Undergraduate Colloquium

August 30 No Talk

September 6 Bob Bell

A Topological Characterization of Regular Curves in the Plane
Abstract: Suppose two curves (with non-vanishing tangent vector) are drawn in the plane. Can you always continuously deform one curve to the other? What if you insist that at any stage in the deformation the curves still have non-vanishing tangent vector? In 1935, Hassler Whitney gave a complete answer to this question in terms of "winding numbers." We will discuss Whitney's proof and possibly discuss the problem's relevance to a present-day mathematical topic: the study of "Legendrian knots."

Reference: Whitney, Hassler - Collected Works Volume 2 - "On Regular Closed Curves in the Plane."

September 13 Gordan Savin

On Primes in Arithmetic Sequenes
Abstract: A famous theorem of Dirichlet says that there are infinitely many primes in every arithmetic progression a, a+d, a+2d, ..., where a and d are two relatively prime positive integers. Take, for example, d=3. Then a can be either 1 or 2, and the corresponding arithmetic progressions are (with primes in bold face)

1, 4, 7, 10, 13, 16, 19, ...
2, 5, 8, 11, 14, 17, 20, ...

In this lecture we shall present a complete and simple proof of the theorem of Dirichlet in this case, based on the fact that the alternating series

1 - 1/2 + 1/4 - 1/5 + 1/7 - 1/8 + ...

is convergent and non-zero. (Note that the denominators of fractions in this series are precisely the terms of the two arithmetic progressions given above.) The general proof follows the same idea, but is technically a bit more involved, and requires the use of complex numbers.

September 20 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 27 Meagan McNulty

From Romeo and Juliet to Respiratory Disease
Abstract: Seven years ago Steven Strogatz proposed using Romeo and Juliet's tragic love story as an example to describe the behaviors of a system of coupled ordinary differential equations. Using these two lover's feelings for each other, we will explore different behaviors of a general linear autonomous system. For instance, would this tragic tale have ended differently if Romeo and Juliet had been more cautious? We will then try to expand this idea to a simple non-linear system of coupled equations, derived from a model of respiratory disease. Perhaps, Romeo and Juliet can help us discover mechanisms behind respiratory disease.

October 4 Klaus Schmitt

Iteration of Rational Functions, Julia-Fatou Sets, and Chaotic Dynamics
Abstract: During the 1870's Cayley and Schroeder, independently, observed some very complex dynamics of the Newton iteration schemes for solving polynomial equations. Their work was the motivation of some very beautiful work by Julia and Fatou on the iteration of rational functions during the years 1918-1920. The lecture will describe what Julia and Fatou found, relate it to chaotic dynamics of some iteration schemes and give their description of the Mandelbrot set.

October 11 Nelson Beebe

Pseudo-Random Numbers: mostly a line of code at a time
Abstract: Random numbers have an amazing range of applications in both theory and practice. Approximately-random numbers generated on a computer are called pseudo-random. This talk discusses how one generates and tests such numbers, and shows how this study is related to important mathematics and statistics --- the Central-Limit Theorem and the X² measure --- that have broad applications in many fields. Come and find out what the Birthday Paradox, Diehard batteries, gorillas, Euclid, French soldiers, a Persian mathematician, Prussian cavalry, and Queen Mary have to do with random numbers.

October 18 No Talk

Abstract:

October 25 Stewart Ethier

Benford's Law
Abstract: In 1881 Simon Newcomb observed that books of logarithms in the library were dirtier in the beginning and progressively cleaner throughout. From this he inferred that users of log tables were looking up numbers with first significant digit 1 more often than numbers with first significant digit 2, those with fsd 2 more often than those with fsd 3, and so on. This led him to conjecture in the American Journal of Mathematics the probability law

P(first significant digit is d) = log₁₀(1 + 1/d)

for d=1,2,...,9. In 1938 Frank Benford independently conjectured the same result, and tested it with a data set of over 20,000 observations. His paper, in the Proceedings of the American Philosophical Society, was more widely read, so the law was named after him.

Attempts to prove Benford's law remained unsuccessful until about 10 years ago when Theodore Hill discovered the key idea. We will discuss Hill's results. We will also discuss modern applications of Benford's law, which include testing for tax fraud.

November 1 Lajos Horváth

Models for Volatility
Abstract: Economists have been interested in models explaining volatility for nearly 50 years. They wanted to explain the following features of markets, exchange rates and financial instruments:

(i) volatility is not constant, it depends on the previous volatility and market values
(ii) a random shock should persist measured by volatility
(iii) volatilities of the same order appear in clusters

Granger and Engle (Nobel prize winners in 2003) suggested a time series approach to volatility. Their model and its various modifications have become the focus of research in econometrics. We discuss if the properties (i)--(iii) could hold assuming the Granger--Engle model exists at all.

November 8 Henryk Hecht

Regular Polyhedra
Abstract: We can construct regular polygons with an arbitrary number of sides. However, there exist only five distinct regular polyhedra. Why is it so? We investigate this, and related questions, about polyhedra. In particular, we outline a beautiful construction of a regular polyhedron with twenty sides (icosahedron) due to Luca Pacioli, who was a friend of Leonardo da Vinci.

November 15 Christopher Hacon

Can We Turn the Sphere Inside Out?
Abstract: We will illustrate some of the mathematical ideas that arise in attempting to turn the sphere inside out.

November 22 Charlotte Hansen

The Minimal Polynomials of Trigonometric Values at 2*pi/p
Abstract: We will present and prove the minimal polynomials of tangent, cotangent, secant, and cosecant at 2(pi)/p, where p is a prime. This is motivated by the paper on the minimal polynomials of sine and cosine at 2(pi)/p written by Beslin and Angelis.

November 29 Marian Bocea

Numbers
Abstract: Well-known results and several open questions regarding the properties of some remarkable real numbers will be discussed.

December 6 Video Presentation

The Right Spin
Abstract: "The story of a dramatic rescue in space and the mathematics behind it." Told by astronaut Michael Foale.