UUMath - Undergraduate Colloquium Spring 2005

Undergraduate Colloquium Spring 2005

January 11 No Talk
January 18 No Talk
January 25 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 numbers (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.
February 1 Aaron Bertram: Waring's Problem for Integers and Polynomials
Abstract: Waring's Problem is one of those marvelous problems in elementary number theory that is easy to state and (seemingly) impossible to solve. It goes as follows. If you wanted to write every natural number as a sum of nth powers, how many nth powers would you need? For example, if you are working with perfect squares, then:
7 = 4 + 1 + 1 + 1 really requires 4 squares, as do all the numbers that have a remainder of 7 when divided by 8. In the 18th century, Lagrange proved that 4 squares is always enough. When we pass to cubes, things get more interesting. Turns out that 9 cubes are enough, but that as far as anyone knows (apparently), only the two numbers:
23 = 8 + 8 + 1 + 1 + 1 + 1 + 1 + 1 + 1
239 = 64 + 64 + 27 + 27 + 27 + 27 + 1 + 1 + 1 can't be done with 8 cubes or less. So one is led to ask: If you are allowed to leave out finitely many "bad" numbers, how many cubes do you need? This problem is wide open! There is some suspicion that 4 cubes will be enough, but it has only been proven that 7 cubes are enough.

Waring's problem also makes sense for polynomials. If you wanted to write every polynomial of degree n as a sum of nth powers of linear polynomials, how many nth powers would you need? If the polynomials have one variable, and if you allow complex numbers as coefficients of your polynomials, the answer is very simple and beautiful. But if you look at polynomials in two or more variables the problem becomes hard again!
February 8 Stewart Ethier: Game Theory and Games of Chance
Abstract: Von Neumann's minimax theorem for finite two-person zero-sum games does not apply to most casino games, because in most games, such as blackjack, the dealer's strategy is fixed. Two card games to which it does apply are le her and chemin de fer. We will show how to find the optimal mixed (or randomized) strategies for both games, as well as for a simpler contrived game just to get warmed up. The talk is a preview of the summer 2005 Utah REU program, "The Mathematics of Games of Chance," for which the application deadline is Feb. 15.
February 15 Klaus Schmitt: How can one generate a "brocciflower" using simple mathematics?
Abstract: Fractals are intricate mathematical objects which are self-similar on various scales. In the lecture it will be explained how one may generate a large class of fractals using several functions, each of which is a contraction (as mappings, each strictly decreases the distance between points), as a limit of an iteration process defined by these contractions. Several concrete examples will be discussed, such as the Cantor set construction, the Sierpinski triange, and other similar objects. A good undergraduate project would be: Find an iterated function system of the type described to generate a brocciflower!
February 22 No Talk
March 1 Graeme Milton: There is No Such Thing as a Fair Election
Abstract: An election procedure is an algorithm for ranking candidates given the voter preferences. A fair election is an election procedure which satisfies a number of axioms, all of which are highly reasonable. Taken together, however, they lead to the startling conclusion that no election procedure for more than two candidates is fair. This is known as Arrow's Impossibility Theorem.
March 8 Dan Margalit: Fixed Point Theorems
Abstract: Take a cup of coffee and stir it around as much as you want. The Brouwer fixed point theorem says, amazingly, that there is at least one point in the coffee that didn't move at all. A related idea, the Borsuk-Ulam theorem, tells us that there is always a pair of diametrically opposite points on the Earth where both the temperatures and barometric pressures are the same. I will explain the precise mathematical content of these theorems and give intuitive, visual proofs.
March 15 No Talk (Spring Break)
March 22 Gordan Savin: The Banach-Tarski Paradox, or How to make Two Oranges from One
Abstract: It is possible to break up a ball into finitely many pieces (essentially four), and then to rotate them to assemble two balls of the same radius! The construction is based on the axiom of choice, using a free group with two generators, which appears as a subgroup of the group of all rotations of the ball.
March 29 Fletcher Gross: How did Archimedes do it?
Abstract: Mathematicians often publish elegant logical proofs of their theorems that may not give any idea how the theorem was discovered in the first place. This is particularly true of the Ancient Greeks and Archimedes, the greatest of them all. So, how did Archimedes actually find his results? A document first found in 1907, then lost again, and found again in the 1990's goes a long way to answer this question. The history of the document will be discussed along with Archimedes' original method used to find the volume and surface area of a sphere. Nothing harder than the Pythagorean theorem is required and yet Archimedes is revealed to be hundreds of years ahead of his time.
April 5 Davar Khoshnevisan: Counting After Cantor
Abstract: In the mid-to-late 1800's, Georg Cantor invented a number of extremely original ideas that revolutionalized all of mathematics. Many of his ideas continue to have quite surprising consequences. I will describe a few of them.
April 12 Special event sponsored by Pi Mu Epsilon: The Joy of Mathematics
Please note the room change: JWB 335
Abstract: Two of our faculty members, Nick Korevaar and Dan Margalit, will talk briefly about the joy they find in mathematics. This will be followed by a showing of "Donald Duck in Mathmagic Land." There will, of course, be refreshments. Everyone is invited!
April 19 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.
April 26 Sandra Spiroff: Unique Factorization
Abstract: Starting from the familiar factorization of integers into primes, we extend the concept of unique factorization to polynomials and beyond. In particular, we will discuss how unique factorization, or lack of it, influenced early attempts to prove Fermat's Last Theorem, and we will explore how it can be used to determine the probabilities associated with rolling a pair of dice.