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...

#
Members Seminar

I revisit the basic statistical problem of estimating the mean of a real-valued distribution. I will introduce an estimator with the guarantee that "our estimator, on *any* distribution, is as accurate as the sample mean is for the Gaussian...

New types of symmetries have been considered in algebra and algebraic geometry and a higher analog of representation theory has been developed to answer questions of classical representation theory. Geometric representation theory can be viewed as...

A few months ago, a group of theoretical computer scientists posted a paper on the Arxiv with the strange-looking title "MIP* = RE", impacting and surprising not only complexity theory but also some areas of math and physics. Specifically, it...

A few months ago, a group of theoretical computer scientists posted a paper on the Arxiv with the strange-looking title "MIP* = RE", impacting and surprising not only complexity theory but also some areas of math and physics. Specifically, it...

Most of the visible matter in the Universe is a plasma, that is a dilute gas of ions, electrons, and neutral atoms. In many circumstances, the dynamics of this plasma can be modeled in the continuum limit, using the equations of fluid mechanics...

A 1948 theorem of de Bruijn and Erdős says that if nn points in a projective plane do not lie all on a line, then they determine at least n lines. More generally, Dowling and Wilson conjectured in 1974 that for any finite set of vectors spanning a...

The celebrated Brunn-Minkowski inequality states that for compact subsets XX and YY of ℝdRd, m(X+Y)1/d≥m(X)1/d+m(Y)1/dm(X+Y)1/d≥m(X)1/d+m(Y)1/d where m(⋅)m(⋅) is the Lebesgue measure. We will introduce a conjecture generalizing this inequality to...

Physics inspired mathematics helps us understand the random evolution of Markov processes. For example, the Kolmogorov forward and backward differential equations that govern the dynamics of Markov transition probabilities are analogous to the...

Low-dimensional topology and geometry have many problems with an easy formulation, but a hard solution. Despite our intuitive feeling that these problems are "hard", lower or upper bounds on algorithmic complexity are known only for some of them...