Undergraduate Colloquium Fall 2003
August 26     NO COLLOQUIUM

September 2     Gordan Savin
Square-Triangular Numbers
Abstract: Some numbers can be laid out in some shape. For example, a square number, n^2, can be arranged in the shape of an n n square. Similarly, a triangular number is a number that can be arranged in the shape of a triangle, such as the number 10:
   x x
  x x x
 x x x x

A square-triangular number is, of course, a number which is both a square and a triangular number. A natural question to ask is whether there are any such numbers other than 1. Simple calculations show that the first few triangular numbers are: 1, 3, 6, 10, 15, 21, 36, 45, 66, 78, 91. In particular, we see that 36 is a square-triangular number. Can you find the next one? It is surprisingly large! Come to this lecture and learn how to find all square-triangular numbers.

For more information on this topic:
Joseph H. Silverman, A Friendly Introduction to Number Theory, 2e, Prentice-Hall 2001.

September 9     Peter Trapa
Six Degrees of Kevin Bacon and other Small World Phenomena
Abstract: Here's a version of a popular parlor game. Consider a graph whose vertices consist of all actors and actresses ever to appear in a movie. Connect two vertices by an edge if the corresponding actors or actresses appeared in the same movie. Now take any actor or actress in the connected component of Kevin Bacon and find the shortest path to Kevin. For example, start with Rudolph Valentino; he has a Kevin Bacon number of 3: Valentino was in Beyond the Rocks (1922) with Gloria Swenson who was in Airport (1975) with Erik Estrada who was in We Married Margo (2000) with Kevin Bacon.

Of the 594,683 actors connected to Kevin Bacon, only 94 have a Kevin Bacon number greater than 6. It turns out that there is something special about the number 6, and we'll explain this and other interesting features of graphs like the Kevin Bacon graph. In particular, we'll formalize that nagging question of whether it really is a small world or whether it is an enormous world that is so highly stratified that it gives the false impression of being small.

For more information on this topic:
D.J. Watts, Small Worlds: The Dynamics of Networks Between Order and Randomness, published by Princeton University Press (Princeton), 1999.

September 16    Jesse Ratzkin
The Roundness of Bubbles
Abstract: Most bubbles are round. Why? As we investigate this question we will see a beautiful, geometric way to characterize bubbles and a powerful use for the second derivative test you might not have known.

For more information on this topic:
Protter and Weinberger, Maximum Principles in Differential Equations
H. Hopf, Differential Geometry in the Large
R. Osserman, A Survey of Minimal Surfaces

September 23    Peter Alfeld
Infinity is Different
Abstract: There are as many prime numbers as there are natural numbers, and there are as many natural numbers as there are rational numbers. There are more real number (points) in an interval than there are rational numbers anywhere, but there are no more points in a square than there are in an interval. No matter how big infinity is, you can make it bigger. How can all that be? Come to this talk and find out.

For more information on this topic:
Patrick Suppes, Axiomatic Set Theory, Dover 1972, ISBN 0486616304
Paul Bernayes, Axiomatic Set Theory, Dover 1991, ISBN 0486666379

September 30     Graeme Milton
Bubbly Fluids and Stealthy Submarines
Abstract: The propogation of sound in bubbly fluids is quite different to the propogation of sound in either air or water. The speed is much slower than in either medium and the waves are highly damped. This has application to muffling the noise of submarines. Curiously elementary theory predicts the damping is larger the smaller the volume fraction of bubbles!

For more information on this topic:
1. Wood, A.B. 1995. A Textbook of Sound: Being an Account of the Physics of Vibrations with Special Reference to Recent Theoretical and Technical Developments (Third revised ed.).
2. Taylor, G.I. 1954. The two coefficients of viscosity for an incompressible fluid containing air bubbles. Proceedings of the Royal Society of Londong, Series A, Mathematical and Physics Sciences 226:34-39.
3. Solna, K. and G.W. Milton 2000. Bounds for the group velocity of electrmagnetic signals in two phase materials. Physica B, Condensed Matter 279(1-3):9-12.
4. Solna, K. and G.W. Milton 2001. Can mixing materials make electromagnetic signals travel faster? SIAM Journal on Applied Mathematics.

October 7    Daniele Arcara
Can't define a function? Just blow it up!
Abstract: After a quick review of polar coordinates and implicitly defined functions, we shall explore a process called blow-up (which generalizes polar coordinates). In particular, I shall explain how blow-ups are used to make sense of a function at the points where the function is not defined, and what happens if you blow-up an implicitly defined function. The only background needed is the notion of function, and we shall actually only use polynomials.

October 14    David Hartenstine
A Million Dollar Problem: Solve the Navier-Stokes Equations
Abstract: In 2000, the Clay Mathematics Institute announced that seven mathematical problems had been selected. These aren't just any old problems: attached to each is a prize of $1,000,000 for the first correct solution! I will begin by discussing these problems in general. Then I will focus on one of them in particular, solving the Navier-Stokes equations; these partial differential equations describe the flow of an incompressible viscous fluid.

For more information on this topic:
Visit the Clay Mathematics Institute website: www.claymath.org
Devlin, Keith, The Millenium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, published by Basic Books
More books are listed in the notes for the talk, located at www.math.utah.edu/~hartenst/ugradcollslides.pdf

WEDNESDAY, Oct. 22     Robert Lazarsfeld of the University of Michigan
How Many Times Does a Polynomial Vanish at a Point?
Abstract: Suppose that f(z) is a polynomial with complex coefficients that vanishes when z=0. Then there is a natural way to measure how many times z=0 occures as a root: one counts how many consecutive terms of f(z) vanish, starting with the constant term.

The question becomes much more interesting if one considers instead a polynomial f(z,w) in two variables. Then the zeroes of f define a "curve" in C^2. The naive generalization of the one-variable case leads to the notion of the multiplicity of a curve at a point. We will discuss the geometric meaning of this invariant.

However, the multiplicity fails to distinguish between polynomials such as z^2 - w^2 and z^3 - w^3. We will conclude by introducing a more subtle invariant, which involves studying for which real numbers c>0 the function 1/|f(z,w)|^2c is integrable.

October 28    Klaus Schmitt
Kepler's Laws on Planetary Motion
Abstract: The mathematical derivation of Kepler's laws (using Newton's law of gravitation) is one of the early great triumphs of the Calculus, yet very few calculus books and even fewer calculus courses concern themselves with this matter. This lecture will address the highlights of the derivation and also some of the historical background of the laws.

November 4    Fletcher Gross
Counting with Groups
Abstract: If we have q colors with which to paint the faces of a cube and we allow colors to be used on more than one face, there are q^6 different ways to paint the cube. Suppose now that we consider two ways to paint the cube the same if one can be obtained by rotating the other in some fashion. Then the number of distinct colorings is more complicated to compute. Fortunately, there is a method using group theory to solve such counting problems.

November 11    Jingyi Zhu
From Parts Failure in a Car to Enron's Default
Abstract: Suppose you have a complex system that consists of many components connected in some complicated ways and each of the components can fail on its own. How can we estimate the overall reliability of the system? The mathematical theory dealing with such issues is called reliability theory, and it's an essential tool in engineering designs. Now this theory is applied to financial modeling, in particular, the modeling of credit risk associated with a company, an industry, or some correlated systems. In this talk, I will give some simple examples to show how these risks can be studied using mathematical tools, and how the information generated from the model can be used to position ourselves in a better situation in case some uncertain events come true.

November 18   Ken Golden
The Mathematics of Sea Ice, Global Climate, and Extraterrestrial Biology

November 25   Mladen Bestvina


December 2   NO COLLOQUIUM