Seminars archive

There has been a great deal of recent research on the arithmetic of K3 surfaces, some of which has centered around the Brauer-Manin obstruction to the Hasse principle. In this talk, I'll give some background and discuss some of the results and open questions in this area.

It is a classical problem in geometry to find the densest sphere packing in n-dimensional Euclidean space. Analogous problems of packing among lattices, or on compact spaces such as the sphere or Hamming space, are widely studied in number theory, discrete geometry, coding theory and combinatorics. I will talk about some recent work which puts these problems in the framework of potential energy minimization. This leads to experimental and theoretical techniques to approach these optimization problems (and their inverse problems), and leads to some surprising new results in high dimensions.

A pseudo-reductive group is a smooth connected affine algebraic group over a field k which does not contain any nontrivial smooth connected normal unipotent subgroups defined over k. Such groups arise naturally as the quotient of any smooth connected affine algebraic k-group by the maximal smooth connected normal unipotent subgroup defined over k. For study of general affine algebraic groups it is important to know the structure and classification of pseudo-reductive groups. In a joint work with Brian Conrad and Ofer Gabber we have determined the structure and classification of these groups. In my talk I will explain the classification, and also mention group theoretic and arithmetic applications.

A Rogers-Ramanujan type identity is a series=product identity which relates certain partitions into parts satisfying difference to partitions into parts satisfying congruence conditions. Rogers-Ramanujan type identities arise in a variety of settings ranging from Lie algebras to statistical mechanics. A supreme example of such a partition identity is the deep 1967 theorem of Gollnitz. We shall discuss a new approach to Gollnitz's theorem in which this partition result and its generalizations will emerge out of a remarkable $q$-hypergeometric identity in three free parameters. This leads to crucial connections with several fundamental results on partitions and $q$-series, a new combinatorial understanding of Jacobi's triple product identity for theta functions, and to some partition congruences modulo powers of 2. The talk will be accessible to non-experts.

Dyson's rank and crank statistics have long been important in the combinatorial study of integer partitions. Both statistics were introduced in efforts to better understand the famous Ramanujan congruences for the partition function $p(n)$, and there is no apparent relation between their simple combinatorial definitions. The main results of this talk (essentially) prove a remarkable conjecture of Garvan that the moments of the crank are always larger than the moments of the rank. More generally, these inequalities are an important example of relations for the asymptotics of hypergeometric $q$-series that are governed by modularity properties. In this case, the crank is associated to a modular form, while the rank is associated to a mock theta function/harmonic Maass form. If time permits, I will also briefly discuss some ideas of Zagier for deriving asymptotic expansions in these settings.

Recently, M. Marshall answered a long-standing question in real algebraic geometry by showing that if $f(x,y) \in \mathbf{R}[x,y]$ and $f(x,y) \geq 0$ on the strip $[0,1] \times \mathbf{R}$, then $f$ has a representation $f = \sigma_0 + \sigma_1 x(1 - x)$, where $\sigma_0, \sigma_1 \in \mathbf{R}[x,y]$ are sums of squares. In this talk, we give the background to this result, which goes back to Hilbert's 17th problem, and our generalizations to other non-compact basic closed semialgebraic sets of $\mathbf{R}^2$ which are contained in strip. We also give some negative results.