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Fall 2007 schedule

Date Speaker Title
September 5 Ben Polletta SU(2) and the Hydrogen Atom
This is a continuation of a talk I gave last semester in the Applied Math Brown Bag. Last semester I spent 45 minutes talking about groups and their representations, and 5 minutes handwaving about SU(2) and the hydrogen atom. In this talk, I hope to spend five minutes talking about group representations, and 45 minutes talking about Lie groups, SU(2), and the hydrogen atom (so maybe it's an inversion of my previous talk, instead). The idea is the following. The electron orbitals of hydrogen are labeled with three quantum numbers - energy (n=1,2,3,...), azimuthal (l=1,...,n-1), and magnetic (m=-l,...,l). As with most beautiful and regular things in nature, there is a group behind these numbers, and that group is SU(2). In this talk, I will use representation theory and Lie theory to explain how, if not why.
September 12 John Kerl Is "2" a Random Number?
This is one of those questions where you get to push back on the questioner. The modern notion of randomness is grounded in measure theory, but revolves around the simple notion of What would happen if I repeated my experiment many times? Tabulating various expected outcomes (which may vary from one application to another) in the form of so-called distribution functions allows us to meaningfully phrase our questions about randomness. In this multidisciplinary talk, I will show how to create a computer program to generate samples from various distributions (things such as the famous bell curve). Along the way, I will sketch ideas from several different disciplines: Probability, Statistics, As far into number theory as I dare to tread (which as you will see isn't far), and Software
September 19 Tom LaGatta What is a Random Function?
Just as the title question's cousin, "What is a random number?" does not have as simple an answer as it may seem, the notion of a random function requires a lot of mathematical clarification to make sense. In its simplest formulation, a random function is a collection of random variables tied together in the right way. The choice of the random variables' distribution is crucial to the "right way" actually producing a well-defined family of random functions. Miraculously, the choice of Gaussian distributions makes the whole process go through smoother than perhaps it otherwise should. In this talk, I will give the mathematical preliminaries to these so-called Gaussian random fields, detail some of their nice properties, and discuss some applications. I hope to make this talk worthwhile to both pure and applied students, since Gaussian random fields are a very robust tool as well as mathematical interesting. This talk will be accessible to a general mathematical audience.
October 3 Alex Petrov The group law of an elliptic curve (de)mystified
When I was an undergraduate elliptic curves seemed to occupy an enigmatic focal point of a lot of questions. How do we factor numbers? Well, one way is with elliptic curves. How do we do cryptography? Elliptic curves. I heard that this guy Wiles proved Fermat's Last Theorem?!? Yes and he used elliptic curves (along with some other things)... I will not talk about any of the things above. What I will talk about is the mysterious group law that the points on an elliptic curve satisfy. I will try to give the necessary background from algebra, I will state the famous Riemann-Roch Theorem and use it to derive the law in a purely algebraic way. The Riemann-Roch will play a mystery role on which the derivation of the group law depends. Hence the duality in the title. If you are willing to accept facts on faith then you will be demystified. On the other hand you may leave the room wondering what really happened there... then you will be just like me. It is good that there will be bagels regardless.
October 17 Yijun Shao Blow-ups and Resolution of Singularities
Blowing-up is a type of geometric surgery, particularly applied in algebraic geometry. It not only allows one to construct a new variety (or manifold) birational to the old one, but also provides a way to resolve singularities. It is conjectured that the singularity of any singular variety can be resolved by a finite sequence of blowing-ups. In 1960's, Hironaka proved it in all dimensions over fields of characteristic 0. However, the problem remains open for positive characteristics. In this talk, I will try to do several examples to show what blowing-ups are and how they resolve singularities, without assuming much knowledge in algebraic geometry.
October 31 Casey Warmbrand The Lindstrom-Gessel-Viennot method for weightings of non-intersecting paths
We will discuss particle configurations of a zig-zag path in a domino tiling of an Aztec diamond. The Lindstrom-Gessel-Viennot theorem, which counts the weight of a collection of non-intersecting paths using a determinant, will be presented and then proved. I will then use this result to show an equivalence with the probability of a given particle configuration of a random Aztec tiling and the measure on Krawtchouk ensemble of Orthogonal Polynomials, involving the square of a Van der Monde determinant; Hooray for Linear Algebra!

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