The SPAMS Schedule - Fall 2007

Differential equations provide the mathematical backbone of physics. Specifically, understanding equation invariant continuous transformations allows one to gain physical intuition, introduce conserved dynamical quantities and generalize ones problem. Although all branches of modern physics use these ideas, I will focus on a sequence of examples originating from quantum mechanics. Starting with angular momentum, I will show how this lead to the bold, yet logical introduction of spin: an intrinsic particle property corresponding to the irreducible representations of SU(2). In keeping along the same line of reasoning, I will then show how these ideas extend to understanding the particles in relativistic quantum mechanics.

Why does the piano keyboard look the way it does? What is the origin of the famous "do re mi" major scale? Dating back to the Babylonians and Pythagoreans, civilizations have been aware of the notion of "consonance", i.e. that two frequencies sound pleasing together when they form simple integer ratios. In this talk, we expain what is truly a "mathematical journey" that leads one from a simple notion like consonance to the remarkable structure of the keyboard and the variety of scales it encompasses. The results follow rather quickly from three steps and almost no added assumptions: 1) use consonance to define a tuning scheme, 2) use the tuning scheme to construct a chromatic scale, 3) use the chromatic scale to define other scales. Along the way, we will see why consonant intervals are physiologically pleasing, how the 12-note chromatic scale is in fact the solution to an optimization problem, and how Bach's mathematical incite is the reason why we can seamlessly change keys on a piano. No prior knowledge of music theory will be necessary.

Abstract: When is a given position of a Rubik's cube derivable from the solved position? How can one solve the Rubik's cube from a random position? What is the group of all (legal) moves of the Rubik's cube, and in particular how big is it? I will try to answer these and other similar questions.

The Greene-Kleitman theorem is a surprisingly elegant and very non-trivial result in the combinatorics of partially ordered sets. Here is its statement. Let P be a poset on n elements, and let c_i (resp. a_i) be the largest number of elements of P in a union of i chains (resp. antichains). Then (c_1, c_2 - c_1, c_3 - c_2, ...) and (a_1, a_2 - a_1, a_3 - a_2, ...) are conjugate partitions of n. (Even the fact that they are partitions, meaning that numbers are in descending order, is not trivial.) In the talk I will do my best to present the proof.

Abstract: In general, in order to reconstruct an image (or any other signal) exactly from a set of measurements, one needs at least as many measurements as there are variables in the signal. However, if our image can be compressed nicely, say that it is much smaller in JPEG than in bitmap, our intuition would be that we really need measurements on order of the size of the JPEG, not the size of the full image. Recent work has yielded a number of closely related techniques for doing exactly such reconstruction in a variety of contexts. This talk will go over several notable examples as well as the mathematics behind these techniques.

A common problem in computational probabiilty is generating a random sample from a state space, for example, a random spanning tree of a graph. One popular way to do that is to devise a Markov chain. However, it might be slow due to the large number of states. We will discuss some general approaches to tackle this problem as well as specific techniques that speed up the running time. Knowledge of naive probability theory is desired, though not necessary.

Abstract: Puzzles and paradoxes are always crowd pleasers (at least I hope they are...). In this talk, I'll discuss a couple of paradoxes of a probabilistic nature that I think are particularly interesting. In particular, I'll talk about the two-envelopes problem and show why most people's explanation is actually incomplete, and I'll show how being a loser twice over can make you a winner.

It turns out that knowing some elementary representation theory for SO(3), SU(2), and SO(4) can get you a long way in understanding the electron structure of the hydrogen atom. I will briefly discuss some concepts from representation theory of semi-simple Lie groups, then some tenets of quantum mechanics, and finally apply them to predict the sizes of the electron shells. Perhaps we'll also clear up some of the fuzz that remains around our memories of the chemistry we learned in high school. The approach I will take has the advantage of being fairly non-technical: for instance we won't need to solve the Schroedinger equation explicitly, as is often done in intro physics courses.

Abstract: A well-known theorem states that beauty is in the eye of the beholder. In this talk, we will show that the intersection of the eyes of all beholders is non-empty. Indeed, who can look at Gessel-Viennot proof of the Binet-Cauchy formula without bursting into tears of awe and gratitude? Do you know of anyone who has heard the combinatorial proof of the central limit theorem and is still a cynic? Can anyone truly understand the RSK algorithm and not dedicate a poem to it? And can a sentient human being see the Greene-Nijenhuis-Wilf proof of the hook-length formula and not write a love letter to the authors?