I am interested in the development of numerical methods for approximating partial differential equations. In particular, I am concerned with the development, implementation, and analysis of finite difference methods, finite element methods, and discontinuous Galerkin methods for approximating viscosity solutions of fully nonlinear PDEs. I am also interested in the design and analysis of numerical solvers for algebraic systems that result from the discretization of PDEs.
My research is in computational number theory. I am particularly interested in algorithms for local fields and computational class field theory. Recently I also investigated the distributions of the derivatives of the Riemann Zeta function.
My research generally consists of building and analyzing mathematical models of biological phenomena. I use computers as substitutes for experiments on animals and plants. At the early stages, I usually write a computer simulation to get a better grip of the problem I need to model. During later stages, I use usually a simulation to test the properties of the model as well as to try to verify the model predictions.
I study arithmetic quotients of symmetric spaces. These locally symmetric spaces stand at the intersection of various topics in number theory, geometry, and topology. In particular they are closely related to the theory of automorphic forms. I use explicit reduction theory coming from quadratic forms over number fields in order to construct polyhedral tessellations that can be used to compute cohomological modular forms.
- Ricky Farr is working on project "Evaluation of the Derivatives of the Riemann Zeta Function on the Left Half Plane" under direction of Dr. S. Pauli.
- Brian Sinclair is working on project "Polynomial Factorization over Local Fields" under direction of Dr. S. Pauli.
Past Graduate Students
- William Ely completed a project "Pricing European Stock Options using Stochastic and Fuzzy Continuous Time Processes" under direction of Dr. J. Rychtář.