Mathematical Science with Specialized Knowledge
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge.
I use techniques from the fields of dynamical systems, probability and statistics to develop and analyze mathematical models of real world (often biological) systems. I use those models to address specific questions related to population ecology and evolution, epidemiology (infectious diseases) and immunology. I also use those models as a basis for studying the relationship between process and pattern in a more general context, and sometimes pursue interesting mathematical questions that arise in these applications. When working on specific applications, I use different mathematical tools as the need arises, however I primarily use methods from applied nonlinear dynamics and bifurcation theory, including computational methods.
I study Navier-Stokes equations, dynamical systems, fractal dimensions and turbulence. Techniques used include functional analysis and large scale numerical simulation. I am currently working on the normal form of the Navier-Stokes equations, the Langrangian averaged Navier-Stokes alpha model of turbulence, numerically determining modes, the Bouligand dimension and the bioremediation of contaminated soil.
My research centers on the modeling, simulation and control of complex nonlinear systems, multiscale computing, integration of asymptotic and numerical techniques, abstraction and reduction techniques for large nonlinear models, modeling and control of bifurcation and chaos phenomena, and robust, impulsive and observation control of nonlinear systems.
Specific current projects include a) Multiscale modeling and simulation of molecule systems, b) Modeling and simulation of flow of magnetized nanoparticles in blood vessels, c) fast algorithms for modeling of atmospheric chemistry phenomena, d) Image tracking and processing algorithms, e) Stability bounds for partially uncertain nonlinear systems, and f) Robust control of bifurcation phenomena.
My research is driven by a desire to understand the roles of stochasticity, structure, and evolution in shaping the dynamics of biological systems. I develop and analyze mathematical models, combining methods from probability and statistics, dynamical systems, and random graph theory to shed light on biological issues while generating new mathematical questions. In particular, I study stochastic processes on networks with applications in neuroscience and stochastic models in genetics.
Currently, I am working on optimal reduction techniques for complex ion channel gating models, which can be represented as a stochastic (Markov) process on a graph. I am also looking at the relative contributions of network structure and node dynamics in determining the collective dynamics of a network, thinking specifically about neuronal networks involved in sleep-wake regulation. Lastly, I am broadly interested in mathematical and statistical applications in population and evolutionary genetics.
My research group is developing novel computational methods for challenging multiphysics problems in various areas of engineering and science, including nuclear engineering, civil engineering, electrical engineering, mechanical engineering, molecular design (quantum chemistry), climate modeling, and others. We collaborate with numerous researchers from national labs and universities both in the U.S. and in Europe. All our results are freely available online -- check out our flagship open source projects Hermes and FEMHub. Students working in my group have various backgrounds ranging from theoretical physics, mathematics, computer science, to various engineering areas. This is absolutely necessary for successful completion of challenging interdisciplinary projects. Working together with people from other fields and national labs is an excellent experience for our students. We are looking forward to working with outstanding students -- the best way to join our team is by getting active in the projects mentioned above.
My research interests include numerical analysis, approximate solution techniques, mathematical modeling, and mechanics of flows through porous media. Some specific applications in these topics are contaminant transport, modeling of underground flows in enhanced oil recovery and saltwater intrusion. In some cases analytical or approximate analytical solutions can be constructed as a result important qualitative information on nature of solutions can be obtained, but in general equations are solved numerically. There are many numerical methods that can be used for analysis, such as, finite differences, mixed finite elements and Eulerian-Lagrangian methods. In modeling of flows through porous media multiple physical phenomena occur that complicate the numerical solution. Here are some of the difficulties: sharp fronts, which traditional upwind numerical schemes do not resolve due to the excessive numerical diffusion; a grid orientation effect; equations may include chemical reactions, consequently nonlinear thermodynamic constraints must be solved taking significant computational time. All these make these problems interesting and challenging.