Topology

In 2020–21, Allen Yuan, Visitor in the School of Mathematics and a recent graduate from MIT, is studying problems in homotopy theory and algebraic topology.

I like to give a simple-sounding...

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 overarching and...

As part of Ideas: Celebrating 2017–18, Clay Cordova, Marvin L. Goldberger Member in the School of Natural Sciences, gives a talk on Topology and Physics with Charles Simonyi Professor Edward Witten.

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 computer without...

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.

In...