Members’ Colloquium

Human learning outstrips modern machine learning and AI in at least three abilities: rapid robust learning, in effectively open worlds, in near-real time with very little energy. Mathematical formalization of signature human abilities has the...

How does the dimension of the first cohomology grow in a tower of covering spaces? After a tour of examples of behaviors for low-dimensional spaces, I will focus on arithmetic manifolds. Specifically, for towers of complex hyperbolic manifolds, I...

The question in the title was one of the founding questions in symplectic topology 40 years ago, and despite a lot of progress since that time, it remains widely open. In the talk I will discuss the initial questions, the progress, and the remaining...

Despite the fact that the 3-body problem is an ancient conundrum that goes back to Newton, it is remarkably poorly understood, and is still a benchmark for modern developments. In this talk, I will give a (very) biased account of this classical...

The honest answer to the question is that I actually do not know. I will therefore rather talk about several famous examples that are widely called "h-principle results" and try to explain some of the ideas behind the ones I am most familiar with.

Given the current knowledge of complex representations of finite simple groups, obtaining good upper bounds for their characters values is still a difficult problem, a satisfactory solution of which would have significant implications in a number of...

Earlier this semester we heard a fascinating talk by James Stone describing how the equations of compressible magnetohydrodynamics (MHD) can help us understand the Cosmos. Today we will return to Earth and describe a mathematical model, derived from...