Scientific

Within criminology there continues to be wide disagreement over the importance the individual, formal and informal social structure and the environment in driving crime patterns. In person-based theories, individuals are assumed to either innately possess the capacity to commit crime, or learn such capacities from their interactions with others. In structural theories, it is generally assumed that individuals are constrained by static social, economic or political organization, which makes crime a necessary or acceptable alternative to non-crime activities. In environmental theories, the built environment creates abundant, if unevenly distributed opportunities for crime that are easily exploited. While each of these theoretical perspectives finds some justification in empirical studies, they are not equal practical from the point of view of crime control. This talk will review several key ideas underlying crime and crime pattern formation and argue in favor of modeling of short-term, local crime processes because it is these processes that are most easily disrupted and are likely to yield practical results.

N.B. Due to problems with our camera, there is some audio distortion on this file and a small portion of the video has been removed.

In the local Langlands program the (smooth) representation theory
of 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 Hecke
algebras 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.

The study of quadratic forms is a classical and important topic of algebra and number theory. A natural example is the trace form of a finite Galois extension. This form has the additional property of being invariant under the Galois group,leading to the notion of "self-dual nornal basis", introduced by Lenstra. The aim of this talk is to give a survey of this area, and to present some recent joint results with Parimala and Serre.