Seminars Sorted by Series

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...

Define the Collatz map Col on the natural numbers by setting Col(n) to equal 3n+1 when n is odd and n/2 when n is even. The notorious Collatz conjecture asserts that all orbits of this map eventually attain the value 1. This remains open, even if...

The rigorous study of spin systems such as the Ising model is currently one of the most active research areas in probability theory. In this talk, I will introduce one particular class of such models, known as lattice gauge theories (LGTs), and go...

Zeros of L-functions have been extensively studied, due to their close connection to arithmetic problems. Despite several precise conjectures about their behavior, our unconditional understanding of them remains limited. In this talk we will discuss...

How do we describe the topology of the space of all nonconstant holomorphic (respectively, algebraic) maps F: X--->Y  from one complex manifold (respectively, variety) to another? What is, for example, its cohomology? Such problems are old but...

The Zimmer program asks how lattices in higher rank semisimple Lie groups may act smoothly on compact manifolds. Below a certain critical dimension, the recent proof of the Zimmer conjecture by Brown-Fisher-Hurtado asserts that, for SL(n,R) with n...

Let $p$ be a prime number. Roughly speaking, rigid analytic geometry is a counterpart of complex analysis where one replaces the field $\mathbf{C}$ of complex numbers by the field $\mathbf{Q}_{p}$ of $p$-adic rational numbers (or some extension...

Motivated by a discovery by Radchenko and Viazovska and by a work by Ramos and Sousa, we find conditions sufficient for a pair of discrete subsets of the real axis to be a uniqueness or a non-uniqueness pair for the Fourier transform. These...

The finite field Kakeya problem asks about the size of the smallest set in (F_q)^n containing a line in every direction.  Raised by Wolff in 1999 as a ‘toy’ version of the Euclidean Kakeya conjecture, this problem is now completely resolved using...

The theory of matroids provides a unified abstract treatment of the concept of dependence in linear algebra and graph theory. In this talk we explain Bergman fans of matroids, and we investigate isomorphisms of Bergman fans for different fan...

The Higgs mechanism is a part of the Standard Model of quantum mechanics that allows certain kinds of particles to have nonzero mass. In spite of its great importance, there is no rigorous proof that the Higgs mechanism can indeed generate mass in...

Translational tiling is a covering of a space (such as Euclidean space) using translated copies of one building block, called a "translational tile'', without any positive measure overlaps.  

Can we determine whether a given set is a translational...

Let G be an infinite discrete group. Finite dimensional unitary representations of G are usually quite hard to understand. However, there are interesting notions of convergence of such representations as the dimension tends to infinity. One notion —...

Abstract: I will discuss a result with Bonatti and Crovisier from 2009 showing that the $C^1$ generic diffeomorphism f of a closed manifold has trivial centralizer; i.e. fg = gf implies that g is a power of f. I’ll discuss features of the $C^1$...

The problem of control of large multi-agent systems, such as vehicular traffic, poses many challenges both for the development of mathematical models and their analysis and the application to real systems. First, we discuss how conservation laws can...

Any non-negative univariate polynomial over the reals can be written as a sum of squares.  This gives a simple-to-verify certificate of non-negativity of the polynomial. Rooted in Hilbert's 17th problem, there's now more than a century's work that...

The Fourier uniformity conjecture seeks to understand what multiplicative functions can have large Fourier coefficients on many short intervals. We will discuss recent progress on this problem and explain its connection with the distribution of...

A deep result of Furstenberg from 1967 states that if $\Gamma$ is a lattice in a semisimple Lie group $G$, then there exists a measure on $\Gamma$ with finite first moment such that the corresponding harmonic measure on the Furstenberg boundary of...