Kolloquium Wilhelm Killing

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Given some large positive integer N, what is the largest possible size of a subset of {1,...,N} which does not contain a three-term arithmetic progression (i.e. without three distinct elements x,y,z satisfying x+z=2y)? Similarly, given a prime p and a large positive integer n, what is the largest possible size of a subset of the vector space F_p^n which does not contain a three-term arithmetic progression (i..e without three distinct vectors x,y,z satisfying x+z=2y)? These are long-standing problems in additive combinatorics. This talk will explain the known bounds for these problems, give an overview of some of the proof techniques, and discuss additional applications of these techniques to other additive combinatorics problems.

Problems involving multiple scales are ubiquitous in applications. The typical situation, for which analytical as well as numerical methods are quite well developed, is a partial differential equation (PDE) posed on a (macroscopic) domain with coefficients that oscillate on a much smaller (microscopic) scale. In this talk, we consider two areas where additional scales of oscillations occur and the inter-relation of the different scales crucially influences the overall behavior. The first example is wave propagation in spatial multiscale media with a high contrast, where resonances can cause unusual wave phenomena such as all-angle band gaps. The second example are spatial multiscale media with additional temporal variations, in the heat or wave equation. For both examples, we discuss the design and analysis of numerical methods to capture these effects.

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