Kempner Colloquium Abstracts

Fall 2008

From the simplest point of view, transseries are a new kind of expansion for real-valued functions. But transseries constitute much more than that: they have a very rich (algebraic, combinatorial, analytic) structure. The set of transseries is a large ordered field, extending the real number field, and endowed with additional operations such as exponential, logarithm, derivative, integral, composition. Over the course of the last 20 years or so, transseries have emerged in several areas of mathematics: analysis, model theory, computer algebra, surreal numbers. This talk will be an introduction for the non-specialist mathematician.

A derangement is a fixed point free permutation. By an old elementary result of Jordan, every transitive finite permutation group contains a derangement. We will discuss some recent refinements of this result on the proportion of derangements together with applications to Brauer groups and rational maps between curves. In particular, we will discuss the Fulman-Guralnick solution of the Boston-Shalev conjecture on a lower bound for the proportion of derangements for finite simple groups.

Manifolds have a long and distinguished history going back to their Greek roots in Euclidean space. In the twentieth century there were major advances in the understanding of the relationship between abstract manifolds of various categories and submanifolds of Euclidean space. This lecture will present some highlights of this historical development.

Let G be a subgroup of finite index in K^*, the multiplicative group of the field K. We proved in [1] the following the result:

Theorem: if the characteristic of K is 0, or it is large enough compared to the index of G in K^*, then G-G = K, that is, every element of K can be written as the difference of two elements of G.

We also proved that in general G+G ≠ K. For a positive integer m we denote by m × G to be the set of sums of exactly m elements of G. If -1 belongs to the additive closure of G then we also proved that n × G = K, where n is the index of G in K.

The additive dimension of G is defined to be the least positive integer m such that m × G = K. Examples of groups where this additive dimension coincides with the index of the subgroup in the field were also given. In fact, they were given for infinitely many different values of the index. These subgroups G are called of maximal additive dimension.

In [2] Bergelson and Shapiro showed that the Theorem above holds for any infinite field K and extended the scope of the result.

They also noticed that, for all examples of maximal additve dimension subgroups G of Q^* presented in [1], the quotient group is a cyclic group, and they conjectured that this is always the case.

We shall present a proof of this conjecture.

We also give properties of the additive dimension for a larger family of examples.

[1] P. Berrizbeitia, Additive properties of multiplicative subgroups of finite index in fields. Proc. Amer. Math. Soc. 112, no. 2, 365-369, (1991).

[2] V. Bergelson and D. Shapiro, Multiplicative subgroups of finite index in a ring, Proc. Amer. Math. Soc. 116, no. 4, 885-896, (1992)

Starting from the electric-magnetic duality of the classical Maxwell theory, we explain the role of Langlands dual in non-Abelian gauge theory. We then describe the quantum duality conjecture and its manifestations in various supersymmetric gauge theories. An example of such is the Vafa-Witten theory, whose partition functions are the generating functions of the Euler numbers of the instanton moduli spaces. When the gauge group is simply laced, their transformation under the modular group is consistent with the known topological calculations. It is also related to theta functions and quadratic reciprocity. In this talk, we propose the transformation law of the partition functions under the Hecke group when the gauge group is non-simply laced.

This talk will be an introduction to permutation patterns. A permutation p in S_n is said to contain a pattern permutation q in S_m if there exists a subsequence of p that is order-isomorphic to q. If p does not contain q, then p avoids q. One of the central questions in the study of patterns is: given a pattern p in S_m, how many permutations in S_n avoid p? This question has a beautiful answer for the case m=3 and we will prove the result in this case. When m=4, the problem gets extraordinarily more complicated. We will survey what is known for m=4 and discuss what is still left to be solved.

The theory of association schemes has proven useful in several areas of discrete mathematics such as coding theory, finite geometry, and design theory, to name a few. Much attention has been paid to association schemes which are generated by distance-regular graphs; however, the formal dual to this type of scheme, known as a "Q-polynomial" scheme, remains poorly understood. In this talk some introductory material on association schemes will be presented, followed by a description of recent progress toward understanding the structure of Q-polynomial schemes.