Members’ Colloquium

Homological stability has emerged over the past decades as an organizing principle in topology and beyond. Broadly speaking, many sequences of moduli spaces exhibit the striking phenomenon that their homology stabilizes as the underlying complexity...

This talk will recall the Birch--Swinnerton-Dyer Conjecture and describe highlights of the progress towards it, both pre- and post-millennium.  This will largely be a repeat of the speaker's plenary talk from the 2025 Clay Research conference on the...

The study of descriptive set theory in the context of determinacy axioms began nearly 60 years ago. The context for this study is now understood to be the Axiom AD+, which is a refinement of the Axiom of Determinacy (AD).  The objects of this study...

Singularities are local properties of algebraic varieties. For example, the solution set of a polynomial in several variables such as $y^2=x^3$ has a singularity at the origin $(0,0)$. In algebra, one often studies singularities through the quotient...

We prove that in any dimension n there exists an origin-symmetric ellipsoid of volume c n^2 that contains no points of Z^n other than the origin. Here c > 0 is a universal constant. Equivalently, there exists a lattice sphere packing in R^n whose...

Essentially optimal estimates have been obtained for mean values of Vinogradov’s exponential sum as a consequence of the decoupling method (by Bourgain, Demeter and Guth), and the efficient congruencing method (by the speaker). Such work makes...

The P vs. NP problem was formulated about 50 years ago, and was chosen to be one of the seven Clay millenium problems 25 years ago. In this period our understanding of the depth, breadth and impact of the problem has changed dramatically. I plan to...

The fundamental group π1(X) is an important invariant of a complex algebraic variety X.  Despite its topological nature, it is closely connected to the geometry of many algebraic structures on X.  In this talk I want to discuss two elementary...

Since Hironaka's famous resolution of singularities in characteristics zero in 1964, it took about 40 years of intensive work of many mathematicians to simplify the method, describe it using conceptual tools and establish its functoriality. However...

Riemannian metrics are the simplest generalizations of Euclidean geometry to smooth manifolds. The Ricci curvature of a metric measures, in an averaged sense, how the geometry deviates from being flat. The tensor −2Ric can be viewed as a Laplacian...