# Undergraduate Colloquium Fall 2000

- Aug 29 Jim Carlson
**The Mathematics of Vibrations and Waves**

*Abstract:*In the mid-nineteenth century, Charles Fourier showed how any wave, no matter how complicated, can be built up from simpler ones (sine waves). His ideas help us to understand heat, light, and sound, to record and reproduce music with high fidelity, and much more. This talk will explain Fourier's ideas in elementary terms and show how they can be used.- Sep 5 Herb Clemens
**Exploring Simple Figures in Higher Dimensions**

*Abstract:*Simplexes are the simplest geometric objects in each dimension. We will break up some 2, 3, and 4-dimensional objects into simplexes in order to visualize them.- Sep 12 Nat Smale
**Minimal Surfaces, Soap Films, and the Plateau Problem**

*Abstract:*Minimal surfaces have a long, rich history, dating at least as far back as the studies of the Belgian physicist Plateau in the 1940's. He demonstrated, using the laws of surface tension, that a soap film spanning a closed wire contour must have smaller surface area than any nearby surface spanning the same contour (thus the term 'minimal surface'). In this talk, I will survey some of the classical results on minimal surfaces, and try to show how the subject bridges several areas of mathematics, such as geometry, analysis and partial differential equations.- Sep 19 Fletcher Gross
**The Distribution of Primes**

*Abstract:*Some standard facts of first year calculus are used to say something about how many primes there are and how often they occur.- Sep 26 Peter Alfeld
**Hotel Infinity**

*Abstract:*You are the owner of Hotel Infinity. It has infinitely many rooms, and it's full. A new guest arrives and insists you give her a room. How do you accomodate her? The next day, a family with infinitely many members arrives, each of whom wants a private room. The next day the Olympics start. Infintely many families, each with infinitely many members, arrive. Each family member insists on a prive room. You can do it! Infinity is different.- Oct 3 Jim Carlson
**The Mathematics of Internet Security**

*Abstract:*Business on the internet depends on being able to scramble a message (like your credit card number) so that only the intended recipient (the internet store) can read it. The scrambling method is so good that eavesdroppers cannot unscramble messages even with the best supercomputers. Come learn how this works --- it depends on a simple but powerful mathematical idea that was discovered in 1977. See if you can break the code!- Oct 10
**MATHEMATICS CAREER DAY**

There will be no undergraduate colloquium today. Instead Mathematics Career Day will be held in the Aline Wilmont Skaggs Building (ASB), room 210, beginning with refreshments in the lobby at 1:30. Lori McDonald from Career Services will speak about career preparation, and we will have a panel discussion/question and answer session with mathematicians working in industry.- Oct 17 Hugo Rossi
**Topology from Euler to Alexander**

*Abstract:*Euler's Formula and his solution to the Konigsberg Bridge Problem are the first influential results in the area of mathematics known today as Topology. They lead directly to the branches of Topology known as Homology Theory and Graph Theory respectively. In this talk we will give a survey of the topological ideas of Euler and their development over the succeeding two centuries.- Oct 24 Brandon Baker
**Digital Signal Processing**

*Abstract:*Who would have ever guessed that the discovery of a mathematical concept initially rejected for it's lack of "mathematical rigor" and critically regarded as "too general" would 200 years later be used in so many ways. The Fourier Transform is the mathematical root for many modern devices such as: portable phones, digital cameras, digital TV's, computer image processing, the internet, satellite communications, teleconferencing systems and compact disc players. Come find out how Jean Baptiste Fourier changed the world by discovering the Mathematics behind Digital Signal Processing (DSP).- Oct 31 Ken Golden
**The Mathematics of Sea Ice**

*Abstract:*Sea ice, which plays a major role in global climate, is a composite material of pure ice with brine and air inclusions. It is distinguished from many other porous media, such as sandstones or bone, in that its microstructure and bulk properties depend strongly on temperature. Above a critical value of around -5 degrees C, sea ice is permeable, allowing transport of brine, nutrients, and heat through the ice. These processes are important in air-sea-ice interactions, in the life cycles of sea ice algae, and in remote sensing of the pack. Recently we have used the mathematical theory of percolation to model the transition in the transport properties of sea ice. In this talk we give an introduction to percolation theory and how it explains data we took in Antarctica. We also describe how the mathematical theory of inverse scattering can be used for recovering the physical properties of sea ice via electromagnetic remote sensing. At the conclusion, we will show a short video on a recent winter expedition into the Antarctic sea ice pack.- Nov 7 Mladen Bestvina
**Is it Knotty or is it Nice? -- How to Tell Knots Apart**

*Abstract:*A knot can be represented by a string whose ends are tied. The simplest knot is the "unknot" -- it is represented by a string that lies flat on the table without any crossings. The next simplest knots are the trefoil and the figure eight knots. Two knots are regarded as being equivalent if one can deform one string to the other without cutting it. Knot theory is a branch of topology that tries to classify knots. This talk will introduce methods (some of which were developed in the last 15 years) to tell knots apart.- Nov 14 Nat Smale
**Efficient Road Building, the Steiner Problem and Soap Films**

*Abstract:*The Steiner problem consists of determining the shortest network of roads that connects together a given collection of towns. In this talk, we will look at some aspects of this problem, and also see how one can sometimes find the solution in soap films.- Nov 21 Blake Thornton
**Infinite and Infinitesimal Numbers: Constructing New Number Systems and Why We Care**

*Abstract:*The founders of calculus (especially Leibniz) did not think in terms of limits as we do today. Instead dx represented an infinitely small number. Similary, infinitely large numbers were used as well, and a limit to infinity was thought of as an infinitely large number. The notion of infinitesimal and infinite numbers was declared unfounded and the notion of a limit was used to explain calculus. Then, in the early 1960's, Abraham Robinson constructed the nonstandard real numbers: a number system with infinite and infinitesimal numbers. We will go through a construction of the nonstandard real numbers and then talk about some applications to calculus.- Nov 28 Robert Hanson
**The Reconstruction Conjecture**

*Abstract:*One of the most interesting problems in Graph Theory is the Reconstruction Conjecture. First posed almost sixty years ago, the Reconstruction Conjecture says: "A finite simple (unlabeled) graph G on at least three vertices is determined uniquely by its deck of cards." What's all this mumbo jumbo about a graph having a deck of cards? Come find out. We will use this particular problem as a platform for defining many of the most basic useful terms in Graph Theory. We will also discover some easy to prove facts regarding this Conjecture.- Dec 5 Dan Stevens
**Brownian Motion and Stock Option Pricing**

*Abstract:*What in the world does the motion of dust particles in liquid have to do with the pricing of a stock option? We'll discuss how and why Brownian motion is a good model for stock behavior as well as give some results used in the financial industry. An example of odds betting, coupled with an introduction of the Arbitrage Theorem (resulting from Linear Algebra's separating Hyperplane) will be used as a starting point. If time permits, some discrete state space stochastic processes will be discussed at an elementary level along with their applications to the insurance industry.