Undergraduate Colloquium Fall 2004
August 31     No Talk

September 7     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.

The fourth edition of the American Heritage Dictionary has a pithy description: a slide rule is a device consisting of two logarithmically scaled rules mounted to slide along each other so that multiplication, division, and other complex computations are reduced to the mechanical equivalent of addition or subtraction.

But there is more! 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. Apart from pizza, you will be given a basic but genuine, mint, NIB ("new in the box") forty years old and never used slide rule to start your own personal collection.

If you are curious, before the colloquium you may want to explore the slide rule exhibit in our library donated to our department by Chris Smith of the University of Utah School of Music.

Here's a couple of homework problems. You can do them before or after the talk. Let me know your answers:
1. Why didn't they build slide rules that can be used for addition and subtraction?
2. What's the base of the logarithm used for the design of any specific slide rule?

September 14     Andrejs Treibergs
Symmetrization and the Isoperimetric Inequality
Abstract: The isoperimetric inequality says that among all closed plane curves of a given length, the circle has the greatest area. (Thus, for any curve, 4 pi A <= L^2, where A is the area enclosed and L is the length of the boundary.) I'll discuss one of the most interesting and useful methods to prove the inequality, Steiner symmetrization.

September 21     Gordan Savin
What can you expect if you do an REU?
Abstract: Research Experiences for Undergraduates (REUs) have been designed by the National Science Foundation to promote participation of undergraduates in scientific research. While in some sciences, such as biology, undergraduates can certainly participate in data gathering and processing, is it realistic to expect that undergraduates can produce original mathematical research? Probably not, but in some rare cases it can happen. This was the case with Michael Hofmann, whose almost two years long research culminated in an original work on continued fractions. In this lecture I will describe his results, and try to give a timeline of events that led to his discovery and write up of results.

September 28    David A. Vogan, MIT
Diagonalizing Group Actions
Abstract: One reason that diagonalizing matrices is useful is that it makes algebra so easy: you can compute powers of a diagonal matrix very easily, and therefore compute polynomial functions. It's also very easy to solve systems of linear equations with diagonal coefficient matrices.

Representation theory seeks to get these same advantages for group actions. I'll explain what that means, and look at a series of (closely related) examples: the permutation group Sn acting on the set {1, ..., n}; the group GL(n) of invertible matrices acting on the projective space of lines in n dimensions; Sn acting on k-element subsets of {1, ..., n}; and GL(n) acting on the Grassmann variety of k-dimensional planes in n dimensions.

Here are two warmup questions: (1) How many k-dimensional element subsets are there in {1, ..., n}? (2) How many k-dimensional planes are there in an n-dimensional space? (The second question is actually in two parts: first, how can this question possibly make sense? Next, what's the answer?)

October 5    Jim White, Career Services
Getting Ready for the Career Fair
Abstract: The University Career Fair is coming up on October 14-15. Come find out from Jim White, the career counselor for mathematics, how you can take advantage of it!

October 12     Fletcher Gross
Rubik's Cube
Abstract: How many different configurations can a Rubik's Cube have? What is the relationship between this number and the number of configurations obtained by disassembling and reassembling the cube? What does all this have to do with group theory?

October 19    Stewart Ethier
The Mathematics of Trente et Quarante
Abstract: Trente et quarante ("thirty and forty" in French) is an elegant card game that dates back to the 17th century and is still popular in Monte Carlo. Numerous authors have attempted to evaluate the probabilities in the game. The first essentially correct evaluations were by well-known mathematicians Poisson (1825) and De Morgan (1838). We say "essentially correct" because these authors, as well as all subsequent ones, were forced to make certain unrealistic simplifying assumptions due to the complexity of the calculations. In this talk we will present, for the very first time, exact probabilities for the game of trente et quarante.

October 26    No Talk

Abstract: The undergraduate colloquium will not be held this week. Instead, students are encouraged to attend the free lunch for math majors held Wednesday, Oct. 27, from 12:00 - 1:30 at the Alumni House. This is a great place to meet other math majors, get some advising, and talk to professors.

November 2    Bob Bell
Combinatorial Gauss-Bonnet Theorems
Abstract: Suppose I give you a surface and you partition it into triangles. (For instance, if I give you a sphere, you might view this as an octohedron or an icosahedron.) If you compute the number of vertices minus the number of edges plus the number of triangles, you get a number called the Euler characteristic. (For the octohedron we get 6 - 12 + 8 = 2. What do we get for the icosahedron? Don't know what I'm talking about?....Check out http://mathworld.wolfram.com/Icosahedron.html.)

There is a classical theorem of Gauss and Bonnet which relates the Euler characteristic to the curvature of the surface. We will prove a combinatorial version of this theorem and discuss some related results from geometry.

November 9    Fred Adler
Why are there so many Common Colds?
Abstract: Over 100 serotypes of rhinoviruses, one of the primary causes of the common cold, co-circulate in the human population. This high diversity makes it effectively impossible to develop a vaccine, even for those at risk of complications due to asthma or cystic fibrosis. Why are there so many colds? We use methods based on random walks on finite spaces to estimate the number of colds that would exist due solely to their insanely high mutation rates, and compare the predictions with real data.

November 16    Jeremy Morris
How Big is the University of Utah?
Abstract: For the last two semesters I have been working on an REU project in network theory. After hearing a colloquium talk about 'small world' networks and a radio program on the Global Small World experiment, I became interested in what the small world phenomena might have to say about the student body of the U. I wondered what the criterion for a small world are and if the U. qualifies.

I was also interested in what other network properties might be studied to further understand the interactions students have with one another. Some linear algebra made popular by the website Google is quite interesting. Google uses some linear algebra to rank websites in terms of how 'authoritative' they are. I have used this same bit of linear algebra to rank University of Utah students. This gives us some indication of how popular any given student is. Who is the most popular person on campus? Come and find out.

I will discuss some basic definitions from network theory and then present my results. You may be as surprised as I was.

November 23    Marian Bocea
One-Dimensional Variational Problems
Abstract: The beginning of the modern era of the Calculus of Variations is marked by Johann Bernoulli's solution of the brachistrochrone problem in 1696. The problem was first formulated by Galileo in his 1638 "Discourse on two new sciences." The statement is as follows: Find the path (brachistochrone) joining two points P and Q in a vertical plane along which a heavy particle starting from rest at P (and moving without friction) will reach Q in the shortest time under its own gravity. Bernoulli circulated the problem to several mathematicians, among whom Jacob Bernoulli, de L'Hopital, Leibnitz and Newton correctly identified the brachistochrone as an arc of a cycloid.

In this talk the brachistrochrone problem will be discussed in some detail, along with other examples related to the classical indirect approach in the Calculus of Variations.

November 30   NO TALK

December 7   Henryk Hecht
What Is A Linear Function?
Abstract: Linear functions f(x)=mx are probably the simplest functions imaginable. They are additive: f(x+y)=f(x)+f(y), and it seems at first glance that every additive function must be linear. But... is it really so?