Most systems of interest, whether they arise from the physical or life sciences, engineering, or the social sciences, evolve with time. Often these systems are nonlinear and thus detailed analysis of specific systems typically requires calculations of one form or another. The fundamental mathematical challenge is how to provide useful qualitative and/or quantitative descriptions of these systems, which of course determines the type of computations that need to be done. Our group focuses on three (not necessarily unrelated) types of problems.

1. An explicit analytic model is given. This is perhaps most common in applications associated with physics and classical engineering. In this case we are interested using the computer to obtain mathematical rigorous existence results and bounds on solutions.

2. Experimental and/or numerical simulations of high dimensional systems. We used and develop tools based on topological methods to simplify the data and to extract features about the dynamics.

3. Our understanding of the system of interest is based on heuristic principles and imprecise experimental data. This is common to many multi scale systems. Of primary interest to us are signal transduction/gene regulatory networks where phenotypic expression is the result of multiple complicated interactions ranging from the atomic to the cellular level. Our goal is to develop a coarse theoretical approach to nonlinear dynamics that focuses on accurate as opposed to precise statements about the dynamics and is amenable to efficient large scale computations. The theoretical framework for this approach is based on combinatorics and algebraic topology and we are collaborating with experimental biologists to apply these ideas to the understanding of regulatory networks.

Fall 2019: Our group has regular meetings Tuesday 4:00-6:00 pm.