The SPAMS Schedule - Fall 2000

Abstract: Alberto, Alex, Francois and Peter are moving into their new 4-bedroom apartment this week, which they all got to see before the summer break. They have already divided the total rental cost of the apartment between the 4 bedrooms and have priced each bedroom according to its merits. Each of the 4 guys will choose a bedroom as soon as he returns to Boston and, more specifically, will take the most expensive one that is still available but fits within his budget. Alberto, who wants to spend as long as possible at home in Italy, will be the last to return to Boston. Will the last available bedroom fit within his current budget or will he be forced to cut back on his wild drinking and partying lifestyle?

We will address this and other related questions. Our approach will be in terms of parking functions and will be largely combinatorial.

Abstract: Theoretical computer science has devised simple algorithms which, given access to a source of randomness, probabilistically verify fundamental mathematical properties. We can quickly check if a number is prime or a multivariate polynomial is identically zero-- assuming we are willing to tolerate some small chance of error. Computationally these tests are celebrated for their efficiency and accuracy. From a mathematical perspective, however, they are less than satisfying. These randomized tests do not reveal, say, the locations of roots of polynomials, nor do they give an explicit characterization of primality.

Derandomization is the attempt to eliminate randomness from a randomized computational test while preserving its correctness. For example, enumeration over all possible random choices constitutes one, albeit naive, approach to derandomization. Nontrivial derandomizations yield nontrivial structural characterizations of mathematical properties; a deterministic algorithm for primality almost certainly implies a breakthrough in number theory. In return, structural considerations suggest tantalizing possibilities for new randomized algorithms-- can we test if a multivariate polynomial is identically zero using a number of random bits related to the total degree rather than the number of variables? We will survey recent attempts at derandomization, most of which involve exploiting intriguing connections between mathematical problems and their computational complexity.

Abstract: Sunspots are relatively cool areas that appear as dark blemishes on the face of the sun. They are formed when magnetic field lines just below the sun's surface are twisted and poke through the solar photosphere. Scientists became aware that the sun went through cycles and changes by observing sunspots.The number of sunspots can be an indication of the degree of solar activity. The part of the cycle with low sunspot activity is referred to as "solar minimum", the portion with high activity is known as "solar maximum".

Solar activity gives rise to the variations of the solar-terrestrial environment, which may affect weather patterns, radio communication and the orbital lifetime of satellites.

Much effort has been devoted to forecasting the properties of future cycles, in particular, the amplitude and phase of sunspot maximum. Scientists still do not completely understand all of the aspects of the solar cycle and it is difficult to predict just how strong the solar maximum will be.

Modelling and prediction of the sunspots numbers will be discussed in the talk.

Throw a bunch of planes into space, and you'll find a lot of interesting combinatorics. My talk will illustrate this by discussing the Linial arrangement and some combinatorial problems related to it.

While the title contains far too many mathematical words, I will assume no previous knowledge of any of them.

Abstract: To encompass the above I will give one of the following talks

a) A proof of the Littel theorem for unstable systems and a short proof of the 4-colour theorem and an analysis concerning the existence/implications of "the new economy".

b) Putting m balls into k boxes.

Abstract: We study the phase diagrams of a variety of $SO(M)\otimes SO(N)$ symmetric spin models. These arise as effective field theories in a number of contexts. In particular, for $M=1$, $N=2$, this theory describes an anisotropic antiferromagnet in an external magnetic field. For $M=2$, $N=3$, the theory unifies superconductivity and antiferromagnetism. Finally, the case $M=4$, $N=6$, describes a model that unifies chiral symmetry breaking and color superconductivity in QCD. We describe some theoretical predictions for the structure of the phase diagrams and present preliminary results of numerical simulations of these models. Pretty pictures will be shown.

Abstract: The Support Vector Machine (SVM) idea has attracted recent attention in solving classification and regression problems. As an example based method, SVMs distinguish two point classes by finding a separating boundary layer, which is determined by points that become known as Support Vectors (SVs). While the computation of the separating boundary layer is formulated as a linearly constrained Quadratic Programming (QP) problem, in practice the corresponding dual problem is computed.

In this talk, we discuss how the solution to the dual problem depends on the geometry. When examples are separable, it can be shown that the Lagrange multipliers (the unknowns of the dual problem) associated with SVs can be interpreted geometrically either as a normalized ratio of simplex volumes or in terms of three angles, and at the same time a simplex volume decomposition relation must be satisfied. We finish this talk with a random matrix model of Lagrange Multipliers.

Abstract: Ever wondered what would happen to a drop if we were to put an electric charge on it? Or what would happen to the surface of water in the presence of an electric field? Of course, we all have. The short answer is that if the charge or the field are relatively small then surface tension will win and nothing interesting will happen. However, if the electic charge (or field) are sufficient then the surface tension will lose to the electrostatic forces and the drop (surface) will begin deform.

We will provide a slightly longer answer. We will start with a refresher of calculus of variation and re-derive the Euler-Lagrange equation. We will then demonstrate how to generalize the approach to the case of a deforming boundary and more complicated integrands. We will show how to incorporate iso-perimetric constraints (i.e. mass conservation) into the calculations. We will then apply these techniques combined with idea of energy minimization to charged bubbles and determine evolution equations and stability criteria.

We gaurantee that every attendee will leave with a firm understanding of the presented material, abilty to apply variational techniques to a problems relevant to their own research, but at the very least with a stomach full of Indian food flown in directly from Deli.

PS: Did you say "Deli" or "a deli", Francis?

Abstract: Sometimes we are faced with multidimensional data where there are a few data points that don't look like the others. How many of these points can there be? How can we identify them? We present an answer involving volumes, ellipsoids, slabs, duality, and other neat features of n-dimensional geometry that any lay person could understand.

Abstract: Imagine for a moment that it is 11:50am, and we are locked in a room containing a huge board and a switch; on the board, there are thousands and thousands of little dots.

At 11:55am, our guardian tells us that what we see on the board is the random graph $G(1,000,000,p)$; he tells us that, starting at 12:00pm, and over the course of the next $1,000,000$ hours, $p$ will slowly evolve from $0$ to $1$. As it does, edges connecting the dots will appear on the board. The moment the very first triangle appears on the board, the switch will be turned on automatically, the door will open, and we will be free to go. "But", he says with a devilish smile, "don't hold your breath. You will be stuck here for a long, long time."

This talk will prove that who laughs last, laughs best -- and that we will be home in time for dinner.

Abstract: The genome projects are providing biologists with a wealth of protein sequences for which no functional or structural characterization exists. When a three dimensional structure can be assigned to a sequence, useful biological conclusions can often be drawn about the function of the protein in the absence of any lab work. For this reason, computational methods for predicting protein structure from sequence alone can be of great value. I'll describe some of these methods, which exploit similarities between the protein in question and others for which structural data is known.

Say it's your birthday and you want to share your big, square-shaped, triple fudge brownie chocolate chip cake with your friends. Apparently given the consistency of the cake the best idea is to cut it into triangles, and in the spirit of fairness you want all the triangles to have the same area (let's not argue about the fact that more triple fudge goes onto the pieces around the boundary). Being a mathematician, you don't just start choppong away at the cake and hoping for the best - instead you ask yourself: does such a fair cutting exist? Unfortunately you wonder about this aloud and your archenemy (who somehow got invited to the party) promptly interjects:

"Of course, dummy! There's 10 of us, so just cut the cake along a diagonal, then cut each of the two resulting triangles into 5 pieces of equal area by dividing the diagonal into 5 equal segments."

Your honour deeply hurt, you reply in the most menacing tone you can muster up:

"Ok, but what if there were only 9 of us?"

Your nemesis doesn't seem to care about the threat:

"Even easier, dummy! Just cut it into 10 triangles of equal area, and keep one for later."

At this point, you're positively furious. Not only you were called a dummy by your fiercest adversary (twice, and on your birthday!), but his latest answer doesn't satisfy you at all. So you eat your piece of cake, go home and think about how to cut a square into an odd number of triangles of equal areas.

In case all this nonsense isn't discouraging you from attending SPAMS this week, I'll prove that there are no odd equidissections of the square. Time permitting, I will discuss some of the open problems related to equidissections of polygons. Practically no prerequisites needed. 99% fat free. Natural and artifical flavors. High fructose corn syrup.

Abstract: In our daily lives we have all seen vortices in motion. Whether its due to water emptying out of a kitchen sink or the swirling of leaves behind the Green building on a windy day. Besides being fun to watch, vortices are also very important structures that form in the worlds atmosphere and oceans and they determine a lot of the weather systems that we experience. This talk will present basic ideas of geophysical vortices and how they may form due to instability processes.