Undergraduate Colloquium Fall 1999

August 31       Jim Carlson
How to Send a Clear Message Through a Noisy Channel
Abstract: CDs and digital cell phones give remarkable audio quality, even when the CD is scratched and the air is full of radio static. The technology that makes this possible is based on the mathematics of finite fields that grew out of the work of Gauss published in 1801. The aim of the talk is to explain the technology and the mathematics which makes this possible.

September 7    Sean Sather-Wagstaff
(Commutative) Rings and Things
Abstract: Algebra is one of the most fundamental subjects in mathematics. It is usually the first subject encountered by young mathematicians on their paths to other subjects. Many students fail to realize that algebra is a beautiful area which is not only interesting in its own right but also incredibly useful as a language and tool for working in a variety of other fields. In particular, commutative ring theory is one of the powerful tools used by algebraic geometers in the study of modern geometric questions. In this talk, I will introduce the basic objects of study in commutative algebra, especially focusing on rings of polynomials and their geometric counterparts.

September 14   Fletcher Gross
The Rubik's Cube
NOTE: This colloquium will be held in JFB B-1 (Physics building, basement)
Rubik's Cube is a group theorist's toy. This talk will be about some mathematical questions concerned with this toy such as (1) how many positions are really possible? (The Ideal Toy Company advertised that over 3 million positions are possible, apparently thinking that no one would believe the true number.) (2) How do the legal positions (those obtained by not cheating) compare with the number of positions obtained by disassembling and reassembling the cube? (3) Why is it impossible to twist a single corner and leave all other parts of the cube alone? The answer to (2) is equivalent to one of the first nontrivial results of group theory while the answer to (3) involves the important idea of homomorphism. While this talk may introduce some ideas from Group Theory, no prior background in the subject is needed or assumed.

September 21   Mark Lewis
Wolf Territories and Prey Survival
Abstract: Field studies in Northeastern Minnesota indicate that wolf (Canis lupus) territory patterns are clearly defined and that the spatial distribution of white-tailed deer (Odocoileus virginianus) is strongly affected by the wolf territories. In this work, wolf interactions and movement are modeled with mathematical equations. No assumptions are made about the territories themselves. Analysis of the model, however, indicates that territorial patterns arise naturally as stable stationary solutions. Lastly, the effect of resulting wolf distributions on the deer population is considered. Results from the mathematical model reflect field observations: deer are found primarily in buffer zones between the pack territories.

September 28   Hugo Rossi
Egyptian Multiplication, Medieval Usury and Installment Loans
Abstract: The ancient Egyptians had a technique for multiplying two integers which required them to know a) how to double a number and b) that every integer can be written as a sum of powers of 2. Their technique is precisely that used by modern computers and, after some adjustment, that we learn in school.

An abacus with B rings on each spindle works because any integer can be expressed (uniquely) as a polynomial in B with coefficients 0, 1,..., B-1. This was understood by the ancient Greeks, and leads to Euclid's proposition on geometric progressions (but not Euclid's proof). This, together with the usurer's rule for calculating new balances on loans, brings us to the formulae for installement loans embedded in business calculators.

October 5         Eric Cytrynbaum
Understanding and modeling the nerve axon
Abstract: Up until the middle of this century, it was known that neurons played a key role in the propagation of various signals through the body but no mechanistic understanding existed. In 1952, A. L. Hodgkin and A. F. Huxley published a series of papers that made for the largest single step in our understanding of this signal propagation and eventually won them the Nobel prize in medicine and physiology. A brilliant synthesis of experiment and mathematical modeling allowed them to explain observed results including an accurate numerical reproduction of the form and speed of signal propagation. Their work laid the foundations (practical and conceptual) for a wide variety of modeling contexts in physiology, including not only nerve cells but also cardiac cells, pancreatic beta-cells and endoplasmic reticulum (ER) driven calcium dynamics. In this talk, I intend to outline the work of Hodgkin and Huxley, focusing on how they inferred mechanisms from their experiments and how they translated these mechanisms from physical descriptions to differential equations.

October 12      Stewart Ethier
How to Win at Blackjack
Abstract: Blackjack (also known as 21) is the most popular table game played in gambling casinos today. Part of that popularity is undoubtedly due to the fact that blackjack is the only such game in which the skillful player enjoys an advantage over the house, notwithstanding the fact that only a very small percentage of blackjack players could be described as skillful. As discovered by Thorp in 1960 and subsequently refined by others, a winning strategy requires that the player keep track of the cards already played, thereby allowing him or her to identify favorable deck compositions and to respond with increased bet sizes. The aim of the lecture is to explain the mathematics behind all this, as well as some of the practicalities.

October 19      Undergraduate Career Day
Career Day will be October 19 from 1:00-3:30 in INSCC 110. We will begin with refreshments, then a counselor from Career Services will talk about looking for a job and creating a resume. We will also have a panel discussion with representatives from industry, and you may stick around for a while to talk one-on-one with the panel members. This will be fun and informative. You won't want to miss it!

October 26      Jim Carlson
Fixed Points of Mappings: Ancient, Modern, and Postmodern History
By 1700 BC the Babylonian scribes had found ways of efficiently computing square roots. This was likely the first use of the "fixed point method" to find approximate solutions to equations. We will describe the Babylonian method as well as later developments due to Isaac Newton and others. The method works for many kinds of equations (algebraic, matrix, differential). The idea is to find a (hopefuly simple) rule for generating a sequence of better and better approximations to the actual solution from some crude starting value. We will also discuss more general sequences which are generated by repeatedly applying a fixed rule, e.g., sequences of "random" numbers.

November 2     Michael E. Gage, University of Rochester
The Peano Kernel, Combining Abstract and Applied Mathematics
In 1913 Giuseppe Peano published a uniform procedure for obtaining estimates for a wide variety of numerical approximations, including the trapezoid rule and Simpson's rule for integration, Lagrange interpolation schemes, Euler's method for solving differential equations and many others. While well known to experts, Peano's procedure has been largely overlooked by numerical analysis textbooks, even though, using Peano's theorem as an organizing principle, it is possible to understand nearly all the error formulas from first semester numerical analysis in a uniform manner. I'll give a few examples of how this is done, illustrating in the process that using a few abstract ideas from linear algebra and functional analysis can conceptually simplify concrete problems.

November 9     Brad Peercy
Excitable Media: Qualitative Behavior
The goal of this talk is to introduce the idea of excitability through common examples as well as examples in biology. Ordinary differential equations can be used to describe excitability, and I will use phase plane analysis to understand these ODE's. The biology of the nerve and the mathematical description, again using ODE's, developed by Hodgkin and Huxley will be described. Spatial considerations, in the form of diffusion, will be added into the ODE's. I will briefly discuss concerns in cardiology which are being addressed by students in the mathematical biology program at Utah. Some of the concerns include mechanisms of defibrillation, atrial arrhythmias, excitation-contraction coupling, calcium flow in cardiomyocytes, and effects from ischemic heart. A short video will allow visualization of numerics describing some cardiac arrhythmias.

November 16   Jingyi Zhu
Option Pricing and PDEs
It is not an overstatement that financial derivatives have become part of everyone's life, with the continual growth of the economy and the longest bull market in recent decades. However, it is probably not widely known that the reason words like "options" have become ubiquitous is that back in 1973, a mathematician, Fischer Black, and economists Myron Scholes and Robert Merton discovered some beautiful mathematics that revolutionized the security market even to this day. The so-called Black-Scholes formula made it possible for Wall Street to trade all these securities. Today, mathematical finance has evolved to encompass diverse fields including stochastic calculus, PDE, and numerical analysis. In this lecture, I will attempt to describe how each of these fields contributes to the principle of option pricing as well as some personal perspective.

November 23      David Eyre
A Mathematical Look at Snow and Avanlanches
Snow in a winter snow pack is a remarkable material. It is a composite of air and ice that is continually evolving. This evolution depends on the environmental conditions and can either strengthen or weaken the material properties of the snow. The strength of snow to loading is a particularly difficult problem to understand and model. It is also a problem of practical interest because of snow avalanches. In this talk, I'll describe the basics of a winter snow pack and avalanche dynamics, and I'll present some mathematical models of both.

November 30      Blake Thornton
A Short Introduction to Nonstandard Analysis and Nonstandard Constructions
The concepts of infinitesimal and infinite numbers have been used by a great number of mathematicians including Newton, Leibniz, and Euler. Reading their work today looks a bit nonsensical. In the 1960's Abraham Robinson developed a framework for calculus using infinitely small and infinitely large numbers, thus making sense of what Newton and Leibniz did many years earlier. I will discuss a bit of the history of this and then construct the non-standard real numbers: a field with infinitesimal and infinite numbers. If there is time, I will discuss some applications.

December 7      Nat Smale