Undergraduate Colloquium Spring 2008
January 9     No Talk

January 16     Andrejs Treibergs
Steiner Symmetrization and Applications
Abstract: Steiner symmetrization about an axis transforms a domain in the plane to one that has perpendicular slices of the same length as the original domain but is mirror symmetric with respect to the axis. The transformed domain has the same area but less boundary length. By applying symmetrization several times, Steiner gave an argument for the isoperimetric inequality, that the area of any domain in the plane is always less than the area inside a circle with the same boundary length. Symmetrization can also be used to estimate the diameter as well as the fundamental tone of a domain.

January 23     Bob Palais
Euler's Theorem on the Axis of a Rotation: Proofs Old and New
Abstract: A rotation in two dimensions (or other even dimensions) does not in general leave any direction fixed. Even in three dimensions it is not immediately obvious that the composition of rotations about distinct axes is equivalent to a rotation about a single axis. However, in 1775-1776, Leonhard Euler published a construction for finding a fixed axis of any rigid motion of a sphere about its center. This important fact has a myriad of applications in pure and applied mathematics, and as a result there are many known proofs, which we survey, and add a new one that is elementary and contructive. We include and discuss Euler's demonstration (in the original Latin with an English translation) and not that it contains elements of our new proof.

January 30     Peter Trapa
Convergence of Difference Boxes
Abstract: Consider the following construction of a "difference box":

1. Draw a (large) square, and label each vertex with a real number.

2. On the midpoint of each side, write the absolute value of the difference between the two numbers at its endpoints.

3. Inscribe a new square in the old one, using the new numbers to label the vertices.

4. Repeat this process, and continue inscribing new boxes until reaching a square that has all four vertices labeled 0.

One can ask if there is a starting box so that this process never terminates. For instance, you can't do it using integer labels: any initial configuration of integers converges to the zero box in a fairly small number of steps. (As a warm-up, we'll prove this.) But what about noninteger values? Surprisingly the answer has something to do with the tribonnaci sequence, 1, 1, 1, 3, 5, 9, 17,..., where each term is the sum of the previous three.

February 6    Dylan Zwick
Knots, Unknots, and Polynomials
Abstract: In this talk I'll give an introduction to the basics of knot theory: what do we mean by a knot, what questions knot theory tries to answer, and what techniques knot theorists use to answer these questions. I'll then introduce some slightly more advanced concepts in knot theory like how to multiply two knots together, and end with a discussion of the Jones Polynomial, one of the more powerful techniques to come out of knot theory in the last thirty years. This talk should be accessible, seriously, to anybody who's made it through calculus.

February 13     Peter Alfeld
What can you do with a slide rule?
Abstract: Back in the days when people first went to the moon, electronic calculators did not exist. Instead we used slide rules. They were indispensable for professionals, and students were required to own one and know how to use it. There were courses on the proper use of a slide rule. Just like calculators today, slide rules were mostly everyday and commonplace instruments, but some were fancy, expensive, and treasured by their owner.

I'll describe how slide rules work, why they work, and what you can do with them. A typical slide rule has anywhere from ten to thirty scales, rather than just two, and there are thousands of mathematical expressions that you can evaluate just as easily as you can multiply or divide two numbers. On the other hand, you can't use a slide rule to add or subtract two numbers, and you need to understand your problem well enough to be able to figure out on your own the location of the decimal point in your answer.

You'll be able to examine several slide rules, and I'll tell you what's involved in being a slide rule collector. You may even decide to have a fun weekend and attend the upcoming slide rule convention of the Oughtred Society on February 23 at the Atomic Testing Museum in Las Vegas: http://www.oughtred.org/WinterMeeting.shtml

Here's a couple of home work problems. You can do them before or after the talk. Let me know your answers:

Why is it so hard to find slide rules that can be used for addition and subtraction?

What's the base of the logarithm used for the design of any specific slide rule?

February 20    Tommaso Centeleghe
p-adic numbers
Abstract: We are all familiar with R, the set of real numbers that can be defined by means of sequences of rational numbers satisfying the Cauchy condition. Its very definition involves, therefore, the notion of distance on Q. Mathematicians soon discovered, however, that there are many other distances that can be put on the set of rational numbers! There are indeed infinitely many of them, and they are naturally parametrized by prime numbers. The construction of the so called completion of Q with respect to this exotic distances leads to the p-adic numbers Qp. In this talk we would like to learn how these distances are defined and we would like to present a few basic properties of Qp, underlying analogies and differences with R. We'll soon realize that some psychological adjustment is need in order to work with these distances. I'll try to convince you that the sum of pi converges in Qp!

February 27    Distinguished Colloquium Series: Benson Farb, U of Chicago
Topological Invariance of Non-Topological Invariants
Abstract: There are many objects in math that at first glance have no relationship between them. This talk will give 3 examples of objects which seemingly have no relationship with topology, which is the study of the shape of space.

Sometimes in math there are deep and surprising connections that can escape our first glance, but exist nonetheless, and that's certainly the case with the examples we'll see.

March 5    Stewart Ethier
The Mathematics of Texas Hold'em
Abstract: Texas hold'em is the most widely played form of poker today. Each player receives two hole cards face down. Then three community cards are dealt face up (the flop), then a fourth one (the turn), and finally a fifth one (the river). There are four betting rounds, pre- flop, post-flop, post-turn, and post-river. A showdown follows, with the best five-card poker hand winning. As they say on television, "It takes a minute to learn and a lifetime to master."

We begin with some elementary considerations and then turn to the difficult question of how to rank the 169 distinct initial hands. We then use this information to illustrate how a particular hand might be played by a mathematically inclined player.

March 12    Kevin Wortman
Hyperbolic Space
Abstract: Euclidean space provides the setting for the geometry most commonly learned in early math curriculum. In this talk we'll see another alternative setting for geometry called hyperbolic space. It might seem unusual at first, but it's no less important in higher mathematics than its more well known counterpart. And aside from being useful, it's pretty interesting on its own.

March 19    No Talk - Spring Break


April 2    Daniel T. Gillespie, Distinguished Colloquium Series
Continuous Markov Processes
Abstract: Ordinary differential equations describe the continuous, past-forgetting (Markovian), deterministic way in which many physical processes evolve in time. But suppose we wanted to describe a process that is continuous, past-forgetting, and stochastic. What changes to ordinary differential calculus would we have to make? And why? This talk will take a peek into the fascinating world of Langevin and Fokker-Planck equations.

April 9    Aaron Wood
Abstract: A partition of a positive integer n is a non-increasing sequence a_1, ..., a_k of positive integers such that n = a_1 +...+ a_k, and the number of possible partitions of n is denoted p(n). For example, the partitions of 4 are 4, 3+1, 2+2, 2+1+1, 1+1+1+1, so p(4)=5. To study the partition function p(n), many tools and techniques have been developed using combinatorics, generating functions, and modular forms. Using combinatorics and generating functions, we will discuss various properties of p(n) like Euler's Theorem and the Ramanujan congruences, and we'll mention the exact formula for p(n) due to Hardy and Ramanujan.

April 16    Daniel Allcock, Distinguished Colloquium Series
The Jump to Light Speed
Abstract: Everyone knows that when your starship jumps to light speed: you see the stars suddenly rush past, so that it looks like an explosion of stars in front of you. But this isn't what actually happens: really, they appear to all rush together in front of you. So it looks like an *implosion* instead. Find out why. Dot products and visual imagination are all that is needed.

April 23    Dan Ciubotaru
Riemann's Zeta Function
Abstract: One of our favorite infinite series from calculus is the sum from one to infinity of zeta(k) = 1/n^k, and we all learn that this series is convergent for k>1, and it is divergent for k <= 1. I will show you how to compute the series for every positive even integer k, for example zeta(2) = 1 + 1/2^2 + 1/3^2 + ... = pi^2/6, or zeta(8) = 1+ 1/2^8 + 1/3^8 + ... = pi^8/9450. Riemann found a way to extend these series to a function defined over the complex numbers. I will try to explain how this function is related to prime numbers, and to introduce you to one of the most famous unsolved problems in mathematics, the Riemann Hypothesis.