# Scientific

## Quasilinear systems and residential burglary

In this talk we will present some results for systems of equations modeling residential burglary.

For the parabolic system model proposed by Andrea Bertozzi et-al, we study the equilibrium case. By using bifurcation theory we show that this system does support pattern formation. We also give some results concerning stability of the bifurcating patterns. These results correspond to a joint work with Chris Cosner and Steve Cantrel from the University of Miami.

The model has been recently modified by Pitcher giving rise to a new parabolic system of equations. We show some results for this system that contain a condition for existence of global solutions. This work corresponds to a collaboration with Philippe Souplet and Quoc Hung Phan from Paris 13.

## The Shape of Data

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.

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## The Causes of Crime and the Practical Limits of Crime Control

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.

## Mathematics of Crime

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.

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## Hot Topics in Computational Criminology

## Iwahori-Hecke algebras are Gorenstein (part II)

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.

## Iwahori-Hecke algebras are Gorenstein (part I)

**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.**

This lecture is the first of a two part series (part II).

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.

## Alan Turing and the Patterns of Life

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.

## Quadratic forms and finite groups

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.

## 24th Canadian Conference on Computational Geometry Proceedings

This volume contains the official proceedings of the 24th Canadian Conference on Computational Geometry (CCCG’12), held in Charlottetown on August 8-10, 2012. These papers are also available electronically at http://www.cccg.ca and at http://2012.cccg.ca.

We thank the staff at Holland College, and in particular Tina Lesyk, Marsha Doiron and Tracey Campbell, for preparing the conference site.

We are grateful to the Program Committee for agreeing to a rigorous review process. They, and other reviewers, thoroughly examined all submissions and provided excellent feedback. Out of 75 papers submitted, 49 are contained in these proceedings. We thank the authors of all submitted papers, all those who have registered, and in particular Günter Ziegler, Pankaj Agarwal and Joseph Mitchell for presenting plenary lectures.

We also thank Sébastien Collette, who prepared these proceedings, as well as Perouz Taslakian and Narbeh Bedrossian who designed the conference logo.

Last but not least, we are grateful for sponsorship from AARMS, the Mprime Network, PIMS and the Fields Institute. Their financial support has helped us to cover many costs as well as provide significant funding to over 50 students and postdocs, including waivers of their registration fees.