Scientific

The notion of higher dimensional shape has turned out to be an important feature of data. It encodes the qualitative structure of data, and allows one to find useful distinct groups in data sets. Topology is the branch of mathematics which deals with shape, and in recent years methods from topology have been adapted for the study of data. This talk will survey these developments, with examples.

There is an extensive applied mathematics literature developed for problems in the biological and physical sciences. Our understanding of social science problems from a mathematical standpoint is less developed, but also presents some very interesting problems. This lecture uses crime as a case study for using applied mathematical techniques in a social science application and covers a variety of mathematical methods that are applicable to such problems. We will review recent work on agent based models, differential equations, variational methods for inverse problems and statistical point process models. From an application standpoint we will look at problems in residential burglaries and gang crimes.
Examples will consider both "bottom up" and "top down" approaches to understanding the mathematics of crime, and how the two approaches could converge to a unifying theory.

n the local Langlands program the (smooth) representation theoryof p-adic reductive groups G in characteristic zero plays a key role. For any compact open subgroup K of G there is a so called Hecke algebra H(G,K). The representation theory of G is equivalent to the module theories over all these algebras H(G,K). Very important examples of such subgroups K are the Iwahori subgroup and the pro-p Iwahori subgroup. By a theorem of Bernstein the Heckealgebras of these subgroups (and many others) have finite global dimension.

In recent years the same representation theory of G but over an algebraically closed field of characteristic p has become more and more important. But little is known yet. Again one can define analogous Hecke algebras. Their relation to the representation theory of G is still very mysterious. Moreover they are no longer of finite global dimension. In joint work with R. Ollivier we prove that over any field the algebra H(G,K), for K the (pro-p) Iwahori subgroup, is Gorenstein.

In 1952, Turing published his only paper spanning chemistry and biology: "The chemical basis of morphogenesis". In it, he proposed a hypothetical mechanism for the emergence of complex patterns in chemical reactions, called reaction-diffusion. He also predicted the use of computational models as a tool for understanding patterning. Sixty years later, reaction-diffusion is a key concept in the study of patterns and forms in nature. In particular, it provides a link between molecular genetics and developmental biology. The presentation will review the concept of reaction-diffusion, the tumultuous path towards its acceptance, and its current place in biology.