Undergraduate Colloquium Spring 2000

- January 18 Jim Carlson
**Infinity: Its History, Uses, and Abuses**

*Abstract:*A brief history of infinity from Zeno's paradoxes in the 5th century BC to the revolutionary work of Cantor and Cohen in the 19th and 20th centuries AD, with applications to numbers, logic, and computation.- January 25 Grisha Mikhalkin
**Visualizing Surfaces in Four-Dimensional Space: How Complex Numbers Help**

*Abstract:*Equations like y^2 = x^3 - x define curves in the plane that are easy to visualize. But if x and y are complex numbers, the same equation defines a surface in four-dimensional space. Come and learn how to visualize surfaces in it. Learn how to visualize complex equations!- February 1 Chris Staskewicz
**Approximating Roots to Polynomials in Several Complex Variables**

*Abstract:*This was an REU (Research Experience for Undergraduates) project I completed at the Pennsylvania State University (PSU) while I was an undergraduate in Math at UC Santa Cruz. From Calculus we try to find unknown roots (i.e. zeros) to a polynomial in one real variable using Newton's Method, which relies on a guess for the root. When we consider polynomials of several complex variables, making a good guess can be difficult, if not impossible. And how do we know we get all the roots when we are done with the Newton Procedure? The method I'll introduce is a "homotopy" proceedure where we use a polynomial with "known" roots to find the "unknown" roots of the given polynomial.- February 8 Klaus Schmitt
**Fractals and Ways to Generate Some**

*Abstract:*Fractals are intriguing geometric objects that exhibit self-similarities (or other types of similarities) on various scales. Space filling curves, the Koch curve, the Sierpinski gasket, Julia sets, and many natural objects, such as trees, clouds, landscapes, etc., are examples. In the lecture I shall discuss various fractals and their properties and shall show how many of these objects may be obtained by the iteration of some simple mappings. In particualr, I shall illustrate how the contraction mapping principle (a result which has wide applicability in many branches of mathematics) may be used to obtain fractals as sets that remain fixed by certain types of transformations.- February 15 James Stanard
**Complex Adaptive Systems: The Ying and Yang of Chaos and Order**

*Abstract:*For centuries, scientists have recognized that from relatively simple systems, chaos emerged. Systems could only be tracked for a short time before accumulated error blew up to enormous proportions. It seemed that at last Nature had finally won with its impossible indeterminism. Now, Science is showing that a balance exists between Chaos and Order at which highly interesting patterns emerge. In fact, it seems that systems gravitate towards this balance, thus creating emergent behavior in many systems ranging from Ecology to Economy. This is about these very Systems.- February 22 Peter Alfeld
**Unsolved Problems in Mathematics**

*Abstract:*Mathematics is driven by good problems. Particularly fascinating are problems that are easily understood even by undergraduate students, and yet that are difficult to solve. In this talk I will present a number of problems. All of them are easily understood. Some of them are famous and still unsolved, a couple are famous and were solved only recently after a great deal of effort by many people, one is unsolved and obscure but of great interest in my own work, and one is famous and "solved", but its "solution" is very strange indeed.- February 29 Grant Gullberg
**Applications of Tensor Tomography in Medical Imaging**

*Abstract:*Tensors of diffusion, deformation (stress and strain), conductivity and photoelasticity are physical quantities of biological tissue, which can contribute to better diagnosis of various diseases through the development of more accurate models of the properties of biological tissue. Tensor tomography is being investigated as a possible imaging technique for measuring these imaging modalities such as magnetic resonanace imaging (MRI), emission computed tomography (PET, SPECT), magnetoencephalography (MEG), magnetocardiography (MCG), ultrasound, optical imaging, or photo-acoustic imaging.- March 7 Hugo Rossi
**Regular Solids, Cages, and Buckyballs**

*Abstract:*The Greeks knew the regular solids, that there are only five of them, and Kepler thought that they described the Universe. Today we see the icosahedron in domed buildings, soccer balls and superconductors, and the mathematics of the symmetries of the icosahdron explains why.- March 21 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.- March 28 Nate Jones
**Topological Graph Theory: What's So Special About These Donuts?"**(Join us in a Peace Officer's dreamworld).

*Abstract:*A fundamental problem in topological graph theory is to draw a graph on a surface without any edge crossings. We will investigate a certain special case of this problem and discuss its solution.- April 4 Stewart Ethier
**How to Win at Video Poker**

*Abstract:*In video poker, after inserting some money into the machine, you are dealt 5 cards. You then have the option, for each card, of holding it or discarding it. Thus, there are 32 ways to play the hand. If you discard k of the 5 initial cards, you draw k new cards from the remaining deck. You then receive a certain specified return based on the amount of money bet and the rank of your final hand. Perhaps surprisingly, several versions of this game offer a long-term return of over 100 percent to the player with a nearly optimal drawing strategy. We will focus on the popular version "Deuces Wild" (100.76 percent return), explaining the optimal drawing strategy and how it was derived. We will point out that the correct play is not always the intuitively obvious one.- April 11 Shawna Haider
**Air Quality Modeling for Future Growth Scenarios**

*Abstract:*Achieving and maintaining acceptable air quality is a major challenge and a possible constraint to future growth in Utah. Mathematical models to predict the effect of different growth scenarios on Wasatch Front air quality would be valuable to regulatory, policy, and environmental research groups as well as to the general public. I will review the current state of Wasatch Front air quality modeling and highlight ways in which this modeling might be made more effective.- April 18 Aaron Bertram
**The Fundamental Theorem of Algebra**

*Abstract:*The fact that every polynomial has a complex root is called the fundamental theorem of algebra. Even though it is in general not possible to find the root exactly, we still know it is there. This is already pretty amazing. But even more amazing is Liouville's proof, borrowing ideas from analysis, in particular (sub)harmonic functions. I want to talk about this proof as an excuse to propagandize about the surprising connections that arise among different areas of mathematics.- April 25 Jim Carlson
**The Spiral of Archimedes**

*Abstract:*Two of the classical Greek problems are to construct a square whose area is that of a given circle and to trisect a given angle, both using ruler and compass alone. In the late nineteenth century new methods of algebra and analysis showed that no such construction exists. However, in the third century BC Archimedes discovered that both problems could be solved using a spiral-shaped curve with many remarkable properties.