Applied Math
Project List
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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.