The SPAMS Schedule - Spring 2005

Black, Scholes and Merton got 1997's Nobel prize for their 1973's theory on option pricing, in which they elegantly described the price in terms of the time-reversed diffusion PDE. Although the equation admits closed-form solutions for simple cases, it was yet in an esoteric language of ``advanced'' mathematics. Over 30 years afterwards, economists and financiers sought various ideas to intuitively understand the BS equation and extend it to more complicated derivatives. In this seminar, I'll talk about "risk neutral probability" and "change of numeraire" method among others.

Humans, some of them anyway, are skilled learners. Using enigmatic powers of induction, people can often make accurate generalizations based on few examples. Current machines, however, often cannot. The study of the automation of induction by machines constitutes Machine Learning. This field is too broad to cover in a single talk, but we will survey a small portion of it, with an eye towards computability and computational efficiency. Most likely, we will discuss (to some degree of accuracy) the Probably Approximately Correct (PAC) model, which defines the "learnable" concepts in terms of polynomial time computation.

Abstract: We prove Stanley's plethysm conjecture for the $2 \times n$ case, which composed with the work of Black and List provides another proof of Foulkes conjecture for the $2 \times n$ case. We also show that the way Stanley formulated his conjecture, it is false in general, and suggest an alternative formulation.

Melanie went to a fruit-seller at Haymarket to buy a (convex, centrally symmetric) watermelon. He had two of them for the same price, so Melanie wanted the one with more volume. The fruit-seller didn't know which one had more volume ... but he could tell something for sure -- for any pair of planar cuts through their respective centers, the area of the cross-section was always greater for the first watermelon.

Abstract: Diffusive tracers tend to develop intriguing patterns when mixed by fluid flows. These complex patterns, or strange eigenmodes, appear in a broad class of flows, ranging from simple laboratory mixing experiments to simulations of atmospheric dynamics. In this seminar, we will talk about the relatively new dynamical systems approach which view the "strange" eigenmodes as inertial manifolds in the infinite-dimensional function space as well as some perturbation techniques to obtain their shapes approximately.

The sandpile model is defined as a directed multigraph, which is usually assumed to be symmetric, with each vertex containing one or more grains of sand. Every vertex has certain capacity, normally equal to the out-degree of the vertex. If at some point the number of grains of sand at a vertex $v$ exceeds its capacity $D$, we can perform a toppling operation by removing $D$ grains from $v$ and adding one grain to every neighbor of $v$. A stable configuration is a configuration in which no vertex exceeds its capacity. In this talk, we represent the sandpile model algebraically and prove some of its properties. In particular, we discuss the set configurations which inevitably reoccur in the process of random addition of grains to vertices and performing the necessary toppling; such configurations are called recurrent. We establish a group structure on the set of recurrent configurations. We also show that the number of recurrent configurations is equal to the number of spanning trees of the underlying graph, both algebraically and combinatorially.

Chaos theory is one of the most exciting branches of twentieth century mathematics. In this talk, I will present the main concepts behind fractals and chaos, illustrating them both with simulations, and examples from physics. I will discuss population dynamics, strange attractors, Mandelbrot/Julia sets, and fractal self-similar dimension; and I will also report on some recent investigations into three dimensional fractal clusters, created by diffusion limited aggregation.

Abstract: Maybe factorization and prime numbers are of the most interesting properties of natural number. I guess some of us may have encountered with the question of "is there any "fast" way that figures out whether a given number is a prime or not?" Recently Manindra Agrawal, Neeraj Kayal and Nitin Saxena obtained a deterministic algorithm that checks primality of the input number n in polynomial time (over the size of the input bits, here, log n.) In this talk, I'll present this algorithm and show that it determines the primality in polynomial time (roughly, O(log^(7.5) n) time, well, rather a big polynomial lol) So.. now we can check the primality with our pencil and paper, like calculating GCD! (really?)

Abstract: Soap films have recently received quite a lot of attention from researchers because of their natural beauty, and their theoretical applications. In particular, it is thought that they can be used as an experimental realisation of 2D flow. This is intuitively quite natural because soap films are very thin, and one would expect the cross film velocity to be very small compared to the through film velocity. Typically, one is able to exploit this property of soap films to obtain an asymptotic approximation to the flow.

In my talk, I will show that there is a limiting case where the momentum equation can be considered as analagous to that of non-newtonian, viscous, compressible flow, with the effective viscosity depending on the film thickness. I also make an attempt to solve the Chomaz equations on an overset grid using classes from the Overture Library, which is a finite difference framework for solving PDEs numerically on structured, overlapping grids.

Abstract: How can you prove the shortest path between two points is a straight line? Though it makes sense intuitively, is there a rigorous mathematical way to SHOW that a sphere is the shape which minimizes the surface area for a given volume? These questions can be swiftly and elegantly answered thanks to variational calculus.

The calculus of variations has proven itself as an incredibly useful tool for both theoretical and applied scientists. In this short tour, we will derive the basics methods for finding the stationary points of integral functionals (Euler-Lagrange Equation, confining multipliers) while pointing out the amazing usefullness of this topic with several cute geometric examples. Among other physically relevant examples, we will delve into the least action principle of dynamics and Fermat's principle of time minimization for the motion of light rays. We may also consider more advanced theoretical arguments concerning the second variation of integral functionals and functionals with higher order derivative dependence. Get ready for some serious FUN with the calculus of variations!!!

Abstract: The card game SET 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 and answer questions inspired by the game, such as the problem of finding the largest SET-less collection of cards. We will also consider some generalizations of the game and open problems.

Abstract: This question (minus the "discrete") was posed by Marc Kac in an influential 1966 lecture. His conjecture was not without cause: it was known at the time that the sound spectrum of a drum beat encodes the drum's area, perimeter and genus. Although John Milnor had exhibited a 16-dimensional counterexample in 1964, the 2-dimensional problem was not solved until 1992, when Carolyn Gordon, David Weber and Scott Wolpert found two different drums that sound exactly alike.

The original question involves diffusion processes and differential geometry. That's much too hard, so I will present a discrete (simple person's) version of this problem. What's the difference, you ask? Well, differential geometers make drums from leather, whereas we'll make do with fabric (conveniently approximated by a graph on a lattice).

Come find out if two cloth drums can sound the same!