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## Undergraduate Colloquium 2001 - 2002

August 28, 2001
Speaker: Klaus Schmitt (Professor of Mathematics)
Title: Iteration of Mappings and Newton's Method
Abstract: During the 1870s, Schroder and Cayley (independently) studied Newton's method applied to polynomial equations and attempted to determine the domains of attraction of the zeros of polynomials relative to the Newton iteration scheme. In the lecture, I shall discuss the questions and results of these studies and subsequent work by Julia and Fatou on iteration of rational functions.

September 4, 2001
Speaker: Peter Trapa (Professor of Mathematics)
Title: Counting to n! with Tableaux
Abstract: Define a standard young tableau of size n to be a left justified array of n boxes labeled with the numbers 1,...,n such that the labels increase across rows and down columns. It is a remarkable fact that the number of same-shape pairs of standard Young tableaux of size n is exactly n!. We'll give two proofs of this fact, each with its own merits. The second is especially interesting not only because it's the product of an immensely satisfying shuffling construction reminiscent of the famous "15 Puzzle", but also because it leads to some open problems whose answers are predicted to have important geometric applications. Incidentally, if you've never encountered the 15 Puzzle, you might enjoy playing it at this site before attending the colloquium.

September 11, 2001
Speaker: Fred Adler (Professor of Mathematics)
Title: Mathematical Bioeconomics and the Tragedy of the Commons
Abstract: Why do people and other organisms tend to overexploit resources, often in ways harmful to themselves? Simple models combining the mathematical description of resource dynamics and the mathematical description of competitive behavior (economic game theory) show that this "tragedy of the commons" results not from stupidity, but form rational greed.

September 18, 2001
Speaker: Jim White with the University of Utah Career Services
Title: Prepare for the Career Fair!
Abstract: The University Science and Engineering Career Fair is Tuesday, September 25, 2001 from 9:00 a.m. - 3:00 p.m. in the Union Ballroom. Jim White from Career Services will talk about resources available at Career Services and how to make the most of the Career Fair.

September 25, 2001
Speaker: Gordan Savin (Professor of Mathematics)
Title: Square and Triangular Numbers and Pell Equations

October 2, 2001
Speaker: Nathan Smale (Professor of Mathematics)
Title: Minimal Surfaces and the Calculus of Variations
Abstract: In this talk, we will discuss some geometric problems that can be formulated as "Calculus of Variations" problems. That is, the solution is found by minimizing some function, which is defined on a domain consisting of a set of functions. In particular, we will look at minimal surfaces, which model soap films spanning a curved piece of wire.

October 9, 2001
Speaker: Hugo Rossi (Professor of Mathematics)
Title: The Regular Solids, Their Symmetry Groups, and the Icosahedron in Particular
Abstract: The soccer ball has 32 faces (20 hexagons and 12 pentagons), 60 vertices, and 90 edges. 30 of the edges lie between hexagons, and 60 are between a pentagon and a hexagon. This information can be codified in a 60 x 60 matrix which is left invariant under any symmetry of the soccer ball. Study of this action led to the theoretical realization of the possibility of the carbon molecule known as fullerine, which was only later synthesized, and then even later discovered to exist in nature.

October 16, 2001
Speaker: James Carlson (Professor of Mathematics)
Title: Elliptic Curves
Abstract: An elliptic curve is the solution set of a cubic equation in two variables. We will learn a little about the number theory, geometry, and applications of elliptic curves, which range from the proof of Fermat's last theorem to the cryptosystem used in devices such as the Palm VII.

October 23, 2001
Speaker: Fletcher Gross (Professor of Mathematics)
Title: Inclusion-Exclusion: A Subject that Really Counts

November 6, 2001
Speaker: Graeme Milton (Professor of Mathematics)
Title: The Mathematics of Shape Memory Materials
Abstract: Shape memory materials have the curious property that they remember their shape. You bend them and twist them into whatever shape you like. Then, when you place them in hot water, they pop back to their original shape.

How does such an effect occur and what does this have to do with mathematics? The study of this problem leads to some beautiful problems in mathematics, many of which remain unsolved. Come to Tuesday's undergraduate colloquium to find out more.

November 13, 2002
Speaker: Alastair Craw (Professor of Mathematics)
Title: How on Earth Does a Graduate Student Start Doing Original Work?
Abstract: Aspiring mathematics PhD students typically ask themselves two things before committing to a PhD program. First, will my body still work after several years of cheap food? Second, how will I solve a problem in mathematics that hasn't been solved before?

I plan to tackle the second question by discussing an apparently simple question that I was asked a couple of years ago: Given a collection of points drawn inside a triangle, is there a nice method to join the dots in the "best" way? I'll begin by explaining precisely what I mean by this (after all, what is the "best" way?), then I'll ask the audience to solve the problem. Pencils and erasers will be provided if necessary! I hope this will give the audience some feeling for what graduate students really do while studying for a PhD.

November 16, 2001
Speaker: Carl Cowen (Professor of Mathematics at Purdue University)
Title: The Farmer's Legacy: An Isoperimetric Problem
Note: Professor Cowen is also interested in talking informally with students about graduate school in mathematics, both in general and at Purdue.
Abstract: This talk will concern the problem of dividing a region in the plane into pieces with specified areas by using curves of the shortest total length. For example, of the curves that divide a triangle into two pieces of equal area, which has the shortest length? This problem can be formulated in the classical calculus of variations, and was studied in that context at the turn of the century by several mathematicians including Norbert Wiener.

In this talk, I will give qualitative information about the solution of the problem that is sufficient to answer the question about the triangle completely and make reasonable guesses for harder problems. We will not use the calculus of variations; rather, the techniques used to develop this qualitative information will be from high school geometry. Thus, the talk will be accessible to those interested in mathematics who have had high school geometry.

November 20, 2001
Speaker: Andrejs Treibergs (Professor of Mathematics)
Title: My Favorite Proofs of the Isoperimetric Inequality
Abstract: For a planar region bounded by a closed curve of fixed length, the enclosed area can never exceed the area of a circle with that boundary length. Computing the area of the circle gives the isoperimetric inequality in the plane: For any closed curve in the plane whose length is L and which encloses and area A, 4(pi)A is less than or equal to L2. If equality holds, then the region is a circle.

There are many proofs. I will present a few of my favorites. I hope to discuss proofs that depend on more primitive inequalities. The four hinge proof of Steiner depends on Brahmagupta's inequality for quadrilaterals in the plane. Hurwitz's proof depends on the Wirtinger inequality from Fourier series. The proof of Minkowski depends on Brunn's inequality from the theory of convex sets.

November 27, 2001
Title: Not Theory for Mountaineers; An Introduction to the Mathematical Theory of Knots - None of Which is Applicable to Climbing Mountains
Abstract: In this talk, we will introduce the concept of a knotted circle in three-space and regular projections onto the plane. We will learn the three basic moves that get us from one projection of a given knot, K, to another projection of K. One activity that many mathematicians enjoy is coloring things, so we will color some knots to figure out that they are actually different knots. Hopefully, we will also travel to higher dimensions and look at knotted circles in four-space as well as knotted spheres (whatever THAT means).

December 4, 2001
Speaker: Davar Khoshnevisan (Professor of Mathematics)
Title: Stirling's Formula and Laplace's Method, or How to Put Your Calculus to Good Use
Abstract: It has been known for a long time that the number of jumbles of N distinct objects is N(N - 1)(N - 2)...1. This quantity is sometimes written as N! and read "N factorial" for brevity. For example, there are
• 2 jumbles of "OF" (namely, "FO" and "OF");
• 6 jumbles of "OFT" ("OFT", "FOT", "TOF", "OTF", "FTO", and "TFO"); and<
• 24 jumbles of "SOFT"...
The number of jumbles of "ABCDEFGHIJKLMNOPQRST" is 2,432,902,008,176,640,000, while there are something like 403.29 x 1024 ways to jumble the entire English alphabet! (Not a factorial sign.)

One might guess that N! is quite large, even when N is a moderately large number (try 400! on your favorite hand calculator).

In this talk, I will present a result - due to A. DeMoivre (Miscellanea Analytica 1730, improved later by J. Stirling circa 1738) - that is called "Stirling's Formula"; it accurately describes how N! grows as N grows. In fact, I will only show one way of getting to this formula; it relies on aspects of elementary combinatorics, probability, and - most important of all - the calculus of real functions.

January 15, 2002
Speaker: Robert Guy (VIGRE Graduate Fellow)
Title: Linear and Non-Linear Circuits: Oscillations, Bifurcations, Excitability, and Chaos
Abstract: I will begin by reviewing how simple circuits made up of resistors, capacitors, and inductors give rise to systems of differential equations. I will analyze a simple system and demonstrate with the circuit. Interesting behavior occurs when a bit of non-linearity is introduced via a non-linear system. The circuit was designed to model the electrical impulse in a nerve cell, and it provides an excellent example of an excitable system. I will demonstrate the results with the actual circuit. Finally, I will discuss how a simple modification can give rise to chaotic dynamics.

January 22, 2002
Speaker: Gordan Savin (Professor of Mathematics)
Title: Continuous Fractions and Pell Equations
Abstract: In this talk, I will define continuous fractions and give some examples. Then, we shall write down quadratic irrational numbers as continuous fractions, and apply it to solve the Pell Equation.

January 29, 2002
Speaker: Andrej Cherkaev (Professor of Mathematics)
Title: Damage and Failure: A Mathematical Perspective
Abstract: Even if a structure fails, most of its material stays undamaged. The failure is mostly attributed to an instability of the stress distribution. A structural morthology can increase the stability of the loading process. The talk deals with the corresponding mathematical principles, problems of description and control of the damage, and simulation. Video clips of destructing construction, made by Liya Zhornitskaya, will be demonstrated.

March 5, 2002
Speaker: Paul Bressloff (Professor of Mathematics)
Title: What Visual Hallucinations Tell Us About the Brain
Abstract: Geometric visual hallucinations are seen by many observers after taking hallucinogens such as LSD or cannabis, on viewing bright flickering lights, on waking up or falling asleep, in "near death" experiences, and in many other syndromes. We present a dynamical theory of the origin of hallucinations in the visual cortex, based on the assumption that the form of the eye-brain map and the architecture of the cortex determine their geometry.

March 12, 2002
Speaker: Jingyi Zhu (Professor of Mathematics)
Title: First Exit and Company Default
Abstract: Given a finite region, we can define the first exit for a Brownian motion starting from any interior point, and the problem is related to the diffusion equation. In financial applications, we would like to model possible company defaults in the future, and one of the approaches is to track some financial index as an indicator of the health of the company. When the index falls below (or above) a certain threshold, we may declare that the company defaults. I will show in this talk how this approach can be formulated, the partial differential equations problem arises, and how it can be solved.

March 19, 2002
Speaker: Mladen Bestvina (Professor of Mathematics)
Title: On the Total Curvature of Knots
Abstract: By the (total) curvature of a polygon in, say, the 3-dimensional Euclidean space, I mean the sum of all exterior angles. For example, for convex planar polygons, the curvature is 2(pi). I will discuss a theorem of Fenchel and Borsuk that, for all other polygonal curves, the curvature is greater than 2(pi). Borsuk raised a question whether, for knotted polygonal curves, one could show that the curvature is greater than or equal to 4(pi). This problem was solved in 1949 by a Princeton undergraduate student named John Milnor (a good friend of John Nash). I will discuss Milnor's solutions. If there is time at the end, I will discuss the relationship between this "piecewise linear" concept of curvature and the more traditional curvature of smooth curves in differential geometry.

March 26, 2002
Speaker: Nick Korevaar (Professor of Mathematics)
Title: Making Minimal Surfaces with Complex Analysis
Abstract: In 1866, Karl Weierstrass discovered an amazing connection between the shapes of soap films and the field of complex analysis. Starting with the microscopic characterizations of minimal surfaces and of complex differentiable functions, we will derive Weierstrass' representation formula. It encodes the local differential geometric information of a minimal surface in terms of a pair of analytic functions. Concrete applications of Weierstrass' formula blossomed in the early 1980s, when David Hoffman realized that computer graphics would allow one to actually visualize the minimal surfaces corresponding to particular Weierstrass data. We will compare some computer creations to what we can make with a bucket of soapy water and wire frames.

April 2, 2002
Speaker: Jesse Ratzkin (VIGRE Assistant Professor)
Title: Geometric Models for Crystal Growth
Abstract: One can determine the growth of a crystal by tracking how the boundary between it and its environment moves, which amounts to determining how a surface in three-space evolves. If the motion of this surface (i.e. growth of the crystal) is determined by geometric properties (e.g. lengths, areas, and volumes) then one can use geometry to track the motion of the surface. I will discuss some of these techniques, including motion by mean curvature, weighted mean curvature and crystalline curvature.

April 9, 2002
Title: The Good and Evil of the Axiom of Choice
Abstract: The Axiom of Choice (AC) is undeniably the most (well, the only) controversial axiom of Set Theory, the foundation of modern mathematics. In this talk, there are three points I would like to make:
• AC is intuitively appealing (a lot of people will disagree).
• AC is everywhere; in particular, a great deal of powerful results in functional analysis and PDEs rely on consequences of AC.
• AC has certain "absurd" consequences for which one might almost want to reject AC as an axiom.
Personally, I find it mildly amusing the be "AC-alert", yet I am starting to feel that I am being perceived as "AC-lunatic" instead. I am planning on using this talk to vindicate myself (it might very well do the exact opposite). Come see for yourself; it is going to be fun.

April 16, 2002
Speaker: Stewart Ethier (Professor of Mathematics)
Title: The Gambler's Ruin Formula
Abstract: Suppose you bet \$1 on 'red' on each spin of a 38-number roulette wheel. What is the probability that you double your \$1,000 bankroll before losing it? The answer, 1.75 x 10-46 comes from the famous Gambler's Ruin Formula, which dates back to the time of Pascal and Fermat (1656). We discuss how the latter two mathematicians likely derived the formula, as well as other aspects of the history, a modern derivation, and applications and extensions of the formula.

April 23, 2002
Speaker: Jim Keener (Professor of Mathematics)
Title: Arrhythmias by Dimension
Abstract: Abnormalities of function of the cardiac conduction system are the cause of death of hundreds of people every day. For that reason, the study of cardiac arrhythmias is of great interest from a medical and scientific perspective. However, cardiac arrhythmias are also interesting for mathematical reasons because the cardiac conduction system can be viewed as a dynamical system and the variety of its behaviors can be studied from the viewpoint of dynamical systems theory. In this talk, I will give a classification of cardiac arrhythmias that is based on spatial dimension, and is therefore useful for mathematicians, but probably not (as much) for physicians. I will describe examples of zero dimensional arrhythmias (abnormalities of single cells), one dimensional arrhythmias (Wolff-Parkinson White Syndrome), two dimensional arrhythmias (atrial flutter), and three dimensional arrhythmias (ventricular tachicardia and fibrillation).

April 30, 2002
Speaker: Peter Alfeld
Title: Infinity is Different
Abstract: The natural numbers are 1, 2, 3,..., and the integers are ...-3, -2, -1, 0, 1, 2, 3,... Since all natural numbers are integers but some integers (all the negative ones and zero) aren't natural numbers, there seem to be more integers than natural numbers. Not so! Indeed, by the end of this talk you will be fully and rightfully convinced that there are no more rational numbers (fractions) than integers, even though there are infinitely many rational numbers between any two consecutive integers (like 3 and 4). But it's far from true that infinity is infinity is infinity... Indeed, there is an amazing hierarchy and structure to infinity, and the talk will conclude with a bizarre fact that may change your life. The talk makes no assumptions about your mathematical background, indeed, you don't even need to understand Calculus. [an error occurred while processing this directive]