Members’ Colloquium

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...

One of the primary goals of the mathematical analysis of algorithms is to provide guidance about which algorithm is the “best” for solving a given computational problem. Worst-case analysis summarizes the performance profile of an algorithm by its...

The SAT (Boolean Satisfiability) problem asks whether a given logical formula on n Boolean variables has an assignment of true/false values to its variables that makes the formula true.  The P vs NP question is equivalent to asking whether SAT has a...

I'll give a brief introduction to o-minimality and how it can be used to prove asymptotic estimates for the number of rational points in definable sets. I'll then show how problems from various areas of mathematics can be reformulated as questions...