The SPAMS Schedule - Spring 2006

Abstract: Classification is a central problem in AI and Machine Learning. The goal of classification is to take a small set of example points, each labelled +1 or -1 by some unknown classification function, and infer the specifics of that classification function. To accomplish this goal, one typically needs to make assumptions about the hidden function, and a popular assumption is that it is a "linear threshold function" (LTF) which separates "+1 points" from "-1 points" by means of a plane.

LTFs have been extensively studied by neuroscientists and computer scientists, who have used them to model neural computation and learning. In this talk we will look at two problems related to LTFs: the "learning problem" and the "testing problem." The former has a long history, dating back to the Perceptron algorithm developed by psychologist Frank Rosenblatt in 1959. The latter is the subject of recent research, and utilizes interesting new techniques in discrete fourier analysis.

No particular knowledge of LTFs, discrete fourier analysis, or Frank Rosenblatt will be assumed. Based on joint work with Ronitt Rubinfeld (MIT CSAIL) and Rocco Servedio (Columbia).

Abstract: The living cell is composed mainly of polymers and water, and many biological processes are affected by the particular conformations polymers are likely to explore while floating in solution. One tractable model of these processes treats the polymer as a so-called "wormlike chain", a semi-flexible rod with a quadratic potential energy in bending and torsion. The likelihood of finding a wormlike chain with endpoints at a given relative position and orientation is a problem that was only recently solved. The solution, which will be the focus of this talk, employs statistical ensembles, path integrals and a 'resummation' to arrive at a closed-form expression suitable for numerical evaluation.

Abstract: As you might suspect, knitting can be used to make nice models of mathematical objects. As you might not suspect, some mathematical objects are nice models of knitting. I'll be discussing relations between mathematics and knitting, and displaying both knitted and mathed models. Plus, special guest star: The Geekiest Sweater Ever Made.

Abstract: Electrochemical systems are most commonly described by the Nernst-Planck equations, which can be derived through an intuitive reasoning of fluxes over a control volume. In this talk, we present an alternative derivation of Nernst-Planck equations using statistical mechanics. As a byproduct, we modify our model in a way to account for the finite size effects of the ions and the water molecules, which result in the so-called modified Nernst-Planck equations. The solutions of this set of equations, unlike standard NP, never exceed the physical limit set by the ion size, eliminating one of the weaknesses of the NP system.

Abstract: Have you ever wondered what would happen if you fell into a black hole, but were afraid to ask? If so, this talk is for you. Starting from a whistle-stop tour of special and general relativity, I shall present an informal discussion of these fascinating objects. I shall discuss their geometry and causal structure, present astronomical evidence for their existence, and speculate on the possibility of parallel universes.

Abstract: A crystal with a small miscut from a plane of symmetry results in a surface covered by steps of atomic height. In the absence of material deposition, crystal surfaces relax to become flat via the motion of steps, and can develop macroscopically flat regions called facets. For axisymmetric surface profiles, the steps are concentric circles with radii that satisfy a system of ODEs coupled because of step interactions. These "step-flow" ODEs have peculiar properties associated with the rapid motion of extremal steps and the formation of facets.

First, I will focus on predictions generated by the step-flow ODEs. These results concern properties of step bunching and - since crystals with an infinite number of steps have been studied frequently in the past - the effects of finite crystal height. In particular, I show that under certain conditions, step bunches can always form by choosing a suitable initial step configuration. The effect of finite crystal height is described quantitatively by tracking the facet expansion.

I will also talk about a PDE model of surface relaxation, focusing on the issue of boundary conditions. A boundary condition that incorporates the discreteness of steps at the facet is implemented and is shown to give good agreement with step-flow data for a wide range of step-interaction parameter values.

Abstract: The latest hype in computational PDEs is called the Discontinuous Galerkin Method (or "DG"). It can be seen as a combination of the popular finite element and finite volume methods, and it is particularly successful for solving wave propagation and convection dominated problems. The solution is represented in a piecewise discontinuous basis, and the discretization is stable for arbitrarily high polynomial orders. The domain is discretized by unstructured meshes (for example with simplex elements), which makes it geometrically flexible and allows for local adaptation.

I will talk about numerical solution of PDEs in general and give an informal introduction to the DG method. I will show the benefits of high order discretizations and unstructured meshes, and illustrate with examples ranging from simple scalar problems to the full unsteady compressible Navier-Stokes equations.

Abstract: In the mythical world of King Arthur, can the wizard Merlin help his king solve the most challenging of problems while not revealing his techniques in doing so?

Abstract: The quantum entanglement is a useful resource that not only strengthens the computational power of quantum computer, but makes the quantum communication efficient. We investigate how to create the quantum entanglement and use it as a resource of communication. No previous knowledge about quantum computation is required.

Abstract: In this talk we will discuss basic topics widely used in mathematical finance such as the binomial model, martingales, brownian motion, and the Ito integral. In addition, we will consider some examples of option pricing, and, perhaps, get into the Black-Scholes model. No special background in probability is needed.

Abstract: In honor of the 10th anniversary of SPAMS we discuss gambling, a topic pervading several applied math fields. We begin by presenting techniques used to customize the distribution of results in a gambling environment, with emphasis on betting systems and bet spreading. We will then analyze the game Craps and show how to reduce the house edge to almost 0. Moving into an entirely different realm of applied math, we will then discuss Roulette as a physical system and derive equations to predict the trajectory of a Roulette ball(!). The history of beating Roulette with physics is very exciting and we will cover some of the major successes of the past. No knowledge of Craps or Roulette needed.