Date | Speaker | Title |
---|---|---|

We will present a proof of the Stone-Weierstrass Theorem.
(Integration Workshop talk 1 of 3.) | ||

We will recreate Eisenstein's proof of Gauss' Theorema Aureum of quadratic
reciprocity.
(Integration Workshop talk 2 of 3.) | ||

We will prove the famous result on the existence of primes in arithmetic
progression.
(Integration Workshop talk 3 of 3.) | ||

Algebraic K-theory is the study and application of certain functors
K from the category of rings with 1 ≠ 0 to abelian
groups. The functors _{n}K_{0}, K_{1} (called the
“lower” or classical K-groups) are easier to define than the
others and have the most immediate applications.
| ||

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.
| ||

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. | ||

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. | ||

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. | ||

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. | ||

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. | ||

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. | ||

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. | ||

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. | ||

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. | ||

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. | ||

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. |

This page is maintained by John Kerl