## Professor James Keesling

Department of Mathematics
University of Florida
P.O. Box 118105
Gainesville, FL 32611-8105
Phone: 352-392-0281 ext. 289
Fax: 352-392-8357
E-mail: jek@math.ufl.edu
http://www.math.ufl.edu/~jek/

Professor Keesling (Ph.D., University of Miami) has been at the University of Florida since 1967 and Professor since 1975. He has published numerous research articles and given numerous invited addresses on his research in national and international forums. He is one of the managing editors of Topology and Its Applications. In recent years, he has also concentrated on talks aimed at undergraduates to increase interest in mathematics.

Fractals: Jagged Geometry

This talk is about the geometry of fractal sets. Two types of fractal sets are the focus: self-similar sets and Brownian motion. Self-similar sets have the same jagged appearance under any magnification. Examples began to appear in the mathematical literature in the early part of the twentieth century. However, the advent of the computer has increased fascination with these objects. This is because they can now be created so easily using algorithms programmed on the computer. These sets have an allure of their own, but also show promising applications.

Fractals based on Brownian motion is the other focus. Brownian motion is a natural phenomenon which has been shown to be intrinsically fractal based on the physical principles governing it. The theory of Brownian motion and fractional Brownian motion has led to algorithms that generate realistic landscapes which are used in high tech cinema. It is also being used to evaluate algorithms processing radar images which pick out the important objects in the display. This talk can be adapted to any level of audience from high school to graduate and research level. Even students with limited background will learn to calculate the Hausdorff dimension of some examples.

The Chaotic f(x)=Ax(1-x)

Whenever one is studying any system, a good task would be to describe it mathematically. If this can be done, the next task is to analyze the behavior of the model. Not so long ago, most scientists had the naive notion that such models would be dominated by stable equilibria or periodic orbits. The quadratic family of functions shows that this is altogether false. For many values of the parameter A, iterations of f(x) will converge to a stable equilibrium or a stable periodic orbit as anticipated. However, for many values this fails. The bifurcation diagram illustrates just how strange the behavior can be for such a simple and familiar family of functions. This diagram is a helpful geometric focus for this talk.

The quadratic family has been used as a model of growth of a population limited by food or other resources. That certain values of the parameter A lead to strange and unanticipated population fluctuations is a discovery that dates back to the early 1970's. It has changed the way biologists look at population changes. It is also changing the way scientists view nature.