In this talk, I will give a brief summary of my current research projects with open problems for students interested in Ph.D. thesis research. These projects are principally in the areas of tissue pattern formation in developmental biology and genetic instability in carcinogenesis. Some details will be given to show the nature of the mathematical and computational problems involved.

A number of important model linear dispersive equations give rise to
interesting curve flows in differential geometry. In this talk we will
discuss some of these including the Schrodinger curve flow on the two
sphere, the Hodge star mean curvature curve flow in Euclidean and
Lorentzian 3-space, and the geometric Airy curve flow on Euclidean
space and the affine space. The equations of curvatures of these
curve flows turn out to be soliton equations. Hence we can use
techniques from soliton theory to study these curve flows. In
particular, we can construct infinitely many families of explicit
solutions and solve the periodic Cauchy problem.

In this talk I will discuss ways in which cells carry out decisions based on available information from the environment.  One of the most common methods is to use proteins that have several modification sites, and which use these sites to increase the all-or-none behavior of the cellular decision.  New mathematical modeling techniques can improve our current understanding of these systems.