UUMath - Undergraduate Colloquium Fall 2001

Undergraduate Colloquium Fall 2001

Aug 28 Klaus Schmitt: Abstract: During the 1870's 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.
Sept 4 Peter Trapa: 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 http://www.javaonthebrain.com/java/puzz15/ before attending the colloquium.
Sept 11 Fred Adler: Mathematical Bioeconomics and the Tragedy of the Commons
Abstract: Why do people and other organisms tend to overexploit resources, often in ways harmful to themseles? Simple models combining the mathematical description of resource dynamics and the mathematical description of competitive behavior (economic game theory) show that this "tradgedy of the commons" results not from stupidity, but from rational greed.
Sept 18 Jim White: Prepare for the Career Fair!
Abstract: The University Science and Engineering Career Fair is Tuesday, September 25 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.
Sept 25 Gordan Savin: Square and Triangular Numbers and Pell Equations
Abstract:
Oct 2 Nat Smale: 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.
Oct 9 Hugo Rossi: 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 60x60 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.
Oct 16 Jim Carlson: 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.
Oct 23 Fletcher Gross: Inclusion-Exclusion: a subject that really counts
Abstract:
Oct 30 Lunch for Math Majors: No Colloquium this day
Abstract: The free lunch for math majors and prospective math majors will be held this Tuesday from 11:00-1:00 in the Union Building Collegiate Room. This a great chance to meet other math majors, find out about our programs, meet with advisors, and have a tasty lunch.
Nov 6 Graeme Milton: 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.
Nov 13 Alistair Craw: 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.
Nov 16 Special Colloquium: Carl Cowen: The Farmer's Legacy: an Isoperimetric Problem
Pizza will be served before this talk. 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 centruy 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.
Nov 20 Andrejs Treibergs: 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 an area A, 4(pi)A is less than or equal to L-squared. 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.
Nov 27 Robert Hanson: 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 knotted spheres (whatever THAT means).
Dec 4 Davar Khoshnevisan: 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 "FO");
6 jumbles of "OFT" ("OFT", "FOT", "TOF", "OTF", "FTO", "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 10^24 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.