Georgia Institute of Technology College of Sciences

On a smooth manifold, we consider a non-autonomous ordinary differential equation whose right side switches between finitely many smooth vector fields at random times. These switching times are exponentially distributed to guarantee that the resulting random dynamical system has the Markov property. A Hoermander-type hypoellipticity condition on a recurrent subset of the manifold is then sufficient for uniqueness and absolute continuity of the invariant measure of the Markov semigroup. The talk is based on a paper with my advisor Yuri Bakhtin.

(algebraic statistics reading seminar; note unusual time)

We consider the modeling and computations of random dynamical processes of viral signals propagating over time in social networks. The viral signals of interests can be popular tweets on trendy topics in social media, or computer malware on the Internet, or infectious diseases spreading between human or animal hosts. The viral signal propagations can be modeled as diffusion processes with various dynamical properties on graphs or networks, which are essentially different from the classical diffusions carried out in continuous spaces. We address a critical computational problem in predicting influences of such signal propagations, and develop a discrete Fokker-Planck equation method to solve this problem in an efficient and effective manner. We show that the solution can be integrated to search for the optimal source node set that maximizes the influences in any prescribed time period. This is a joint work with Profs. Shui-Nee Chow (GT-MATH), Hongyuan Zha (GT-CSE), and Haomin Zhou (GT-MATH).

A large variety of Constraint Satisfactoin Problems can be classified as "computationally hard". In recent years researchers from statistical mechanics have investigated such problems via non-rigorous methods. The aim of this talk is to give a brief overview of this work, and of the extent to which the physics ideas can be turned into rigorous mathematics. I'm also going to point out various open problems.