My research interest is in studying the solutions of nonlinear elliptic PDEs and ODEs. In particular, I study the questions of existence, uniqueness and multiplicity of solutions of nonlinear boundary value problems in ODEs and elliptic PDEs, and stability of solutions of corresponding evolution equations.
I have been working on projects in different areas of mathematical biology including the evolution of cooperation, game-theoretic models of infectious diseases, and theoretical ecology. The mathematics I use in these projects includes game theory, optimization, ordinary differential equations, and stochastic processes.
I am interested in questions related to modeling, control, and approximation of infinite dimensional linear systems. In particular I study systems governed by either delay-differential equations or partial differential equations which arise in modeling flexible structures. The mathematics we use includes functional analysis, differential equations, and control theory.
I am interested in the development, implementation, and analysis of numerical methods for approximating nonlinear partial differential equations. Nonlinear PDEs arise in stochastic optimal control, optimal mass transport, and materials science. My research has focussed on the Hamilton-Jacobi-Bellman equation, the Monge-Ampere equation, and the Cahn-Hilliard equation.
My research has been motivated primarily by problems in ecology, evolution, and other areas of biology, but I am broadly interested in dynamical systems and general transport and/or diffusion models. The mathematics of my work includes functional analysis, ordinary and differential equations, difference equations, linear algebra, and game theory.
I am interested in applications of mathematics in a wide range of fields. I have collaborated with researchers from biology, sociology and computer science. My work consists in building and analyzing a mathematical model of a phenomena of particular interest of the researcher. The mathematics involved ranges from game theory, ODEs, probability and stochastic processes and graph theory.
Study of nonnegative solutions to partial differential equations, arising in the modeling of natural phenomena that is dominated by reaction and diffusion. Recent results focus on singular problems, problems with nonlinear diffusion and effects of nonlinear boundary conditions. His research has been funded by the National Science Foundation (NSF) and by the Simon's Foundation. Currently, he serves as the PI on an NSF Math Ecology Grant. He has authored over one hundred and thirty research papers. He is a member of the Editorial Board of several mathematics journals. To date, he has directed one postdoctoral student, fifteen Ph.D. students (11 graduates, 4 current), fourteen M.S. graduates and nineteen undergraduate research students (18 graduates, 1 current).