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Fall 2008 schedule

Date Speaker Title
August 27 Maienschein, Reilly, Islambekov The Stone-Weierstrass theorem
We will present a proof of the Stone-Weierstrass Theorem.
(Integration Workshop talk 1 of 3.)
September 3 Donahue, Prasad, Marino Quadratic reciprocity
We will recreate Eisenstein's proof of Gauss' Theorema Aureum of quadratic reciprocity.
(Integration Workshop talk 2 of 3.)
September 10 Hinkel, Gottesmann, Lafferty Primes in arithmetic progression
We will prove the famous result on the existence of primes in arithmetic progression.
(Integration Workshop talk 3 of 3.)
September 17 Jordan Schettler The lower algebraic K-groups
Algebraic K-theory is the study and application of certain functors Kn from the category of rings with 1 ≠ 0 to abelian groups. The functors K0, K1 (called the “lower” or classical K-groups) are easier to define than the others and have the most immediate applications.
September 24 Ryan Smith Strange norms: or, How I learned to stop worrying and love the ultrametric
I’ll give an introduction to the p-adic numbers for those without a background in number theory or algebra, along with motivating examples and motivating bagels.
October 1 Derek Seiple Classifying knots ... well, sort of
We will discuss a brief history of knots and why mathematicians care about them. We will then talk about how we can represent these knots in two dimensions as well as what it means for knots to be equivalent. We will then explore some techniques that help determine whether two knots are equivalent. We will see that classifying knots is still an open problem.
October 8 Victor Piercey What the #@?! are sheaves and schemes?
Schemes were invented by Grothendieck in the middle of the last century. The introduction of schemes brought about the much-desired grand unification of number theory and algebraic geometry. The modest goal of this talk is to define affine schemes and to convey a sense of how they unify these two fields of mathematics. The discussion will be driven by examples. I will omit the technicalities that obfuscate the inherent beauty of the theory. I will assume only the basics from the core courses.
October 15 Grethe Hystad Ising model and Ising correlations
I will give a brief review of the history of the Ising model. Then I will discuss the correlation functions for the two-dimensional Ising model and how to compute them. I will show how this problem can be reduced to a representation-theoretic problem associated with the orthogonal group.
October 22 Victor Piercey A discussion with Bill McCallum, candidate for Mathematics department head
Bill McCallum, candidate for the next head of the Department of Mathematics, will speak about his vision for the future of the department and respond to questions submitted by graduate students.
October 29 John Kerl Computational methods in percolation
Lattice percolation, along with the Ising model, is to statistical mechanics as the fruit fly is to biology: easy to produce in large numbers, not too smart or multifaceted, yet with some properties that (one hopes) shed light on more complex systems. I'll describe what bond percolation is and mention some known theoretical results, then walk through some computational approaches to a few questions in the subject. Along the way, I'll discuss some introductory probability, including the elegant inclusion/exclusion principle, and illustrate with ASCII art.
November 5 Michael Bishop Dynamical systems on torii of genus g
In this talk, I will give a basic introduction to dynamical systems, mainly fixed points and the index of a fixed point. Then, after brief review of the Euler characteristic, I will show an interesting result relating the two seemingly unrelated topics.
November 12 Martin Leslie Elliptic curves
An elementary introduction to the arithmetic of elliptic curves with digressions into history, cryptography and million-dollar prizes. I will try to make the entire talk accessible to anybody who knows what a group is.
November 19 Tom LaGatta Probability for the non-probabilist
Probability is the science of quantifying randomness. I will be presenting the backbone of much of the modern subject: Laws of Large Numbers tell us that averages of random numbers tend to their mean, Central Limit Theorems give the fluctuation from that, and Large Deviations Theory puts this all in a nice general framework. Whether an algebraist or applied, no mathematician should leave graduate school without at least a basic understanding of the aims and results of probability theory. This talk will be geared toward first- and second-year students without a background in probability.
November 26 Josh Chesler What is algebra?
Views of mathematicians, mathematics educators, and teachers
A mathematician, a mathematics educator, and a teacher walk into a bar ... Well, not really, but some representatives from each of these professional groups got together to discuss a set of school algebra problems. What did they notice? What did they value? And what does this have to do with their professional identities and their views of algebra and of mathematics? I will provide some insight into these questions and their relevance to algebra classrooms.
December 3 David Herzog Banach limits
It is easy to cringe at the idea that limits do not always exist. In the context of bounded infinite sequences of complex numbers (little l infinity), we have been faced with this issue since the beginning of calculus. For 50 minutes, we will forget what we were taught so many years ago and try to make sense of “limits” on all of l infinity. In this attempt, we will encounter an ugly beast known and accepted almost universally by mathematicians everywhere: the axiom of choice. Interestingly enough if one accepts the axiom of choice, our result (also known as a Banach limit) can be used in practice.
December 10 Bel Polletta Perspectives on quantum field theory
Quantum field theory (QFT) is an attempt to study fields and many-particle systems from a quantum-mechanical point of view. QFT was born around the same time as quantum mechanics, and its successes include the standard model of particle physics. A great deal of current research is devoted to studying two-dimensional QFT, which has connections to two-dimensional critical systems in statistical mechanics. In this talk, I will give two perspectives on quantum mechanics, and show how these can be extended to the study of fields. I will also give an overview of some of the mathematical approaches to QFT, including Segal's axioms and conformal field theory.

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