Undergraduate Colloquium Spring 2003

Jan. 14     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 Chi-squared measure - that have broad applications in many fields.

Curious questions turn up in this study: come and find out the answer to the Birthday Paradox: "How many people do you need in a room before the probability is at least half that two of them share a birthday?"

Jan. 21     Aaron Fogelson
Enzymes and Mathematics: How biochemical reactions can be switched on and off
Abstract: Enzymes are proteins that control most of the important biochemical reactions that allow cells to function and us to live. They do this by allowing reactions to happen that would not happen on their own. Reactions between enzymes and the chemicals on which they act are modeled by ordinary differential equations. I will show how these equations are derived, that is what science goes into making mathematical models of enzyme reactions, and at least one way in which the reactions can be turned on and off by a small change in coefficents in the equations. This kind of switch plays an important role in quickly allowing a blood clot to form when we are hurt, and keeping it from forming (and blocking an artery) when we're not hurt.

Jan. 28    Gordan Savin
The Banach-Tarski Paradox, or How to make Two Oranges from One
Abstract: It is possible to break up a ball into finitely many pieces (essentially four), and then to rotate them to assemble two balls of the same radius! The construction is based on the axiom of choice, using a free group with two generators, which appears as a subgroup of the group of all rotations of the ball.

Feb. 4    Fletcher Gross
Godel's Incompleteness Theorem
Abstract: Is Mathematics consistent? Can all Mathematical questions eventually be decided? Providing answers to these questions was an important goal to many (from Leibniz to David Hilbert and Bertrand Russell in more recent times) but an astonishing result of Kurt Godel shows that their attempts were doomed to failure. On the other hand, mathematicians will never run out of problems to solve.

Feb. 11     Frank Lynch
Introduction to Population Ecology
Abstract: Mathematics can be used to describe dynamics of a population. We will explore two basic models of population growth. In particular, we will derive, analyze and interpret a deterministic model as well as a stochastic model. A bit of experience with differential equations and probability may prove helpful but is not necessary. Beware! I may ask you to use calculus.

Feb. 18    Bobby Hanson
Euler and Lagrange Show Us the Shortest Path
Abstract: This is an introduction to the method of Euler-Lagrange and the "Calculus of Variations" to transform the problem of minimizing a functional into the problem of solving a differential equation. We will derive the method and look at three examples where we can apply it. We will consider the problem of finding the shortest path between two points, the problem of finding the shape soap film takes when it spans two circular wires, and the problem of finding the shape an ideal ski slope should be to give the fastest race times. This talk should be accessible to anyone in their third semester of Calculus. From Calculus we will use critical points, integration by parts, and partial derivatives. Also, we will solve a differential equation by separating variables (after a trick).

Feb. 25    Graeme Milton
Rainbows, Halos and Glories: Mathematics in Nature
Abstract: Rainbows are so spectacular, but do you understand how rainbows occur? The explanation that it is the same effect as occurs when a shaft of light passes through a triangular glass prism and is dispersed into a spectrum of colors is true in some respects, but misses much of the story. It does not explain why the sky is darker on one side of the rainbow, nor the angles at which one sees rainbows, nor why sometimes one sees alternating purple and green bands on one side of the rainbow, nor why a rainbow in a fog bank is white (a fog bow). What is surprising is that rainbows provide evidence that the wavelength of light is finite, even though the wavelength is extremely small, on the order of microns (a micron is a millionth of a meter). An understanding of rainbows allows one to understand the halos that sometimes occur around the moon, and a detailed analysis accounts for glories such as those that ring the shadow an aeroplane casts on a cloud. Come to this lecture to find out more about how mathematics helps us understand nature. Rainbows are beautiful and so too is the mathematics which explains them.

Mar. 4    Stewart Ethier
Three Paradoxes in Probability
Abstract: By "paradox" we mean a mathematical result that conflicts with one's intuition. The first paradox is the famous Monty Hall problem, popularized by Marilyn vos Savant. The other two are less well known.

1. You are a contestant on the game show "Let's Make a Deal." Behind one of three doors is a new car, behind the other two are goats. You choose Door No. 1, let's say. The host, Monty Hall, opens Door No. 2 to reveal a goat. He then asks whether you would like to switch your choice to Door No. 3. Should you switch?

2. Two envelopes each contain an IOU for a specified amount of money. It is known that one IOU is worth twice as much as the other. You are given one of the envelopes unopened. Reasoning that the other envelope contains twice as much with probability 1/2 and half as much with probability 1/2, and noting that (1/2)(2) + (1/2)(1/2) = 5/4 > 1, you figure that you can increase your expectation by switching envelopes. Should you switch?

3. Again, two envelopes each contain an IOU for a specified amount of money, but all that is known is that the amounts are distinct nonnegative numbers. You are given one of the envelopes and allowed to open it. After seeing the amount of the IOU, you are given the opportunity to switch envelopes. Should you switch?

Mar. 11    Brynja Kohler
A Mathematical Look at the Physiology of Movement
Abstract: "[The body is] a marvelous machine...a chemical laboratory, a power-house. Every movement, voluntary or involuntary, full of secrets and marvels!" Theodor Herzl (1860 - 1904)

I will describe the physiology of muscle contraction as well as some of its simple control mechanisms. Through the mathematical formulation of models we can study these mechanisms and better understand certain pathologies. I will describe the stretch reflex and a pathological neuro-muscular tremor called clonus, and show how a study of differential equations which describe components of the physiology can enlighten us.



Mar. 25    Andrejs Treibergs
The Hyperbolic Plane is Too Big for R^3
Abstract: The hyperbolic plane is an abstract space whose geometry satisfies all the axioms of Euclidean Geometry except the parallel postulate. I'll discuss some of its basic properties, including how to give coordinates and to measure lengths. Then I'll consider the possibility of realizing this space concretely by mapping it to a curved surface in Euclidean three space in such a way that lengths of curves are preserved. This can be done on small pieces, for example, to a pseudosphere, but not globally.



Apr. 8   Nancy Sundell
Snow Geese: Modeling the Scourge of the Arctic
Abstract: In this talk I'll discuss an environmental problem created by Snow Geese grazing in Arctic salt marshes. We'll explore how differential equations can be used to both model the system, and provide insight into possible causes of the problem not discernible by biological field work alone.

Apr. 15   Bob Brooks
Probability: Some Early Problems and their Solutions
Abstract: A brief look at the beginnings of probability as a mathematical subject -- at least a look at one version of the story of its origins.