Members’ Colloquium

Morrey’s conjecture arose from a rather innocent looking question in 1952: is there a local condition characterizing "ellipticity” in the calculus of variations? Morrey was not able to answer the question, and indeed, it took 40 years until first...

In this talk we will discuss connections between the geometric and analytic/PDE properties of sets. The emphasis is on quantifiable, global results which yield true equivalence between the geometric and PDE notions in very rough scenarios, including...

I will discuss some questions of interest in neuroscience, seen through the lens of mathematics. No prior knowledge of neuroscience is needed for this talk. Two of the most basic visual capabilities of primates are orientation selectivity, i.e., the...

The classification of geometric structures on manifolds naturally leads to actions of automorphism groups, (such as mapping class groups of surfaces) on "character varieties" (spaces of equivalence classes of representations of surface groups).

I will introduce a new model of randomly agitated equations. I will focus on the finite finite dimensional approximations (analogous to Galerkin approximations) and the two-dimensional setting. I will discuss number of properties of the models...

The spread of a matrix is defined as the diameter of its spectrum. This quantity has been well-studied for general matrices and has recently grown in popularity for the specific case of the adjacency matrix of a graph. Most notably, Gregory...

The regularity theory for the Navier-Stokes equation will be reviewed. Motivations from Kolmogorov’s phenomenological theory of turbulence will be discussed. Rigorous mathematical results are obtained to confirm some of the phenomenologies.

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...