John's research statement

Research statement

                I am interested in number theory and p-adic analysis. My research focuses on studying the relationships between the pseudo differential equations and the Markov processes. To this end, I have used a lot of techniques from number theory and classical theory of differential equations. Some of the equations that I have studied include the p-adic heat equation and some generalizations of this using homogeneous polynomials instead of the classical laplacian. Furthermore, I have applied the theory of the Igusa’s local zeta function for finding a closed form for the solution of the p-adic heat equation. All these models have attracted people’s attention since the mid 80’s because of their relation with alternative models for quantum mechanics. Also, there is a lot of attention from outside of the mathematics because of the connection with some models in statistical physics. For example, people know that some proteins have very complex hierarchical structures and the setting of p-adic analysis is well suited for this. My long term research agenda is to study a broad class of pseudo differential equations that have a probabilistic interpretation and to use the tools of the side of the equation for obtaining new probabilistic models.