Seminars Sorted by Series

We will state a number of problems with completely different origins, and reformulate them in terms of questions about what happens when you multiply integer matrices. (In fancy words, these are called the "Affine Sieve" or a "Local-Global Principle...

We describe a method of Selberg for constructing fundamental domains for discrete groups of \(\mathrm{SL}(n,\mathbb R)\) in terms of trace inequalities. We show how these Selberg domains can be used to play ping-pong, establishing freeness of some...

What is the difference between algebraic and analytic geometry? Is there some way to construct moduli "spaces" in analytic geometry (in the Archimedean or non-Archimedean contexts)? Is there a common language for expressing the foundations of...

Boltzmann defined the entropy, \(S(M)\), of a macroscopic system in a macrostate \(M\) as the "log of the volume of phase space" corresponding to the system being in \(M\). This definition was extended by von Neumann to quantum systems as "the log...

The following questions are quite intimately related. Please consider them before the talk. Some have surprising answers which are highly nontrivial theorems in computational complexity.

• Do you find Tic-Tac-Toe an interesting game? Why?
• Do you...

Starting with a few remarks on hurricanes, I will sketch some basic facts about the physics of the Quantum Hall Effect. Assuming that the longitudinal conductance of a two-dimensional electron gas in a uniform magnetic field vanishes, I will explain...

Projective planes satisfying certain symmetry conditions correspond to non-associative division algebras, by the work of Hilbert, Dickson, Albert, Wedderburn and Veblen. Knuth began the attempt to classify them over finite fields, and very few...

As a prelude to an enjoyable mathematical conversation, I will present both a visual depiction and a combinatorial interpretation of matrix and hypermatrix multiplication. Finally I will discuss how hypermatrix multiplication relates to Sculpting...

Magic sequences were introduced for the first radar measurements of the distance to Venus. They are now used in GPS, satellite and cell phone communications. In many cases their correlation properties can be determined using some crazy ideas of...

One needs no advanced mathematics to understand Borges' stories, but with some mathematical insight is able to see unexpected and nontrivial connections to rigorous math(s). I shall discuss two of those, one combinatorial, the other analytic, and...

I'll give an introduction to some of the new relations between geometry and physics that have arisen in recent years by considering compactifications of "The 6-dimensional (2,0) theory" -- with ties to (and among) instanton counting, Hitchin moduli...

In 1896, Frobenius obtained a remarkable character-theoretic formula for the number of solutions to the equation \(xyx^{-1}y^{-1}=g\), for any finite group \(G\) and element \(g \in G\). While more than a century has since passed, our understanding...

I'll explain a nice way of visualizing the topology of a smooth complex hypersurface in \((\mathbb{C}^*)^n\), by decomposing it into `generalized pairs of pants'. Then I'll explain some useful symplectic constructions arising from this picture, by...

This talk will be an introduction to K.-T. Chen's iterated integrals and to the de Rham theory of homotopy groups. I will give an historical introduction, starting with works of Frank Adams (1956) and John Stallings (1975) that have been lost in the...

Few theorems in the history of mathematics have inspired mathematicians and philosophers as much as Godel’s first and second incompleteness theorems. I will present a new proof for Godel's second incompleteness theorem, based on Kolmogorov...

I will discuss a 1965 conjecture of J. Andrews and M. Curtis---a beguilingly straightforward statement about presentations of the trivial group, which has striking significance in low-dimensional topology: e.g. in relation to the 3-dimensional...

We discuss some problems in group theory, some of which are undecidable. For example, we would like to know whether or not being thin is a decidable property of a group.

I will describe a result of Peter Scholze (reproved a bit later by George Boxer) establishing a remarkable connection between Galois theory and the homology of certain arithmetic hyperbolic 3-manifolds.

I will try to explain why one expects there to be a Galois theory of a certain class of transcendental numbers, called periods, and illustrate with some simple examples.

What is common to the Szemeredi Regularity lemma in graph theory, the Green-Tao result on arithmetic progressions in the primes, the Schapire Boosting algorithm in machine learning and Impagliazzo Hard-Core set theorem in computational complexity...

Introduced by Stallings in '83, core graphs provide a simple and natural combinatorial-geometric approach to the study of free groups. This approach yields simple elementary proofs to classical results and solutions to various algorithmic problems...

One of the central topics in Complex Analysis deals with the issue of identifying natural conditions that uniquely determine a holomorphic function. A prominent role in this regard is played by the one-sided and transmission Riemann-Hilbert problems...

This will be an elementary/intuitive introduction to the $\pi_1$ of smooth algebraic varieties in $\mathbf A^1$-homotopy theory (over algebraically closed fields of char 0). We will give some flavor on the $\pi_0$ and $\pi_1$. Most of the...

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.