The SPAMS Schedule - Fall 2004

Through the ages, mathematics has danced with many partners. Sometimes leading; sometimes following. Occasionally clumsy; often graceful. In this SPAMS talk, we will take a look at the relatively recent dance between mathematics and computer simulation. We'll see how pure and applied mathematics provide a crucial foundation for the validity of simulation methodology and how modern simulation technology can pave the way to new mathematical insight. Several software packages that may be of value to graduate students and researchers in applied mathematics, science, and engineering will be highlighted, and a few open mathematical problems will be discussed.

High-level quantum mechanics simulations aren't the only way to simulate certain aspects of biological reactions; as it turns out, continuum electrostatics--yes, trusty old Laplace and his equation-- is quite capable of accurately modeling the electrostatic interactions between molecules. As a result, to look at some biological problems of interest, we can turn to well-known results and methods from applied mathematics and computational science. The simplicity of these mathematical models has helped shape our understanding of the biological systems, and in turn permitted the development of computational techniques that exploit the mathematical structure of the biological problem. The talk will move from biology and the hot topic of drug design into the mathematical formulation, and from there to numerical methods for its solution. Finally, we will return to biology and see that all the equations and floating point operations mean something back in the real world.

Polya's recurrence theorem states that a point walking on a d-dimensional grid is bound to return to the starting point when d<=2, but has a positive probability not to return when d>=3.

Instead of classical proof of Polya's theorem which contains a complex calculation of integral, we will go over a more simple and intuitive proof using electrical networks. The talk will present the relation between electric networks and random walks and prove Polya's theorem. At the end we will apply the tool to more general graphs.

Given a integral convex polytope P, for any positive integer m, we use mP = { mx : x is in P } to denote the dilated polytope of P. Eugene Ehrhart discovered in 1960s that the number of lattice points in mP is a polynomial function E_P(m) whose degree is the dimension of P. So E_P(m) is called Ehrhart polynomial of P.

I will discuss the motivation of considering counting lattice points in a polytope, survey some results related to this problem and give a sketch of one proof of Ehrhart's theorem.

In this talk we describe the task of touring the moons of Jupiter. Through theoretical and computational means, we analyze the dynamical behavior of the three-body problem at both high and low energy. At high energy, we use a two-body approximation in order to characterize sensitive impact regions. At low energy, we use the so-called "interplanetary superhighway" whose routes originate from special perturbations of periodic orbits. In the end, we apply the results of our findings to NASA's proposed Jupiter Icy Moons Orbiter mission whose goal is to determine whether life exists under the frozen surface of one of Jupiter's icy moons.

Lump type solutions of various nonlinear wave equations (nonlinear Schrodinger equation, Davey--Stewartson equations, Benjamin equation) and their mutual relations are presented. Simple and intuitive ideas are introduced to explain their existence, numerical implementation and physical meaning. Dynamic simulations are provided to verify their stable structures as solutions of unsteady differential equations.

Wouldn't it be nice if we could turn ugly sequences $a_1, a_2, a_3, \dots$ into nice powers $a^1, a^2, a^3, \dots$? And actually prove stuff this way? Well, sometimes we can. We'll see where this works, why it's useful, and how it has applications to topics including orthogonal polynomials, combinatorial species, and invariant theory.

The availability of the genomic sequences of a large variety of organisms has revolutionized the field of evolutionary biology. One new question regards the organization of information within genes. It was discovered in the late 1970s, here at MIT, that many genes are "interrupted" in the genome and have to be "spliced" together in real time before the cellular machinery can use them. These interruptions, known as *introns*, differ by a variety of measures depending on the genome that houses them: length, frequency, or position to name a few.

Why are these interruptions organized differently in different organisms? How did these changes arise? What were the interruptions like, say, 200 million years ago? Why are they present at all?

In order to trace their evolutionary history we traced their patterns of conservation in related genes from different organisms. Any given pattern could be interpreted in a variety of ways, and it takes a probabilistic model to infer what these interruptions looked like in long-gone ancestral organisms.

Having stripped this talk of most of the complicated physics behind Magnetic Resonance Imaging, we discuss in simple terms some aspects of this relatively new technology, focusing mostly on brain fiber tracking. Using water diffusion data, we are able to extract information about fibers and how they connect various regions of the brain. The oldest brain fiber tracking algorithms are presented and we discuss how these can be improved using functionals and the calculus of variations to give better and more reliable results.

Tropical mathematics is the study of the tropical semiring consisting of the real numbers together with the operations of + and min. One can do arithmetic and geometry over this ring, and there are quite a lot of analogies between ordinary mathematics and tropical mathematics. Additionally, there are applications to mathematical biology! In my talk I will give an introduction to this field, touching on subjects such as tropical lines, tropical curves, the tropical Grassmannian, the Whitehouse complex, and phylogenetics. The adjective "tropical" was coined by French mathematicians, including Jean-Eric Pin, in the honor of their Brazilian colleague Imre Simon, who was one of the pioneers in min-plus algebra. There is no deeper meaning of the adjective "tropical". It simply stands for the French view of Brazil.