...

Classical probability theory is set up to handle random variables whose values are a single complex number. What happens if our random variable is instead a multi-set of complex numbers? For example, on the group of n by n orthogonal matrices, you...

In Récoltes et Semailles, Grothendieck explains that, exactly once in his life, doing math had become painful for him. It was at the end of the analytic part of his career, when he was obsessed by the approximation problem. I will explain what this...

The purpose of this talk is to ask a single question: what is the correct definition of intersection theory on varieties? Join me, as we travel through space and time from the ancient origins of enumerative geometry, through Fulton-MacPherson's work...

The classical Birkhoff individual ergodic theorem states that in the presence of an ergodic invariant measure, almost every orbit is uniformly distributed with respect to the measure. For many applications (in particular to number theory), it is...

The permanent and determinant are polynomial functions of the entries of a matrix, differing only in the signs of their monomials. Despite their apparent similarity, these polynomials play very different roles in mathematics and computer science...

In recent years, there has been much work on nonlinear wave equations with random initial data. Most of this work has focused on the behavior of such nonlinear waves on small scales. In this talk, I will pose a problem concerning the behavior on...

What do graph matchings and independent vertex sets have to do with the cohomology of products of projective lines? I will share with you an example in the study of “equivariant log-concavity”, which enriches the notion of log-concavity. By keeping...

A matrix M is rigid if one needs to change it in many places in order to reduce its rank significantly. While a random matrix M (say over a finite field) is rigid with high probability, coming up with explicit constructions of such matrices is still...

Cubic forms are homogeneous polynomials of degree 3 in several 
variables. Number theory is interested in their zeros over the rational 
numbers. Algebraic geometry studies the cubic hypersurfaces defined by 
them (e.g., the 27 lines on smooth cubic...

We discuss a modern perspective on the winding number on $S^1$ for maps that may not be continuous. This reveals a surprising connection to Fourier analysis and motivates the question: is the winding number determined by the moduli of the Fourier...

Consider a right angled cylinder. Glue the ends together after twisting many times to form a flat torus $C^1$-isometrically embedded in $R^3$. What can we say about the global geometry of this embedding?

There is a remarkable parallel, first explicitly enunciated in the 1960s, between algebraic number theory and 3-dimensional geometry; for example, prime numbers are considered analogous to knots. I will only say a few short words about the substance...

Some people think that the brain is something like a (conscious) computer. But if a brain can compute, why can't a rock, or a river stream? This basic question has been considered by philosophers, physicists, and mathematicians.

It is not entirely...

The Alexandrov-Fenchel inequality---the fundamental log-concavity phenomenon in convex geometry---arose from Minkowski's work in number theory in the late 1800s. It has resurfaced in surprising ways throughout the 20th and 21st centuries in the...

You are swimming at the center of a circular lake with a bear waiting on the shore. The bear, unable to swim, moves four times faster on land than you do in water, but once on land, you can outrun it. Can you escape?

This classic riddle has been...

The Ricci curvature of a Riemannian manifold is best viewed as the right replacement for the (nonlinear) laplacian of the metric g, which in particular explains why it so often appears in geometry and analysis.  Most commonly one studies either...