Undergraduate Colloquium Spring 2001
Jan 16     Angelo Vistoli
So You Think You Know How to Count
Abstract: You are given a standard 8x8 chessboard, and three colors, say green, white and red. You have to cover each square with a different color. In how many different ways can you do it?

If the chessboard is nailed to a wall, the answer is not very hard. But now assume that the chessboard is free to rotate; then of course the number will not be the same. For example, if you color the top half in red and the bottom half in white, or the top half in white and the bottom half in red, these two ways of coloring are not distinguishable. This case is considerably more difficult.

The answer will be given in the talk, as an example of a beautiful formula known as Burnside's formula.

I will also introduce very briefly more exoteric ways of counting things, leading to the astounding conclusion the the number of finite sets in the universe is Napier's number e.

Jan 23     Stewart Ethier
The Method of Equal Proportions, or Why Utah Failed to Get a Fourth Congressional Seat
Abstract: After each decennial United States census, the 435 seats in the U.S. House of Representatives are apportioned among the 50 states in approximate proportion to their populations. For example, the results of the 2000 census show that North Carolina's quota is 12.470 seats, Utah's is 3.457 seats, and California's is 52.447 seats. Of course, the actual number of seats allotted to a state must be a positive integer. If each of these quotas (one for each of the 50 states) is rounded to the nearest positive integer, the total number of seats accounted for is 433. Who gets the two extra seats?

If you said North Carolina and Utah, you'd be using the Hamilton method, which was rejected by Congress in 1911 in part because it admits the Alabama Paradox. For the last 60 years, the method of choice has been the method of equal proportions, developed in the 1920s by mathematician E. V. Huntington of Harvard University. With this method, California and North Carolina get the two extra seats.

In this talk we discuss the history and mathematics of the apportionment problem.

Jan 30     Fred Adler
Stupid Ants Don't Fight
Abstract: Some species of ants have little overlap between territories of different colonies, and thus avoid serious fighting. How can we tell why ants do not fight? If they are avoiding dangerous fights, then how do they know that fights are dangerous? And how can scientists observe behaviors that almost never occur? Mathematics provides the way to "see the invisible." I will present mathematical models of ant behavior that imply that ants are not in fact avoiding fights, but are using a simpler "stupid" strategy of avoiding each other.

Feb 6    Fletcher Gross
Godel's Incompleteness Theorem
Abstract: Is Mathematics consistent? Can all Mathematical questions eventually be decided? Providing answers to these questions was an important goal to many (from Leibniz to David Hilbert and Bertrand Russell in more recent times) but an astonishing result of Kurt Godel shows that their attempts were doomed to failure. On the other hand, mathematicians will never run out of problems to solve.

Feb 13    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. To see Maple code for computations with elliptic curves, visit http://www.math.utah.edu/~carlson/research/maple/ellpack.html.

Feb 20     Don Tucker
Can You Chew Gum and Walk at the Same Time?
Abstract: Most mistakes made in classes below the Calculus are attributable to errors in the use of the distributive law. Most of those in the Calculus involve the chain rule. In each case, the student is asked to deal with two different concepts simultaneously. We will talk about the distributive law case.

Feb 27   Ben McKay
Rubber Bands on Potatoes
Abstract: The shortest curve lying on a surface and connecting two points is called a geodesic. For example, on a sphere a geodesic is just a piece of an equatorial-type circle. So all of the geodesics on a sphere are periodic, wrapping around the sphere. Recently, Bangert, Franks and Hingston used a mixture of hard and easy geometry to show that there are infinitely many periodic geodesics (like tightly wound loops of thread) on any ``potato''-like surface; many dynamical systems similar to potatoes are just beyond our grasp, and may yet yield up their secrets to analogous techniques.

Mar 6    Grigory Mikhalkin
Ovals of Real Algebraic Curves
Abstract: Most people have heard of a hyperbola and an ellipse and can draw them on the plane. These curves are given by polynomials of degree 2. What does a picture of a curve look like if the degree is higher? The curves of higher degree might have several ovals and, as it turns out, the mutual arrangement of the ovals is responsible for some 4-dimensional topology.

This problem of oval arrangement was included by Hilbert in his famous list of problems 100 years ago. In the talk we review some classical and recent results in this area. In particular we will learn how to draw curves with as many ovals as possible for a given degree.

Mar 20    Andrej Cherkaev
A Walk Through Calculus of Variations
Abstract: The desire for optimality (perfection) is inherent in humans. The search for extremes inspires mountaineers, scientists, mathematicians, and the rest of the human race. From ancient times, geometers noticed extremal properties of symmetric figures and bodies: the circle has maximal area among all figures with fixed perimeter; the equilateral triangle and the square have maximal area among all triangles and quadrangles with fixed perimeter, respectively, etc. However, regular proofs of these extremal properties usually are not easy. An even more difficult task was to develop a regular theory that was able to search for optimal cuves and surfaces. This theory of extremal problems -- calculus of variations -- has been successively developed for three centuries. The talk discusses some historic problems and modern challenges of the calculus of variations, including the popular question: how do you solve a problem that does not have a solution?

Mar 27    Domingo Toledo
Triangles in Hyperbolic Geometry
Abstract: Hyperbolic geometry is a geometry that satisfies all the axioms of the usual Euclidean geometry except the parallel postulate. This geometry has two peculiarities that may seem contradictory. On the one hand the area inside a circle grows exponentially with the radius of the circle. On the other hand, there is an absolute upper bound on the areas of triangles. In this talk I will describe some of the history of hyperbolic geometry, explain why there are no triangles of arbitrarily large area, and explain how the area of a triangle is related to the sum of its angles.

Apr 3   Klaus Schmitt
The Contraction Mapping Principle and Fractals
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 treees, clouds, landscapes, etc., are examples. In the lecture I shall discuss a specific class of fractals and their properties and shall show how these objects may be obtained by the iteration of some simple mappings. In particular, 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.

Apr 10    Andrejs Treibergs
Geometric Probability and the Isoperimetric Inequality
Abstract: Here are a few problems from the subject Geometric Probability or Integral Geometry. One of the earliest is the Buffon Needle Problem: Suppose a needle is randomly dropped on a floor that is ruled by evenly spaced parallel lines, what is the probability that it will touch a line? This problem can be solved using calculus. Here is another: suppose one convex set is contained in a larger convex set in the plane, what is the probability that a random line which touches the larger set also touches the smaller? A third example is the following. Among all random rigid motions that move one domain to a position that intersects a fixed domain, what is the average Euler characteristic of the intersection? (The Euler characteristic of a plane domain with nice boundary is the number of conncted components minus the number of holes.) It turns out that the answer to the third question is related to the question of how long does the boundary of a domain have to be in order for it to enclose a given area? The answer is the isoperimetric inequality: at least as long as the boundary of a circle with that given area.

Apr 17    Mark Lewis
Wolf Home Ranges and Prey Survival
Abstract: Wolves are now recolonizing much of North America. It is not yet clear how they will impact the ecosystems to which they return. Data from established wolf populations indicate that when wolves are close to prey they interact strongly with them. However, much of the time wolves are far from their prey: radio tracking data shows wolves moving around their large home ranges in complex spatial patterns, using scent marks, trails and other features as movement cues.

With mathematical models it is possible to describe wolf movement, behavior and impact on prey. Here the home ranges arise through the interactions between scent-marking and detailed spatial movements. When incorporated into a nonlinear partial differential equation (PDE) model, these interactions result in distinctive territories with 'walls' of scent marks defining the edges of their territory, and 'buffer zones' found between territories, where prey are common. Direct fit of the model to radio-tracking data shows that it explains space use better than traditional statistical models for home ranges.

Apr 24    Brandon Baker
Engineering Applications of PDEs
Abstract: Did you know that Fourier Analysis, Einstein's Theory of Relativity, Modern Electromagnetic Theory, and Quantum Mechanics all arose from common ground? From simple engineering applications to The Schrodinger Equation, much of modern science and engineering owes tribute to their shared origin, Partial Differential Equations. Come learn about some simple solutions to the father of some of the most powerful Mathematics known to mankind.