Georgia Institute of Technology College of Sciences

In this era of "big data", Mathematics as it applies to human behavior is becoming a much more relevant and penetrable topic of research. This holds true even for some of the less desirable forms of human behavior, such as crime. In this talk, I will discuss the mathematical modeling of crime on two different "scales", as well as the results of experiments that are being performed to test the usefulness and accuracy of these models. First, I will present a data-driven model of crime hotspots at the scale of neighborhoods -- adapted from literature on earthquake predictions -- along with the results of this model's application within the LAPD. Second, I will describe a game-theoretic model of crime and punishment at the scale of a society, and compare the model to results of lab-based economic experiments performed by myself and collaborators.

Ricci flow is a sort of (nonlinear) heat problem under which the metric on a given manifold is evolving. There is a deep connection between probability and heat equation. We try to setup a probabilistic approach in the framework of a stochastic target problem. A major result in the Ricci flow is that the normalized flow (the one in which the area is preserved) exists for all positive times and it converges to a metric of constant curvature. We reprove this convergence result in the case of surfaces of non-positive Euler characteristic using coupling ideas from probability. At certain point we need to estimate the second derivative of the Ricci flow and for that we introduce a coupling of three particles. This is joint work with Rob Neel.

It is well-known that a deterministic dynamical system can exhibit stochastic behavior that is due to the fact that instability along typical trajectories of the system drives orbits apart, while compactness of the phase space forces them back together. The consequent unending dispersal and return of nearby trajectories is one of the hallmarks of chaos. The hyperbolic theory of dynamical systems provides a mathematical foundation for the paradigm that is widely known as "deterministic chaos" -- the appearance of irregular chaotic motions in purely deterministic dynamical systems. This phenomenon is considered as one of the most fundamental discoveries in the theory of dynamical systems in the second part of the last century. The hyperbolic behavior can be interpreted in various ways and the weakest one is associated with dynamical systems with non-zero Lyapunov exponents. I will discuss the still-open problem of whether dynamical systems with non-zero Lyapunov exponents are typical. I will outline some recent results in this direction. The genericity problem is closely related to two other important problems in dynamics on whether systems with nonzero Lyapunov exponents exist on any phase space and whether nonzero exponents can coexist with zero exponents in a robust way.

I'll be presenting bounds on the uniform entropy numbers for VC classes of sets, VC classes of functions, and symmetric convex hulls of VC classes of functions. The material is in Section 2.6 of Van der Vaart and Wellner.