The SPAMS Schedule - Spring 2003

Everyone knows that, when put through thorough questioning, (light in the eyes, hot, stuffy room, etc) any liar will eventually crack. Add to that some constraints to his lying system, and (given enough time) you're sure to find out anything that he's hiding.

But suppose you're under a lot of time pressure (a bomb will explode somewhere in the city in less than two hours unless you find out the password....) and there's only a given number of questions that you can afford to ask. Will it be enough for you to find the right answer from a large pool of possibilities, thus saving the day, or should you just jump on the first plane and make your getaway in shame and infamy?

Come and find out.

Want to see some amazing juggling? Want to hear about some fascinating mathematics? Want to eat some delicious food? Want to get a hot date for Valentine's day? Then walk through Lobby 10 on a Sunday, come to any other SPAMS talk, head to the North End or read one of those MIT/Harvard Love posters that are around everywhere. But if you want one thing that approximates all these functions (well, hopefully three of them anyway) then come to SPAMS this Thursday.

My Guarantee: after attending this talk, you will be able to juggle at least zero balls using either one or two hands, OR YOUR MONEY BACK!

Warning this talk will be performed by a professional on a closed course and may contain physics - do not attempt at home. Terms ans conditions apply.

Electrons trapped in helium can be modeled as very simple mathematical systems. While soap bubbles minimize surface area (that's why they are round!), electron bubbles minimize the natural frequencies of the cavity. This leads to very beautiful shapes. I will talk about how to obtain these shapes and the related intriguing mathematical, engineering and computation problems.

This Friday marks the 50th anniversary of the day Watson and Crick finally figured out the basic structure of DNA. To celebrate, I'll be talking about the topology of DNA, from the double helix and linking numbers to tangles and topoisomerases.

In a graph G with given edge lengths, the minimum spanning forest is a set of edges with minimal total length, providing connectivity between all pairs of nodes, which are connected in the original graph. We consider the situation where some nodes of G are randomly destroyed, and the minimum spanning forest of the remaining graph is needed for some purpose. Is there a small set of edges Q which contains this random minimum spanning forest with high probability? It turns out that we can always find a set Q of size O(n log n), and this is asymptotically optimal.

For a cell complex the f-vector is (f_0, f_1, ... f_d), where f_i is the number of i-dimensional faces. Perhaps the most familiar encounter with this invariant is the Euler-Poincare formula: f_0 - f_1 + f_2 - ... = 1 + b_0 - b_1 + ... (where b_i is the i-th Betti number). In particular, recall the well known result V - E + F = 2 for planar graphs. The f-vector problem seeks to determine relations on the f_i, to classify which vectors appear for various classes of complexes, and generally to understand the larger role of this invariant. I will survey some of the results and open problems in this area.

One of the most important applications of complex analysis to physics would be the invariance of Laplace equation under conformal map. Using the map one can transform a complicated geometry to a simple one where the same equation still holds and can be solved more easily. Recently this invariance was extended to some other non-Laplacian equations representing many interesting problems of physics. In this talk, I'll explain this invariance in a more intuitive way and show an application of it regarding growth phenomena.

EIGEN, MY EIGEN: A tale of forbidden eigenvalues in an era of trees with low path cover numbers.

First there was the adjacency matrix. Then came the Laplacian. Now enters S(G), the class of all symmetric matrices with graph G. (Think of "weighted adjacency matrices.") The collection of eigenvalues associated with S(G) describes certain features of G. In fact...

The path cover number of a tree T is precisely the highest multiplicity of any eigenvalue occurring among matrices in S(T).

We will define path cover number and prove this claim.

The result arose from an inverse eigenvalue problem, and is due to C. R. Johnson and A. Leal Duarte.

Abstract: The game SET was invented by population geneticist Marsha Jean Falco, who was using symbols on cards to find patterns in genetic data. While simple enough to be played by children, the game has a rich mathematical structure and inspires questions on the combinatorics of affine and projective spaces as well as the theory of error correcting codes. In this talk, we discuss the mathematics of SET as well as some generalizations of the game and open problems.

Water striders Gerridae are insects of characteristic length 1 cm and weight 10 dynes that reside on the surface of ponds, rivers, and the open ocean. Their weight is supported by the surface tension force generated by curvature of the free surface, and they propel themselves by driving their central pair of hydrophobic legs in a sculling motion. Previous investigators have assumed that the hydrodynamic propulsion of the water strider relies on momentum transfer by surface waves. This assumption leads to Denny's paradox: infant water striders, whose legs are too slow to generate waves, are incapable of propelling themselves along the surface. We here resolve this paradox through reporting the results of high-speed video and particle-tracking studies. Experiments reveal that the strider transfers momentum to the underlying fluid not primarily through capillary waves, but rather through dipolar vortices shed by its driving legs. This insight guided us in constructing a self-contained mechanical water strider, Robostrider, whose means of propulsion is analogous to, if less elegant than, its natural counterpart.

Schoenberg invented the technique of 12-tone composition in the 1920's, as a possible alternative to the tonal system which had governed the previous 250 years of Western Classical music. The notes of a 12-tone piece are based on a "tone row," (a particular ordering of the 12 pitches of the chromatic scale), as well as related tone rows that are the result of applying various operations to the original.

Since a tone-row is an ordering of 12 elements, it can be thought of as an element of the symmetric group; furthermore, each operation on a tone row is the result of multiplying the tone row by a particular element of the symmetric group. We will examine the question of how to define equivalence classes of tone rows, with a cute graphic notation to represent them, and then use some group theory to calculate how rare symmetric tone rows are.

Hopefully this will lead us to a better appreciation of this unusual music. Complete with live musical examples!

In a story full of mathematical action, Alice and Bob come together once again to reveal a new face of information theory. And using entropy as a tool we will solve some counting and graph theory problems, thus closing the gap between combinatorics and information theory.

Individuals can decide which groups of people they want to belong to... but that decision is constrained by their attributes. This brings up the following questions: Can we look at the distribution of attributes over a populations and predict which coalitions are likely to be formed? And how likely are we to end up with a stable situation? Finally, what do posets and polytopes have to do with it?