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...
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...
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...
