Math 350 - Fall 2003

Mathematics of Reaction-Diffusion Systems

Math 350 (Topics in Applied Math) - Fall 2003

Instructor: Renato Feres - Time: Tuesdays and Thursdays, 2:30 PM - 4:00 PM (place to be announced)

Reaction-diffusion equations are important in a wide range of areas such as cell processes, drug release, ecology, spread of diseases, industrial catalytic processes, transport of contaminants in the environment, chemistry in interstellar media, to mention a few. It has also drawn from, and contributed to, developments in pure and applied mathematics, such as the theory of dynamical systems and non-linear partial differential equations.

Our course will describe some of the mathematical and theoretical ideas involved in the study of these systems. It will have special focus on chemical kinetics, which we view as a prototype of, or metalanguage for many subjects beyond chemistry itself. More specifically, we'll describe a number of mathematical topics that bear on the study of chemical processes --- the algebraic structure of complex networks, deterministic and probabilistic interpretations of kinetic equations, and the evolution in time and space of solutions to these equations. It will include:

  1. Algebra: linear algebra, elements of graph theory, polynomial systems and Grobner basis.
  2. Dynamics: non-linear differential equations and dynamical systems; stability; bifurcation analysis; chaotic attractors.
  3. Spatially extended processes: the linear diffusion equation; diffusion and probability; reaction-diffusion equations; traveling waves.

Lectures and class activity are planned around projects on topics of both mathematical and general scientific interest. Some of these topics are directly related to research being conducted at Wash U. The emphasis will be on collaborative work and on fostering a research "mode of thinking," in which students from a variety of backgrounds, skills, and interests work together on relatively open-ended problems, using both abstract mathematical and numerical tools.

We will have the assistance of Professor Gregory Yablonsky, of the Chemical Engineering School, who is an expert in catalysis. (Gregory was among the first scientists, in the 1970's, to propose realistic chemical mechanisms that exhibit complex dynamical behavior such as oscillations and chaos.) He will give a number of invited lectures and will be available throughout the course to help in class activities as well as help the students with projects and other assignments.

The course is aimed toward undergraduate students in all areas of science who would like to learn some useful and interesting mathematics beyond multivariable calculus, although graduate students are also welcome to attend. Evaluations will be based on class participation and take-home assignments. Prerequisites are Math 233 or equivalent knowledge of multivariable calculus, some familiarity with matrices and linear algebra, as well as some familiarity with general chemistry. If you are interested but are not sure whether you have the background, come talk to me. For this or any other questions, feel free to contact me at or stop by my office at the math department. It is in Cupples I room 17 (phone 5-6752).

A very rough and preliminary set of notes that covers the first month of lectures can be downloaded. (This is a pdf file.)