Lectures: MWF 2-2:50, MS 5233

Instructors:

** Course Content Overview **

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, especially for young researchers. This course 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. The course will blend some basic, graduate level tutorials on mathematical topics with some recent research results on the mathematics of crime and of insurgent activities. From a modeling standpoint, we will cover agent based models, methods in linear and nonlinear partial differential equations, variational methods for inverse problems, game theoretic models, statistical point process models, and graph theory. From an application standpoint, we will cover methods of data comparison as well as statistical filtering. The course 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.

The course will be structured around the following subtopics:

** Agent-based models:** We consider basic lattice models from statistical physics and discuss how one might ``course grain'' such models using nonlinear partial differential equations - we discuss both theory and numerical results for when mean field approximations are appropriate for agent based models. We also cover the differences between Eulerian and Lagrangian formalisms. We will discuss some recent work using agent-based models to study repeat residential burglary offenders.

** Mean field limit: ** We consider continuum limits of mean field models including a tutorial of basic mathematical methods for determining spatio-temporal properties of such. We discuss techniques for analyzing nonlinear mean field PDE models including linear stability theory and weakly nonlinear bifurcation theory, with examples. Current research on formation of crime hotspots will be discussed along with the mathematical theory.

** Data filtering: ** An important issue that arises in predictive policing is the ability to make real time decisions based on streaming data. If a crime upswing is taking place in a neighborhood, when is a decision made to send more patrols to that neighborhood? Such data is often only partially complete and is subject to a lot of random noise. At the same time, this is a basic question that comes up in many different application areas ranging from Internet DoS attacks to weather forecasting. A classical statistics problem is to develop filtering methods that minimize average detection delay while simultaneously minimizing the false alarm rate. We discuss some of the basic mathematics associated with statistical filtering of data ,such as the Kalman filter and change point detection.

** Statistical density estimation:** We discuss spatial inverse problems using variational methods. Statistical density estimation arises in the study of the geographic placement of criminal events. Using data from Los Angeles, we contrast more classical methods such as approximation by Gaussian kernels with more recent methods including total variation regularization and data fusion methods that incorporate additional data, such as census information. We compare and contrast the density estimation problem with some related problems that arise in image processing, including denoising and image inpainting.

** Point process models: ** We review some of the basic mathematics of point process models, starting with the Poisson arrival process and how to test for its presence in real world data. We contrast with self-exciting point process models such as the Hawkes process and discuss the role of memory in such problems.

** Game theoretic models: ** Game theory is often used to explain the choices that ``rational'' actors make in given situations. We will give a brief introduction/review of some results from game theory, then focus on how game theory might be applied to crime in an attempt to explain why criminals do what they do and what strategies law enforcement might employ to decrease crime.

** Graph theory:** By viewing social networks as graphs, researchers have drawn interesting conclusions about several related social phenomena, such as how co-authorship of scholarly articles may follow a preferential attachment process. Such ideas may also be applicable to the study of crime, especially with regards to gangs or other organized criminal groups. We give an introduction to some elements of graph theory that may be useful in this context.

The course will consist of lectures by the instructors and readings from the literature. All students will be required to do a final presentation consisting of a written paper and oral presentation. There will be opportunities to work on publication level material in this course although students who wish to publish papers will likely need more time once the course is finished. This course is meant for advanced graduate students who have passed the applied mathematics qualifying exams.

Link to University College London Lectures on Mathematics of Crime

Course Schedule and Mentor Assignments for Projects