Seminars Sorted by Series

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...

Structures which minimise area appear in numerous geometric contexts often related to degeneration phenomena. In turn, in many situations these structures also reflect the ambient geometry in some way (they are ‘calibrated’) and so they may provide...

The so-called Problem of Compact Quotients asks for which homogeneous spaces G/H there exists a discrete subgroup Gamma of G such that Gamma\G/H is a compact manifold. We will discuss this problem, and describe recent progress on it arising from...

To any unital, associative ring R one may associate a family of invariants known as its algebraic K-groups. Although they are essentially constructed out of simple linear algebra data over the ring, they see an extraordinary range of information...

Triangulated surfaces are Riemann surfaces formed by gluing together equilateral triangles. They are also the Riemann surfaces defined over the algebraic numbers. Brooks, Makover, Mirzakhani and many others proved results about the geometric...

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...