The SPAMS Schedule - Fall 2005

Abstract: Soap films have become increasingly popular as a cheap and practical realization of two-dimensional flows, and have been used by many researchers within the last decade to study a wide range of phenomena in fluid mechanics, ranging from shockwave dynamics and fluid structure interactions to turbulence. In this talk, I will present a numerical study of soap-film flow past an immersed cylinder, paying particular attention to the vortex dynamics, and on the development of von Karman Vortex Streets, reminiscent of High Reynolds Number flows in classical incompressible Navier Stokes. Modelling considerations include bulk viscosity, film elasticity and two-dimensional compressibility effects arising from thickness variations. The constitutive equations are solved using Overture, a C++ library which facilitates the solution of PDEs on complex geometries using overset grids.

Abstract: Expander graphs are highly connected yet sparse graphs. They are used extensively in computer science and have received attention in many disciplines. Just in the last few years alone, they have been utilized to construct efficient error-correcting codes, amplify the success probability of randomized algorithms, and prove hardness of approximation results.

In this talk, we will give a gentle introduction to this wonderful object. We will discuss the notion of expansion, explain some of its properties, highlight some recent applications, and if time permits, discuss a recent explicit construction.

Abstract: A numerical implementation to investigate localized and coherent structures spread over unbounded domains is presented. Spectrally accurate numerical differentiation, integration and interpolation techniques are combined with the following intuitive ideas to enhance performance of computation:

• Stretching and tilted domain transformations
• Exploiting symmetry of solutions
• Spatial dynamics
• Cauchy's integral formula.

It turns out that various nonlinear dispersive wave equations such as KdV, KP, Benjamin, NLS, DS and Euler equations, which occur in nonlinear optics and fluid mechanics, can be solved very accurately and efficiently by the virtue of the illustrated methods.

Abstract: A simple graph H is called a minor of another simple graph G if H can be obtained from G by edge contractions. In this talk we will discuss the famous Graph Minor Theorem of Neil Robertson and Paul Seymour, which states that every infinite set of graphs contains two graphs such that one is a minor of the other. This theorem is one of the most important results in graph theory, and its proof is very long and complicated. We will consider a special case of the theorem, in which the set of graphs is a family of finite trees. We will then give the idea of the proof of the general theorem and discuss its implications.

Abstract: Why do we disagree? We disagree all the time, especially about politics, morality, religion, and relative abilities. Moreover, we often consider our disagreements to be honest, i.e., we respect each other's relevant abilities, and consider each person's opinion to be his/her best estimate of the truth, given his/her information and effort.

Yet according to a well-known theory, such honest disagreement is impossible. Robert Aumann (Nobel prize winner in Economics, 2005) proved this wonderful result in his paper titled "Agreeing to Disagree". We will try to understand his result and related developments.

Abstract: Epidemics have plagued humanity for all of recorded history. Only recently (in the last century) have mathematicians been able to provide insight into the fundamental dynamics of disease spread. This talk will cover the basics of mathematical epidemiology. It will focus on differential models but also touch on time series and spatial models.

Abstract: Free-surface flows are abundant in nature and have been studied for centuries. Yet, many have resisted proper explanation, especially those with broken symmetry. Polygonal hydraulic jumps and cusped fluid sheets are two such examples. In this talk we present the results of a combined experimental, theoretical, and computational study in order to adequately explain these flow phenomena. More importantly, we hope to inspire the audience with the beauty and complexity of viscous free-surface flows.

Abstract: On a classical computer, factoring a long integer is considered a difficult problem. No polynomial time algorithm is known, and this is the heart of the RSA public-key cryptosystem. In this talk, I will describe how factoring can be done on a quantum computer. The algorithm was discovered by Peter Shor in 1994, and I will follow the treatment in the book "Quantum Computation and Quantum Information" by Nielsen and Chuang.

Abstract: This week, we shall pay tribute to pi, one of the most mysterious numbers in all mathematics. This number, describing the ratio of a circle's perimeter to its diameter, has fascinated civilization for millennia, and it occurs in almost every branch of mathematics. I shall present a history of the number, describing the different techniques used for its calculation from the Egyptians, through the middle ages, up to present day. I shall also discuss pi memorization and pi trivia, and we may even have pie.

Abstract: From high school physics, we know that if we drop a sphere, the sphere will accelerate, then it will experience some drag force which tends to slow it down, until at some point the drag force balances the gravitational force and then the sphere falls with a constant terminal velocity. This motion can be captured by an ordinary differential equation and the dynamics is well understood. This is the way people design parachutes too. But then you think, what if I jump off the plane at some point where the air is very thin, and the density of air keeps getting denser when I fall? Then you figure that you have the problem of a sphere (assuming for a second that your shape bears some resemblance to a sphere for simplicity) falling in a stratified fluid. In this case it will be a bit more complicated than a simple ODE and we will see how.

Abstract: The Fibonacci numbers and the golden ratio phi have made an indelible mark on the natural world. This talk will begin by exploring the essential mathematics and interconnections that characterize the Fibonaccis and phi. In explaining the prevalence of phi in biological systems, we will discuss optimization in organisms such as plant stalks and seed heads which have evolved to exploit phi's most unique underlying property: it is the "most irrational" number in the sense that it is the least well approximated by rationals. We will quantify this notion using elementary Diophantine number theory and continued fractions. Along the way, we will discuss such monumental results as Dirichlet's Approximation Theorem, the Algebraic Number Theorem, Hurwitz' Approximation Theorem, and the Lagrange Spectrum. We will conclude with a less rigorous section illustrating numerous examples of where phi/Fibonacci appear in the human body and the human sense of aesthetics: architecture, art, and music.