My current research involves the modeling of certain types of human activity that exhibit regular spatio- and/or temporal patterns. As a case study, we have generally focused on various types of criminal behavior, since there are clear patterns in this activity and we have access to relatively large amounts of data. A large portion of this work aims to model the formation and dynamics of crime "hotspots" - spatio-temporal regions of increased criminal activity. Working with data provided by the Los Angeles and Long Beach police departments, we have developed methods of measuring the repeat and near-repeat criminal events that are the hallmarks of hotspot formation. We have also constructed a family of discrete models that allow for such patterns to develop from natural criminal behavior, and have derived continuum approximations of these discrete models. Some output from one of many simulations (right) illustrates this finding, with "hot" areas in red and "cold" areas in purple. 

In addition to the work on crime hotspots, this overarching project has also included: more accurate predictions of when and where crimes will occur, based on self-exciting point process models borrowed from seismology; the study of gang territoriality, modeled via diffusive Lotka-Volterra equations; gang retaliatory violence, and how the police may be able to solve such crimes using constrained optimization; the evolution of gang rivalry networks in the presence of retaliation and third-party effects; game theoretic models for the levels of both crime and cooperation with the authorities in society; and new methods for finding the "anchor points" of criminals given the locations of crimes they committed, based on models inspired by animal foraging.