Seminars Sorted by Series

String topology, as introduced by Chas and Sullivan 20 years ago, is a product structure on the free loop space of a manifold that lifts the classical intersection product from the manifold to its loop space. I’ll explain how both a product and a...

Earlier this semester we heard a fascinating talk by James Stone describing how the equations of compressible magnetohydrodynamics (MHD) can help us understand the Cosmos. Today we will return to Earth and describe a mathematical model, derived from...

Given the current knowledge of complex representations of finite simple groups, obtaining good upper bounds for their characters values is still a difficult problem, a satisfactory solution of which would have significant implications in a number of...

The honest answer to the question is that I actually do not know. I will therefore rather talk about several famous examples that are widely called "h-principle results" and try to explain some of the ideas behind the ones I am most familiar with.

Several of the most important problems in combinatorial number theory ask for the size of the largest subset of some abelian group or interval of integers lacking points in some arithmetic configuration. One example of such a question is "What is...

Despite the fact that the 3-body problem is an ancient conundrum that goes back to Newton, it is remarkably poorly understood, and is still a benchmark for modern developments. In this talk, I will give a (very) biased account of this classical...

The question in the title was one of the founding questions in symplectic topology 40 years ago, and despite a lot of progress since that time, it remains widely open. In the talk I will discuss the initial questions, the progress, and the remaining...

How does the dimension of the first cohomology grow in a tower of covering spaces? After a tour of examples of behaviors for low-dimensional spaces, I will focus on arithmetic manifolds. Specifically, for towers of complex hyperbolic manifolds, I...

Human learning outstrips modern machine learning and AI in at least three abilities: rapid robust learning, in effectively open worlds, in near-real time with very little energy. Mathematical formalization of signature human abilities has the...

The regularity theory for the Navier-Stokes equation will be reviewed. Motivations from Kolmogorov’s phenomenological theory of turbulence will be discussed. Rigorous mathematical results are obtained to confirm some of the phenomenologies.

The spread of a matrix is defined as the diameter of its spectrum. This quantity has been well-studied for general matrices and has recently grown in popularity for the specific case of the adjacency matrix of a graph. Most notably, Gregory...

I will introduce a new model of randomly agitated equations. I will focus on the finite finite dimensional approximations (analogous to Galerkin approximations) and the two-dimensional setting. I will discuss number of properties of the models...

The classification of geometric structures on manifolds naturally leads to actions of automorphism groups, (such as mapping class groups of surfaces) on "character varieties" (spaces of equivalence classes of representations of surface groups).

Jus...

I will discuss some questions of interest in neuroscience, seen through the lens of mathematics. No prior knowledge of neuroscience is needed for this talk. Two of the most basic visual capabilities of primates are orientation selectivity, i.e., the...

In this talk we will discuss connections between the geometric and analytic/PDE properties of sets. The emphasis is on quantifiable, global results which yield true equivalence between the geometric and PDE notions in very rough scenarios, including...

Morrey’s conjecture arose from a rather innocent looking question in 1952: is there a local condition characterizing "ellipticity” in the calculus of variations? Morrey was not able to answer the question, and indeed, it took 40 years until first...

Let X be a smooth affine algebraic variety over the field C of complex numbers (that is, a smooth submanifold of C^n which can be described as the solutions to a system of polynomial equations). Grothendieck showed that the de Rham cohomology of X...

I will describe how the orbit method can be developed in a quantitative form, along the lines of microlocal analysis, and applied to local problems in representation theory and global problems involving the analysis of automorphic forms. This talk...

The fundamental equations of fluid dynamics exhibit non-uniqueness. Is this a mathematical fluke, or do the equations fail to uniquely predict the motion of fluids? In this colloquium, we present recent mathematical and physical progress toward...

Thresholds for increasing properties of random structures are a central concern in probabilistic combinatorics and related areas.  In 2006, Kahn and Kalai conjectured that for any nontrivial increasing property on a finite set, its threshold is...

In this talk, we will discuss the behavior of the Yamabe flow on an asymptotically flat (AF) manifold. We will first show the long-time existence of the Yamabe flow starting from an AF manifold and discuss the uniform estimates on manifolds with...

Noetherianity is a fundamental property of modules, rings, and topological spaces that underlies much of commutative algebra and algebraic geometry. This talk concerns algebraic structures such as the infinite-dimensional polynomial ring K[x_1,x_2...

Humans have been thinking about polynomial equations over the integers, or over the rational numbers, for many years. Despite this, their secrets are tightly locked up and it is hard to know what to expect, even in simple looking cases. In this talk...

Projective modules over rings are the algebraic analogs of vector bundles; more precisely, they are direct summands of free modules. Some rings have non-free projective modules. For instance, the ideals of a number ring are projective, and for some...

Markov numbers are positive integers that appear in solutions of the equation x^2+y^2+z^2=3xyz. They also appear naturally when one tries to parametrize positive $SL_2(R)$-local systems on a one punctured torus. In this talk I will explain that...

The theory of "random surfaces" has emerged in recent decades as a significant field of mathematics, lying somehow at the interface between geometry, probability, and mathematical physics. I will give a friendly (I hope) colloquium-level overview of...

Let $E$ be an elliptic curve defined over $\Q$.  The $\bar\Q$-points of $E$ form an abelian group on which the Galois group $G_{\Q} = \Gal(\bar\Q/\Q)$ acts.  The usual Galois representation associated to $E$ captures the action of $G_{\Q}$ on the...

The Markoff equation $x^2+y^2+z^2=3xyz$, which arose in his spectacular thesis (1879), is ubiquitous in a tremendous variety of contexts. After reviewing some of these, we will discuss (briefly) asymptotics of integer points, and (in some detail)...

Given a flow on a manifold, how to perturb it in order to create a periodic orbit passing through a given region? While the first results in this direction were obtained in the 1960-ies, various facets of this question remain largely open. I will...

The Lefschetz property is central in the theory of projective varieties, detailing a fundamental property of their Chow rings, essentially saying that the multiplication with a geometrically motivated class is of full rank.

We drop the keyword...

Contact topology is the study of certain geometric structures on odd dimensional smooth manifolds. A contact structure is a hyperplane field specified by a one form which satisfies a nondegeneracy condition called maximal non-integrability. The...

A quasigeodesic in a manifold is a curve so that when lifted to the universal cover is uniformly efficient up to a bounded multiplicative and added error in measuring length. A flow is quasigeodesic if all flow lines are quasigeodesics. We prove...