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...

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,...

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,...

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....

Derek and I had several conversations during lunches about the potential for “The Symplectic Piece.” And he continued to attend occasional lectures on the subject, searching for a way to map symplectic geometry onto a musical score.

I sometimes like to think about what it might be like inside a black hole. What does that even mean? Is it really “like” anything inside a black hole? Nature keeps us from ever knowing. (Well, what we know for sure is that nature keeps us from...

Quantum theory radically transforms our fundamental understanding of physical reality. It reveals that the world contains a hidden richness of structure that we have barely begun to control and exploit. According to quantum theory, what we...

It has been said that the goals of modern mathematics are reconstruction and development.^{1} The unifying conjectures between number theory and representation theory that Robert Langlands, Professor Emeritus in the School of Mathematics...

*What explains “the unreasonable effectiveness” of mathematics, as the late Princeton University physicist Eugene Wigner phrased it, in answering questions about the real world?*

Natural phenomena could have been structured in a way...