University of Wisconsin - Madison
Undergraduate Math Club

Below are all the announcements for talks given during the Fall 2003 semester.

Adam Feldmann gave a particularly amusing talk about the various types of random number generators (RNGs) available. Starting with the Lehmer RNG, which is little more than an iterated linear polynomial modulo some integer, he moved to Blum-Blum-Shub (spelling is most likely wrong) and its convenient characteristic that it is as secure as factor integers. (See: Dixon's algorithm or any of the variants such as the Number Field Sieve.) The final topic was how to seed the RNG in a modern computer system by using the timing of keystrokes and the occurences of network packets. In all, a rather amusing talk.

Prof. Ken Ono gave an overview of many of the interesting results in the theory of partitions. Specifically, he showed and proved the Eulerian product formula, mentioned the Hardy-Ramanujan asymptotic estimate for p(n), and discussed congruences of the form p(An+B) = 0 (mod M). Ramanujan found the three congruences p(5n+4) = 0 (mod 5), p(7n+5) = 0 (mod 7), and p(11n+6) = 0 (mod 11). Prof. Ono's own work shows that there are infinitely many such congruences for M >= 5 (and they form a set of measure zero), but that the three due to Ramanujan are the only such congruences where A = M. All this was presented in the context of a little history regarding Euler, Ramanujan, and Freeman Dyson, and their work on partitions.

Robert Owen gave a whirlwind tour of the current (post-Godel) state of affairs concerning the Generalized Continuum Hypothesis (GCH) which states that the power set of an infinite ordinal is the next infinite ordinal. As it turns out, restricting the universe of discourse to explicitly constructable sets is more than sufficient to prove the validity of the GCH. However, there are other universes in which the power set of an infinite ordinal can be made to equal any larger infinite ordinal. The theory, Mr. Owen explained, only gets weirder from here.

Prof. Assadi spoke a little about modeling brain function. Although the field is not terribly well understood at the moment, he gave some hints about how current research might be able to quantify the decision making capabilities by careful observation of what is commonly referred to as "subjective" behavior. By continuing to match up stimuli with responses, being very careful to take into account contextual information like the time of day, the person's mood, etc. you can begin to predict reactions, leading towards someday modelling the brain's thought processes artificially.

John Vano spoke this week in place of Benjamin Collins who was sick. Mr. Vano spoke about the three body problem and how it relates to the gravitational interaction between objects in the solar system. Specifically, he spoke about KAM theory which allows approximate solutions to the problem in nondegenerate cases. By solving a stable set of equations and then perturbing the bodies, you can get a fair approximation to the actual solution.

Prof. Martin Isaacs gave a lecture concerning tiling a sphere with pentagons and hexagons, and proved that any such tiling must necessarily contain exactly 12 pentagons. Any number of hexagons (except 1) may be used. The reasoning behind this has to do with Euler's formula for the number of vertices, edges, and faces. Specifically, V + F = E + 2. If H is the number of hexagons and P is the number of pentagons, then F = H + P, E = (6H + 5P) / 2, V = 2E / 3. Solving, we get P = 12.

Prof. Jim Propp gave every attendee a hands-on appreciation for knot theory by demonstrating various knot configurations using long chains of human beings. Great for those dull moments at cocktail parties, and very informative about exactly what is needed for a true knot versus what can be deformed into the "unknot".

John Vano spoke about mathematical games and the winning strategies for non-partisan games. The well-known game of Nim was discussed, along with Nimbers and general winning strategies, but towards the end, the discussion shifted to the deceptively simple game of Dots and Boxes. Although usually played by children, the general strategy is deep enough to fill a small book. It turns out that it can be fairly effectively analyzed by treating it as a game under normal play rules (where the last person to play wins), and that many of the common structures can be assigned Nimbers. In short, the general strategy is to eye the opportunities for long chains of boxes and make sure that there is a specific parity of such structures. Then, by forcing your opponent to take a certain chain, you are assured of taking the next shortest one, and so on. John will most likely continue next semester with a further talk on non-partisan games.

Prof. Joel Robbins detailed a proof of the Gauss-Bonnet theorem applied to polyhedra. In short, the Gauss-Bonnet theorem states that the integral of the Gaussian curvature over a surface is equal to two pi times the Euler characteristic of the surface. This can be applied to polyhedra without the use of calculus by mapping the polyhedron to the unit sphere and calculating the area of a triangle embedded in the sphere. Furthermore, the curvature of a vertex turns out to be two pi minus the sum of the angles of the faces meeting at that vertex.

The talk was a gentle introduction to elliptic curves and the group operation defined on them. Then there was an introduction to the ElGamal public-key cryptosystem where the problem of solving the discrete logarithm came up. After that, a basic overview of the Baby Step, Giant Step and Pollard Rho methods of solving the discrete logarithm. If you were there, you could now (theoretically) go on to solve the Certicom ECC Challenge and win lots and lots of money. See the sidebar to join the ECC2-109 effort.

It was...fun! You should have been there.

Professor Sigurd Angenent talked first about medical imaging and the associated image processing to remove noise and other artifacts from the images. Then he spoke about the topology of the brain (roughly a surface with curvature 1), and finally about how mapping the brain to a sphere localizes brain function and makes it easier for neuroscientists to analyze.

The documentary The Math Life gives an in-depth look at what it's like to be a mathematician engaged in the depths of research. The video shows various mathematicians talking about their experiences and why, exactly, the love math. You can check it out yourself from the Math Library in B2 of Van Vleck, call number QA10.5 M37 2002.