In this Research Experiences for Undergraduates project, six students will work with faculty advisors relating advances in computational technology to mathematical research. The emphasis of the work will be on computational aspects of two areas: group theory and dynamics. Following initial orientation, students will break up into teams, concentrating on one of the two themes. The interplay between experimentation and theory will enable the students to deal, at an early stage, with complicated examples which cannot be done by hand. These will lead to a better understanding of groups, dynamical systems and their underlying structure. Students concentrating on group theory will be introduced to fundamental concepts of computational group theory through permutation groups and exposure to CAYLEY and other programming languages capable of symbolic manipulation. Dynamical systems will build on students' basic courses in calculus and linear algebra to investigate properties of orbits such as period doubling and saddle node bifurcations, Mandelbrot sets and the generation of fractal images using iterated functions systems. As the students become more familiar with computer experimentation, they will be exposed to problem areas and unsolved problems on which they will focus their attention for the remainder of the activity. Interspersed with the research program will be an evening lecture series in the sciences, history and literature.
