Seminars Sorted by Series

I begin with a geometric discussion of Wigner's theorem concerning the linearization of quantum mechanical symmetries; it first appeared in a joint paper with von Neumann. This is the starting point for joint work with Gregory Moore in which we...

Spectral curves appear in many integrable systems. Their quantum cousins are equally ubiquitous and describe among others random matrices, the topology of Riemann's moduli space, and hyperbolic knots. Quantum curves are the simplest example of what...

We will present the ABC conjecture, Belyi's mapping theorem and explain how they combine into a powerful tool for diophantine problems, following the ideas of Elkies, Bombieri, Granville, Langevin. Finally we will speculate a bit about the function...

The Penrose tiling (Roger Penrose(1974)) and the "quasi-crystal" made by Ron Schactman (1985) are beginning landmarks here. Our objects today are tilings $T$, of $\mathbb R^d$, [$d = 1, 2$ mostly] which like Penrose's is aperiodic and can be a...

The classification of finite simple groups is a singular event in the history of mathematics. It has one of the longest and most complicated proofs any theorem (indeed just to define the terms in the statement of theorem requires a lot). It has many...

Thurston realized that certain link complements admit a complete hyperbolic metric, which is a complete invariant of the manifold. We'll discuss the volumes of hyperbolic link complements and what is known about them and open questions.

Powerful theorems of Thurston, Perelman, and Mostow tell us that almost every 3-dimensional manifold admits a hyperbolic metric, and that this metric is unique. Thus, in principle, there is a 1-to-1 correspondence between a combinatorial description...

Chromatic homotopy theory is the philosophy that homotopical phenomena should be understood via the periodicities they exhibit. Equivalently, it's the viewpoint that every prime number p hides an infinite hierarchy of "chromatic primes" of...

In 1900 Hilbert asked for a decision procedure to determine solvability of polynomial equations over the integers. Seventy years later, Y. Matiyasevich showed that this problem is unsolvable, building on earlier work of M. Davis, H. Putnam and J...

All 3-manifolds can be decomposed into two simple objects, or handlebodies. Some manifolds have many such decompositions, which are distinct. All, however, are related by the operation of stabilization and destabilization. Given two decompositions...

Computer algebra systems are large software systems and as such they have bugs. A recent issue of the Notices of the AMS features the article "The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them?" in...

"Hyperbolic" ranks highly among the most-abused terms in dynamics. I'll prolong this abuse, and argue for its value, by illustrating a variety of dynamical systems with distinct forms of hyperbolic behavior that have known or conjectured...

One of the outstanding insights obtained by physicists working on "Quantum Chaos" is a conjectural description of local statistics of the spectrum of the Laplacian on a Riemannian surface according to crude properties of the dynamics of the geodesic...

Although the existence of a totally geodesic surface in a finite volume hyperbolic 3-manifold is "rare", when they do exist, their presence seems to have an impact on the geometry and topology of the hyperbolic 3-manifold, as well as number...

Given a smooth knot in the 3-sphere, there are many topologically distinct smooth surfaces in the 4-ball that have this knot as its boundary. However, if the knot is Legendrian, meaning that it satisfies a geometrical condition imposed by a contact...

The $p$-adic numbers are finally entering the realm of engineering. I will give several examples of how they arise in applications.

Schubert's Wandererfantasie is one of the most monumental and revolutionary piano pieces ever composed. I will highlight some of the structural novelties that were adapted later by Liszt, Wagner, Franck and others, and I will play the piece.

The (derived) Fukaya category is a symplectic invariant developed out of work of Gromov, Floer, Fukaya, Kontsevich and Seidel that encodes the rigidity properties of Lagrangian intersections. The purpose of the talk is to discuss the construction of...

A covering space $C \\to M$ is classified by a subgroup of the fundamental group of $M$. If we refuse to choose a basepoint, then $C \\to M$ is equivalent to a functor from the fundamental groupoid of $M$ to $\\mathsf{Set}$. Suppose $(M,S)$ is a...

I will try my best to make homological mirror symmetry into a tautology. The tool to do so is Family Floer cohomology, which produces a "mirror space" from a given Lagrangian torus fibration. Mirror symmetry thus gets reduced to the geometric...

In broad terms, a phase transition is a variation in the qualitative behavior of a system under changes of some parameter. For instance, as the temperature is changed, water goes through a gaseous, a liquid, and several solid phases, each of which...

This talk recalls how Gromov's classic non-squeezing theorem from symplectic geometry was first conjectured, based on a connection between eigenvalue problems from PDE and the uncertainty principle from elementary quantum mechanics.

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.