Undergraduate Colloquium Summer 2000

- May 30 Jim Carlson
**Beyond the Third Dimension**

*Abstract:*How to visualize and think about objects in four and more dimensions.- June 6 Klaus Schmitt
**The Mandelbrot Set and some of its Ancestors**

*Abstract:*In the lecture I shall discuss iteration schemes which are defined by rational functions. In particular, I shall discuss work of Cayley, Julia, and Fatou, classifying the sets of initial values for which particular schemes converge, diverge, or lead to periodic or chaotic motions. The ideas and notions developed will allow for a description of the Mandelbrot set.- June 13 Hugo Rossi
**The Work of Archimedes**

*Abstract:*It is said of Archimedes that, had he not been slain in the fall of Syracuse in 212 BC, he would have invented the calculus. Although this may be an overstatement, he did consider problems of areas and tangents whose methods were adopted in the seventeenth century for more general investigations. I shall present two or three of these problems of Archimedes.- June 20 Jim Carlson
**The Parallel Postulate and NonEuclidean Geometry**

*Abstract:*Since the time of Euclid around 300 BC, mathematicians have felt the axiom about parallel lines was not as basic as the others. Many attempts were made to show that it could be derived from simpler principles, but all failed until 1829 when Lobachevsky showed that the parallel axiom is logically independent of the others. the talk will trace the development and influence of ideas about the parallel postulate, and will show how to construct two non-Euclidean geometries.- June 27 Jim Carlson
**Poincare's Dynamics and the Recurrence Theorem**

*Abstract:*Will a billiard ball come close to its starting position if it rolls long enough? Does the earth's orbit describe a closed path in space? Is the earth's orbit stable? These questions were addressed and partially solved by geometric methods going back to Liouiville and Poincare, and are the subject of today's talk.- July 11 Peter Alfeld
**Hotel Infinity**

*Abstract:*You are the owner of Hotel Infinity. It has infinitely many rooms, and it's full. A new guest arrives and insists you give her a room. How do you accomodate her? The next day, a family with infinitely many members arrives, each of whom wants a private room. The next day the Olympics start. Infinitely many families, each with infinitely many members, arrive. Each family member insists on a private room. You can do it! Infinity is different.- July 18
**No Colloquium**

- July 25
**No Colloquium**

- August 1
**No Colloquium**