In 2017–18, I led a special program about analysis and
topology on locally symmetric spaces as a Distinguished
Visiting Professor in the School of Mathematics. Locally symmetric
spaces are the home of the Langlands program—a set of
During the 2015-16 academic year, the School of Mathematics
hosted a program on the topic of geometric structures in three
dimensions. This article is an adaptation of a talk I gave in fall
2015, as part of the School's biweekly "Mathematical...
Symplectic and contact structures first arose in the study of classical mechanical systems, allowing one to describe the time evolution of both simple and complex systems such as springs, planetary motion, and wave propagation. Understanding the evolution and distinguishing transformations of these systems led to the development of global invariants of symplectic and contact manifolds.
Topology is the branch of geometry that deals with large-scale
features of shapes. One cliché is that a topologist cannot
distinguish a doughnut from a coffee cup: if a coffee cup were made
of rubber, one could continuously deform it to a doughnut...
Topology is the only major branch of modern mathematics that
wasn't anticipated by the ancient mathematicians. Throughout most
of its history, topology has been regarded as strictly abstract
mathematics, without applications. However, illustrating...
Mathematics has proven to be "unreasonably effective" in
understanding nature. The fundamental laws of physics can be
captured in beautiful formulae. In this lecture, given at the
Perimeter Institute for Theoretical Physics, Robbert Dijkgraaf...
The story of the “data explosion” is by now a
familiar one: throughout science, engineering, commerce, and
government, we are collecting and storing data at an
ever-increasing rate. We can hardly read the news or turn on a
For a long time time, I have had a profound interest in studying
“higher order structures” of various kinds. What is a
higher order object? I will not here attempt to give a definition,
but rather illustrate by examples what I have in mind.