Seminars Sorted by Series

Many problems in classical topology can be formulated as decision problems, with yes/no answer and an algorithm as a solution. While such problems often appear to be intuitively hard, we still know little about lower bounds on their algorithmic...

In algebraic combinatorics there well known objects called q-integers, q-binomial coefficients, and q-factorials which for lack of a better word "q-ify" the usual integers, binomial coefficients, and factorials. I will explain these notions (and say...

I will speak, broadly, on, the arithmetic and geometry of quaternion algebras and aspects of the spectral theory of automorphic laplacians. I will describe spectral correspondences between spaces of Maass waveforms on the unit group of a quaternion...

In 1895, Hill discovered a $1$-parameter family of tetrahedra whose dihedral angles are all rational multiples of $\pi$. In 1976, Conway and Jones related the problem of finding all such tetrahedra to solving a polynomial equation in roots of unity...

We touch lightly on the background of four mathematicians over four centuries whose names are famous in mathematics with my personal emphasis on fluid dynamics.

The main object of interest in this talk will be a system of many particles, modeled using the prescriptions of quantum mechanics. A significant challenge to studying such systems is that particles interact with each other, via weak or strong...

Embedded Contact Homology (ECH) is a Floer type invariant due to Hutchings. This talk provides a gentle overview of ECH (in part through a video of the Hopf fibration) and sketches why ECH of a prequantization bundle over a Riemann surface is...

We'll give several concrete examples of how to go from the geometry of surfaces to the study of point processes, following work of Siegel, Veech, Masur, Eskin, Mirzakhani, Wright, and others. We'll discuss how this "probabilistic" perspective helps...

Which spaces can be the fixed point sets of actions of $G$ on finite cell-complexes of a given homotopy type? The general answers to such questions, for $G$ not a group of prime-power order, will be expressed, even for non-simply-connected spaces...

This talk will explain work stemming from a group project that investigated the ellipsoidal embedding capacity function for the family of Hirzebruch surfaces. This problem turns out to have unexpected arithmetic structure, leading to an intricate...

An exceptionally gifted mathematician and an extremely complex person, Floer exhibited, as one friend put it, a “radical individuality.” He viewed the world around him with a singularly critical way of thinking and a quintessential disregard for...

I planned to give a different talk, about recent work I am excited about. But then Helmut asked me to instead talk of "Dreams of mathematics and computer science". And his wish is my command...

I'll describe, mainly through works of some great...

In this talk we discuss two models of a discrete random surface. The first is a Markov process, like a simple random walk, under which the surface is grown according to random updates. The second chooses the surface uniformly at random, after...

Originating in the work of Hadamard in the 1890’s on the coding of geodesic flow, symbolic dynamics has become a key tool for studying topological, smooth, and measurable dynamical systems. The automorphism group of a symbolic system capture its...

In several diverse settings (variational problems and geometric flows, elliptic, parabolic, but also some dispersive PDEs) monotonicity formulas allow to get a first coarse description of singularities, which are commonly called tangent cone. Their...

Convexity plays a central role in several geometric and dynamical problems in symplectic geometry. However, convexity is not preserved under structure preserving isomorphisms and it is unknown whether there exists an intrinsic property responsible...

Gaussian elimination is one of the oldest and most popular techniques for factoring a matrix. The growth of entries in Gaussian elimination is an important practical problem. Modern results as well as practice show that entry growth is not a...

I will give an elementary introduction to the connections between diffusions and stochastic characteristics in $\mathbb R^n$. I will then explain how one might think about what it means to be elliptic or hypoellipticity in an infinite dimensional...

I will remind the audience of Noether’s theorem in the calculus of variations and give a little of the history. An elementary application to integrals of Lagrangians defined on functions with domain a hyperbolic surface will be given, ending with a...

The study of nodal sets, i.e. zero sets of eigenfunctions, on geometric objects can be traced back to De Vinci, Galileo, Hook, and Chladni. Today it is a central subject of spectral geometry. Sturm (1836) showed that in 1D, the $n$-th eigenfunction...

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.