University of Wisconsin - Madison
Undergraduate Math Club

We will have a meeting during the second week of classes which will be largely mathematical. We will plan on having regular, weekly (or biweekly) meetings this semester. We hope all who have been coming will continue to do so and will inform their friends as well. Look for advertisements. Happy New Year.
When: Second Week of Spring Classes
Where: 9th Floor Lounge, Van Vleck

The Math Club will be showing "The Math Life," a documentary about the mathematics profession, to be followed by a panel discussion with faculty members, graduate students, and undergrads.
When: Monday, September 27th, 4:00 PM
Where: 9th Floor Lounge, Van Vleck

Well, it's the end of the semester, do you really need a better excuse to party? Also, the winner of the Undergraduate Lecture Competition will be announced. Who'll it be? Jesse Beder, Wil Brady, Carl Edquist, Emilie Hogan, Sam Lachterman, or Jason Rute? Come and find out!
When: Monday, May 3rd, 4:30 p.m.
Where: 9th Floor Lounge, Van Vleck

The UW math department is planning to roll out a whole series of CURLs, or Collaborative Undergraduate Research Laboratories, over the next few years. Jim Propp gave an overview of what is happening in this semester's version of a CURL, SSL (Spatial Systems Laboratory) as well as some hints as to what you could expect in the future. Look for the advertisements for any upcoming CURLs as well as keep your eye on Math 491 since that course will be oriented towards whatever research topic is offered at the time.

There were groups, there were Rubik's cubes, and somewhere in between a connection betwixt the two. Ha! Just kidding! Everyone knows you can solve the Rubik's Cube through group theory. If you were there, you'd have known how. Also, be on the lookout for those tricky cubes which someone has moved the colored stickers around on, they're usually impossible to solve (Carl Edquist found that one out the hard way).

Wil Brady gave an overview of the life and times of four famous mathematical personages: Euler, Abel, Galois, and Godel. Euler, known for his prodigious output even after becoming blind in mid-life bounced around various courts of Europe, spending a lot of time in both Germany and Russia. Abel, who died young as the result of poverty, was almost completely unknown during his short life. Norwegian by birth, his poverty made it difficult for him to visit the continent and come into closer contact with the more famous mathematicians who could have helped his career. Galois, every bit the revolutionary, was also hampered by a lack of recognition. He was unable to gain acceptance into the Ecole Polytechnic and died after being shot in a duel. Finally, Godel, who is famous for his incompleteness theorem, was constantly in fear for his health. As a consequence, because he believed people were trying to poison him, he eventually starved himslf to death. Now, if anyone asks you what you study or would like to know a little more about math, you can regale them with these stories instead of waxing lyrical about the joys of group theory.

Alejandro Adem gave a rather in-depth talk about the group SL(2,Z) and its geometric interpretation. Specifically, he spoke about its action on the upper half of the complex plane and about various subgroups such as Z/2Z, Z/4Z, Z/6Z as well as the torsion-free subgroups. In particular, any torsion-free subgroup is also free. He also calculated the indices of SL(2,Z/pZ) in SL(2,Z) as well as the Euler characteristic for SL(2,Z). In all, fairly advanced for an undergraduate lecture.

Several visitors from the NSF came and interviewed some of our undergraduate math majors and were suitably impressed. This provided a convenient excuse to have a party. If you weren't there, you ought to feel shame and remorse for having missed such a good time. The next time the math club gives a party, you'd better be there . . .

Jesse Beder gave a talk detailing braid groups, what they are, how to calculate them, the natural homomorphism to the symmetric group on n letters, and how to solve the Word Problem given a word in B_n. The Word Problem is roughly: given a group (infinite groups being the only ones of interest here) and a presentation for the group, can you determine whether a given element, composed of a sequence of generators for the group, is equal to the identity. Though it's impossible to give an algorithm for an arbitrary group, in the case of braid groups, the problem is easily solved. Jesse presented the algorithm for determining this, along with various other interesting properties of the groups B_n.

Sam Lachterman presented three different proofs of the infinitude of primes, and none of them were even remotely as simple or transparent as Euclid's. The first used a combinatorics argument to show that the sum of the reciprocals of the primes diverged. The second showed explicitly that the prime-counting function pi(x) = { Number of primes less than or equal to x } always satisfies ln(x) <= pi(x) + 1. Since ln(x) goes to infinity, so does pi(x). Finally, there was an algebra proof showing that, were the number of primes finite, Lagrange's theorem for the order of elements in a group cannot hold, hence there is no largest prime. You can check out these proofs as well as others in Proofs from the Book by Martin Aigner (QA36 A36 2004).

This meeting was changed to the Smale talk, given by professor Smale for the entire math department. Smale is a Field's Medalist best known for his proof of the generalized Poincare conjecture for dimensions higher than 4. I did not attend, so I really couldn't tell you how it went.

John Vano spoke about some of his favorite mathematical games, concentrating primarily on the children's favorite Dots and Boxes (which actually does have some non-trivial strategy to it) as well as covering some of the better-known games like Nim.

Jason Rute gave and interesting talk about the philosophy of mathematics, in particular, some of the attempts made at completely defining the notion of "number". Since it is perhaps reasonable to ask whether the objects which mathematicians talk about have any grounding in "reality", various people have tried to create air-tight formulations of the natural numbers which do not appeal to any description of their properties or to actual counting. All have failed. Probably the best that can be done is to use the correspondence principal and leave them without any external existence.

Emilie Hogan gave a proof of the fact that the game of Hex (see: http://mathworld.wolfram.com/GameofHex.html) implies the Brouwer Fixed point theorem for the two-sphere S². She learned this strange result while attending the Carleton College summer program for women in mathematics. If you're female and interested in summer programs, check out the undergraduate research link below.

Prof. Ono, Prof. Propp, and Prof. Vano as well as two graduate students, Jackie and Holly, gave their thoughts on research experiences for undergraduates (REUs). The REU program is run through the NSF and involves various institutions around the country which run summer programs for undergraduates to perform research. Most run for about two months and give a stipend of about $2,000-$3,000 dollars. These are great ways to decide if research is for you as well as to get a leg up on the competition for graduate school. Visit http://math.wisc.edu/~maribeff/UResearch/ for more information.

The premise is simple: Give the best lecture to the Math Club and receive a substantial reward! (We're talking large cash prizes here.) To be elligible, you need only prepare a lecture on some math-related topic (approximately 45 minutes to an hour in length), present it at one of the Math Club meetings and walk away with the prize if yours was the best one all semester. Come to the meeting on Jan. 26th to find out more.

This meeting is over, but you can still get involved with the Math Club. Email mathclub_at_math.wisc.edu. Of particular importance, we need people to give lectures and to think of fun and interesting things to do. Please consider helping out.