Applied Math   Project List

Though many of the problems described here are of general utility, we are focussed on biological application. The ultimate goal of much of our work is to be able to analyze the networks that govern cellular and tissue development. The physical phenomena encompassed in such an analysis span many levels from processes governed by just a few molecules homogeneously dispersed in a medium, to those governed by mechanical/chemical coupling and transport localized to a low dimensional surface such as a membrane. All processes are ultimately nonlinear and they occur on a near continuum of time-scales.

Thus nearly all mathematical projects are dedicated to:

1) being able to decide which level of abstraction is appropriate for modeling which phenomena (molecular dynamics, stochastic process, homogeneous chemical kinetic, reaction-diffusion, mechanochemial, thermodynamical/statistical mechanical, discrete?),

2) how to efficiently simulate systems composed of models at different levels of abstraction.

3) how to decompose these large networks of interacting biological objects (DNA, RNA, proteins, metabolites, compartments, membranes, cells) into a hierarchy of subsystems separated due either to weak coupling or by "high impedance" boundaries (boundaries across which little mass or energy flows).

4) how to create a simplified picture of control and dynamics in biological subsystems. In simple, low dimension, nonlinear dynamical systems this is most often accomplished using bifurcation analysis and linear sensitivity theory.