Undergraduate Colloquium Fall 2006

- August 23 No Talk
- August 30 Peter Alfeld
**Multivariate Splines**

*Abstract:*Splines are smooth piecewise polynomial functions. They are used to approximate, represent and interpolate data, and to design shapes like that of a car or airplane. In one independent variable, splines are simple and well understood. In several variables, splines are vastly more complicated. I will describe some of the issues and techniques, and list some unsolved problems.- September 06 János Kollár, Princeton University (THE FIRST TALK IN DISTINGUSHED LECTURES SERIES)
**What is the biggest multiplicity of a root of a degree d polynomial?**

*Abstract:*A one variable real polynomial f(x) has a k-fold root at the origin if |f(x)| is like |x|^k for |x| sufficiently small. We know that k is at most the degree of f. What can one say about several variable polynomials? Come and find out.- September 13 Julian Chan
**An introduction to groups and group theory**

*Abstract:*Getting to know groups S_n, D_n, and C_n. We will also discuss some of the classical theorems such as the class equation, Lagrange's theorem, and the first isomorphism theorem.- September 20 Bob Palais
**Surprising algorithms for performing rotations, and their consequences**

*Abstract:*Objects on computer monitors can be made to appear to rotate in three dimensions when pulled by a mouse. We will explain a little known algorithm which provides the simplest method to implement this task. It also allows us to visualize the composition of three dimensional rotations as easily as we do translations with the parallelogram law or two dimensional rotations with angle addition. It provides a method for interpolating 3D rotations which is more efficient than spherical interpolation on S^3. Finally, we will use these ideas to understand and demonstrate physically the following surprising fact: When an object in the center of a room is connected to the walls by strings, and the object is rotated one turn about an axis, the strings become tangled, but if the object is rotated another turn about that axis, they may be disentangled without moving the object.- September 27 Aaron Bertram
**Triangles in the Plane**

*Abstract:*We first saw triangles - scalene, isosceles and equilateral; acute, right and obtuse - in elementary school. Congruences among triangles (the SSS, SAS and ASA criteria) are taught in high school and date back to the ancient Greeks. But have you ever seen a map of all the triangles? Surprisingly, the triangles that cause the most problems are the ones with the most symmetries. And what about other polygons?- October 04 Zsuzsanna Horvath
**Random Number Generation Using Dynamical Systems**

*Abstract:*Numbers that are chosen at random have numerous applications in many fields, such as simulations, sampling, numerical analysis and computer programming. When we refer to randomness we refer to a sequence of independent random variables with a specified distribution. The main challenge is how do you create a random sequence when each number is dependent upon its predecessor? The answer is that the sequence itself is not random; it just appears to be. We use the Newton-Raphson process to generate a sequence of random numbers; we then show that these numbers are independent. We show the process can generate uniform random variables, which can be transformed into standard normal data.- October 11 No Talk, Gordan is in Schiermonnikoog (where is it?)

*Abstract:*- October 18 Gordan Savin
**The Banach-Tarski paradox or how to make two oranges from one**

*Abstract:*It is possible to break up a ball of, say, radius one into finitely many pieces (essentially four), and then to rotate them to assemble two balls of the same radius one! The construction is based on the axiom of choice, using a free group with two generators, that appears as a subgroup of the group of all rotations of the ball.- October 25 Troy Goodsell
**Problem Solving on the Rhine**

*Abstract:*George Polya formalized a four-step strategy for problem solving that is often taught to students in educational courses. Research mathematicians may not formally think through the steps of Polya's four-step strategy when approaching a problem yet in this talk we will look at an example of a distinguished research mathematician approaching a new problem and how he informally used Polya's problem solving strategy. The problem in our example is simple and elegant yet has a very surprising twist in its solution.- November 01 Robert Hanson
**Cantor Set and the Devil's Staircase**

*Abstract:*A couple weeks ago, Dr. Savin presented the Banach-Tarski Paradox. No collection of paradoxes is complete without a particular set of real numbers called the Cantor Set. We will use this wonderful set to explore various concepts of set "size." In particular, we will see that the Cantor set has just as many points in it as the entire real line, yet its total length is zero! We will then use this fact to create a very interesting function, called the Devil's Staircase. Although we will be using the concept of measure (more accurately, Jordan content), this talk should be completely accessible to anyone who has completed Calculs II (or at least limits of sequences).- November 08 Jacob Grosek
**Trisecting angles**

*Abstract:*By 400 B.C., Greek mathematicians had established the basis of (what is now called) Euclidean Geometry. Plato, a renown philosopher and mathematician of the time, suggested that geometry problems be solved using only a straight edge and a compass. During this period of discovery, mathematicians became intrigued with the question of how to trisect an arbitrary angle using a straight edge and a compass. I will show how Pierre Wantzel (1837) was able to prove that this is not possible for all angles. Here are the notes for the talk.- November 15 Davar Khoshnevisian
**Random Bits and Pieces: An Excursion in Symbolic Dynamics**

*Abstract:*Consider the following four questions that are borrowed respectively from computer science, gene-sequencing, metric number theory, and classical analysis: 1. Is a certain random-number generator actually generating random numbers? 2. Does a certain gene sequence belong to a given family? 3. What does a typical number between 0 and 1 look like? 4. How big is the middle-thirds Cantor set? Mathematics offers good answers to these, and many more, questions of this type. In fact, all four questions are answered by the same circle of ideas that are referred to as the ``symbolic dynamics of 0-1 sequences.'' I will describe one or two of these ideas in very concrete terms. It would be helpful to know some probability theory at the undergraduate level, but that is not necessary as the talk is self-contained.- November 22 No Talk, Thanksgiving Break

*Abstract:*- November 29 Gordan Savin
**Diffie-Hellman Key Exchange in Cryptography**

*Abstract:*Have you ever wondered why your internet purchase is "safe and secure?" Although the need to make information secret or unreadable to the general public is as old as human civilization, some recent ideas have revolutionized the subject. Come to this talk to see for yourself.- December 06 No Talk

*Abstract:*