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.
What happens when we decrease the length of a closed curve in the plane as fast as possible? This seemingly simple question has a very nice answer which involves a beautiful combination of partial differential equations and planar geometry. Come and get a glimpse of the amazing subject of geometric flows!
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.
Crawling cells, including the white blood cells that patrol your body in search of infections, display several distinct dynamical patterns, driven by both biochemistry (diffusion and reactions between chemical species) and mechanics (physical forces between the components inside cells). Our understanding of these spatiotemporal patterns has been aided by mathematical modeling using techniques including partial differential equations (PDEs). Recently, traveling waves have been observed in the protein actin, which powers certain cells’ ability to crawl. Following experimental observation of one type of crawling cell, we hypothesized that traveling waves are excitable waves arising from interactions of three components and developed a mathematical model formulated as a system of PDEs with a nonlocal integral term. Numerical solutions lead to a number of predictions, confirmed in further experiments. Our model also reveals a role for tension in the membrane that surrounds the cell, which would otherwise be difficult to observe directly by experiment.
In this talk I will use a classic moving boundary problem of fluid dynamics to offer some insight into the kind of questions and results researchers in nonlinear partial differential equations are interested in.
