My current research aims to model the formation and dynamics of crime "hotspots" - spatio-temporal regions of increased criminal activity. Working with data provided from 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. Movies from the simulations can be found here.

This work is being done in collaboration with Jeff Brantingham, Andrea Bertozzi, Lincoln Chayes, George Tita, Virginia Pasour, and Maria D'Orsogna.

For many species of aquatic microorganisms (such as Volvox, pictured to the left), diffusion alone cannot supply nutrition at a rate high enough to keep the organism alive if the size of the organism exceeds a critical value.  These organisms can get around this evolutionary bottleneck by using flagella to swim in their aqueous surroundings, thereby bringing advection into the picture.  This advection effectively "stirs" the solution, greatly increasing the nutrient uptake rate, which allows the organisms to continue to evolve to ever greater sizes.  In addition, for the specific example of Volvox, this adaptation necessitates the transition from species with only one type of cell to those in which cell specialization exists.  Our work in this area models these various processes for the many species of Volvox.  We use experimental measurements to validate our models and find that our predicted bottleneck radius, at which cell specialization must occur, is very close to the actual size in which this adaptation is observed.

This work was a part of my PhD thesis and was done in collaboration with Cristian Solari, Sujoy Ganguly, Thomas Powers, John Kessler, and Raymond Goldstein.

Though the general processes underlying the formation of stalactites and icicles have long been known, it was unclear how these processes eventually led to the iconic carrot-like shape often associated with these objects.  This research project modeled the growth of both stalactites and icicles as free-boundary problems, arriving at growth laws for each of these objects.  Interestingly, these growth laws are scale invariant, implicating that there is in fact a single platonic ideal shape for both stalactites (picture at right) and icicles.  Even more interestingly, the mathematical form for the ideal stalactite is the same as that for the ideal icicle far from the tip, even though the growth processes underlying these two structures are vastly different.  Click here to see me explaining the growth of icicles on the television program Arizona Illustrated (link goes to a WMV video).

This work was a part of my PhD thesis and was done in collaboration with Jim Baygents, Warren Beck, David Stone, Rick Toomey, and Raymond Goldstein.