Midwest Partial Differential Equations Seminar: Abstracts and Lecture Materials
We investigate empirically and theoretically the bond price in the framework of a mathematical finance term structure model.
At the restriction point of its first growth phase G1, the cell must decide whether to go into the S phase, apoptosis, or the quiescent phase G0. A similar decision is made just before the cell is ready to go into mitosis. The above decisions are affected by the cell environmental conditions, e.g., hypoxic neighborhood, overpopulation, etc. When some genes are mutated, the decision to go into S may be made in spite of unfavorable conditions, such as hypoxic conditions, and this leads to tumor proliferation.
The multiscale model we shall discuss deals with the effects of gene mutation during the time a cell spends in each phase, as well as during the absolute time. The system, consisting of hyperbolic and diffusion equations, involves, in addition to the two time scales, also the space variable and the variable boundary ("free boundary") of the tumor. I shall state some general theorems, and some qualitative properties of the free boundary.
This talk will discuss a remarkable and quite symmetrical interaction between the applied mathematical subject of nonlinear wave motion and the pure mathematical subject of algebraic geometry. As an example, we will talk about how algebraic geometry can be used to generate solutions of nonlinear wave equations. Then we will discuss how nonlinear wave theory solves a long-standing problem of algebraic geometry. Other topics may also be discussed if there is time.
Shadow systems have been used to approximate 2x2 reaction-diffusion systems. In this talk, I shall discuss the dynamics of shadow systems, and compare them to their original 2x2 reaction-diffusion systems.
We investigate a model for drawn fibers made up of cylindrical, smectic layers of banana-shaped liquid crystal molecules. The molecular shape induces both a spontaneous polarization and a distinctive effect on layer density. These features lead to a model that predicts stable fiber formation. We cast the model as a free boundary problem for the fiber's cross-section and analyze its solutions. This is joint work with P.Bauman.
The manifold of the steady-state solutions of 2d Euler's equation in a domain (and with suitable boundary conditions) is typically infinite-dimensional. The geometric interpretation of Euler's equations suggests a natural local parametrization of the manifold (under some non-degeneracy assumptions). This is established rigorously in some interesting situations. (Joint work with Antoine Choffrut.)
