Mathematics

Ultraproducts as a tool in the model theory of metric structures

Ward Henson (UIUC)

Abstract: L is a signature of continuous first order logic for metric structures and we have a class C of L-structures which we want to investigate from the point of view of model theory. In general, this involves letting T be the L-theory of C, and working to understand the models of T as fully as possible. This means not only knowing which L-structures are models of T, but also understanding the definable predicates and (especially important) the definable sets in models of T. (A valuable byproduct might be an explicit axiomatization of T.) In this talk we will lay out how understanding ultraproducts of members of C can be an important practical tool for understanding the full class of models of T. As much as time permits, we will discuss examples that have been successfully treated in this way, including some new ones, focusing on Banach spaces and Banach lattices. (Most of this work on examples is part of a collaboration with Yves Raynaud.)

Beatty Sequences

Xiaomin Li (UIUC Math)

Abstract: A Beatty sequence is a sequence of the form [a*n], where a is an irrational number and the bracket denotes the floor function. A remarkable result, called Beatty's Theorem, says that if a and b are irrational numbers such that 1/a+1/b=1, then the associated Beatty sequences "partition" the natural numbers. That is, every natural number belongs to exactly one of these two sequences. It is known that Beatty's Theorem does not extend directly to partitions into three or more sets, and finding appropriate analogs of Beatty's Theorem for such partitions is an interesting, and wide open, problem, which has applications to optimal scheduling questions. The goal of this project is to explore different constructions of partitions of integers into perturbed Beatty sequences and possible applications to optimal scheduling algorithms.