Date: December 7, 2009

Time: 4:00-5:00

Room: Rome #204

Speaker: James Alexander/Case-Werster Reserv Unversity/currently at the Nattional Science Foundation

Title: The Gibbs Phenomenon: The Saga

This is a historical talk. The Gibbs phenomenon is a feature of the behavior of Fourier series of a discontinuous function. The story of its elucidation at the turn of the 20 th century illustrates that mathematical research is very much a human activity—including such things as misattribution, disputed claims of priority, rhetorical long knives, vague (and not so vague) insults, uncritical parroting, etc. Josiah Williard Gibbs was only one person in the story. Other luminaries include Albert Michelson, A.E.H. Love, Maxine Bôcher (of the AMS Bôcher prize), Leopold Fejér, Thomas Grönwall, and Henri Poincaré. In this lecture, we review this history from the late 1800s through the 1910s, together with sequels and prequels.

Date: October 26, 2009

Time: 4:00-5:00

Room: Monroe Hall # 451

Speaker: Keye Martin

Title : A free object in quantum information theory.

Abstract : Here are some questions one can ask about communication:

(1) Is it possible to build a device capable of interrupting any form of quantum communication?

(2) Is it possible to maximize the amount of information that can be transmitted with quantum states in a fixed but unknown environment?

(3) The planet loves binary symmetric channels because all of their information theoretic properties are easy to calculate, but what are their higher dimensional analogues?

(4) If we attempt to teleport quantum information using a state that is not maximally entangled, what happens?

(5) Is it possible to do quantum information theory using classical channels?

It turns out that the answer to all of these questions depends on a certain free object over a finite group.

In any collection of mathematical objects, "free objects" are those which satisfy the fewest laws. For instance, if one takes the set of finite words built from symbols in a set S, and uses concatenation to define a multiplication on it, they obtain the "free monoid" over S, since any other monoid over S can be thought of as the free monoid together with additional restrictions imposed on its multiplication. Computer scientists call the elements of the free monoid "lists." They are among the most fundamental objects in computation. In particular, the free monoid over a one element set is the set of natural numbers with addition.

This talk is about a free object that is very important in quantum communication, where its elements are called "channels." It will be accessible to any undergraduate that is comfortable working with 3x3 real matrices.
 

Date: September 14, 2009

Time: 4:00-5:00

Room: Monroe Hall # 451

Speaker: James Yorke

Title: Surprising patterns emerge when "bisecting" proofs of theorems

Abstract: Theorem and proof are conjoined twins but I have always had a secret preference for proof, or rather for those kernels of proofs that recur.

I have been developing an art form in which the goal is to take a proof of any theorem and reorganize it into two parts. Reduce it to two tasks or lemmas to be proved. Of course those parts might in turn be similarly bisected, hopefully not ad infinitum. I was led to this goal by remembering classes that presented complicated proofs requiring many disparate facts to be proved, making it impossible for me to wrap my mind around the collective entity and see it as a unity. For example I encountered the Poincare-Bendixson Theorem in college, to my considerable confusion.

I often find this "bisection" task quite difficult, but when I succeed, I sometimes unearth connections with proofs of other theorems. Here I will report on three eerily connected results.

1) The Poincare-Bendixson Theorem

2) Degree theory in R^n made constructive

3) Why period doubling cascades exist in one-parameter families of maps.

Past Colloquium and Seminar Speakers