The SPAMS Schedule - Spring 1998

A little look at volume computation. Big things have greater volume than little things. Some things have no volume at all. Can we estimate efficiently these volumes for convex bodies? Do we want to?

Let $g(n)$ denote the least value such that any $g(n)$ points in the plane in general position contain the vertices of a convex $n$-gon. In 1935, Erd\H{o}s and Szekeres showed that $g(n)$ exists, and they obtained the bounds \[ 2^{n-2}+1 \leq g(n) \leq {{2n-4} \choose {n-2}} +1. \] Last year, Chung and Graham improved the upper bound by 1; the first improvement since the original Erd\H{o}s-Szekeres paper. We show that \[ g(n) \leq {{2n-4} \choose {n-2}}+7-2n. \] We will also survey other developments since our result and discuss some related Ramsey type problems in combinatorial geometry.

Abstract: We survey some important ideas in theoretical computer science with applications to cryptography. The two major ideas, identification and interaction, have led to surprising applications and interesting new classifications of natural mathematical problems. Among other things, we will show how to

1. Prove colors exist
2. Prevent nuclear war
3. Wipe out fraud over the internet

and if time permits

4. Cut a cake fairly among n people.

What more could you want?

I will describe how can some simple techniques or ideas can be used to solve some of the fundamental problems in atomic physics (actually more). I will describe how a new view on time, i.e. thinking of time as imaginary instead of real numbers can bring an especially interesting and useful tool for treating a physical system and I will describe a practical approach to get the energy levels of a system from simple first principles.

Simple but to-the-point examples will be given. No special knowledge of physics is assumed from the audience. I will start from a $ \leq 10$-minute quick treatment of essential quantum mechanics that will be used in the talk. (We are simple persons!)

The answer is well known: It is 42. The question, however, still eludes us. In the search for the ultimate question, physicists have become sidetracked by the study of the statistical mechanics of fields. It was also realised that there exists an intimate relation between statistical field theory and quantum field theory.

In this talk I will develop some of the ideas of statistical mechanics that help understand phenomena in particle physics, always using the ferromagnet as a workhorse to provide examples and intuition. In particular, I will talk about phase transitions, global and local symmetries and universality.

Prerequisites consist of some familiarity with the following facts:

1. Water exists in three phases (solid, liquid and gas).
2. Matter consist of particles.

Abstract: Little do people know that there are palm trees in Scotland. Why? Is this some clever trick by the Scottish people to attract tourists, or do these occur naturally? The answer is that thanks to the Gulf Stream that transports warm tropical water northeastward it is a natural phenomenon. In this talk I plan to discuss a simple mathematical model that explains why the Gulf Stream rotates clockwise rather then counterclockwise.

What's inside: Have you ever wondered how you could harness the random energy of ocean waves? Have you ever wondered why waves bend in parallel to the shoreline as they approach the beach? I will discuss how a linearized treatment of waves over a slowly varying bottom topography helps to answer both these questions, and many more. It will be seen that surface waves can undergo refraction in the same fashion as light passing through media of variable density. Or in other words, Snell's Law works for water waves too!

An algorithm, Vector ADI, is presented here for solving the Lyapunov equation, $AX+XA'=D$, $A \in \mathbb{R}^{n \times n}$, $X \in \mathbb{R}^{n \times n}$, in the case where $D = BB'$, where $B$ has rank much lower than $n$. It is formulated in terms of finding an orthnormal basis for a Krylov subspace based on nontrivial (ie, both the numerator and denominator are of degree $\geq 1 $) rational functions of $A$, and the vector input $B$. It requires only matrix-vector products and linear matrix solves by $A$ and hence enables one to take advantage of any sparsity in $A$. It is theoretically equivalent to the ADI method, and so optimal parameters and error bounds developed for ADI still hold.

Lyapunov equations with low rank right hand sides occur frequently in linear control systems, the solution of which is necessary to obtain an optimal Hankel-norm model order reduction. Several examples of model order reduction of large linear control systems using approximate solutions to Lyapunov equations obtained via the Vector ADI method are presented.

Abstract: We will describe an approach to analytically continuing functions defined by series outside their domains of convergence by analysing directly the divergent series which arise. The method involves introducing a suitable linear operator on a suitable space of functions and undersrtanding the spectrum of eigenvalues and eigenfunctions of the operator.

We'll apply this approach in two examples:

(i) Analysing the Riemann zeta function, which arises centrally in number theory and questions about the distribution of prime numbers (we'll say something briefly about this), and

(ii) Fourier series, where a notion of "Cesaro summation" of divergent series has already proven useful in theorems dating back many decades. We'll show that a slightly different analysis modeled on the approach above can also be used to obtain these results (and may explain some further issues).

The talk will be machinery-averse, and may even involve occasional short bouts of rampant and tenuous speculation.

If time permits I will make a few pithy comments on the performance of the Patriots offensive and defensive lines.

Abstract: (sung to "I can't smile without you") I will attempt to explain the basic idea behind Godonov's method for solving systems of first order hyperbolic equations. The method requires solving what is known as a Riemann problem (a particular initial value problem) many times. This will be explained and demonstrated for Burgers equation and for linear systems. Godonov's method is 1st order accurate in smooth regions and if time permits I will explain smooth regions of the flow. Hopefully, this talk will be accessible to people with a broad background.

A persistent conundrum in biology has been the so-called $3/4$ law. $B$, the average basal metabolic rate of an individual of a species (measured by, for example, the heat it puts out), has been observed to be proportional to $M^{3/4}$, where $M$ is the average mass of an individual. The curious thing here is that it might be reasonably expected that only integer multiples of $1/3$ would appear. Surface area presumably grows like $M^{2/3}$ so what is going on? It turns out that multiples of a $1/4$ appear for other quantities---life span, heart rate, heights of trees and so on.

I will discuss one possible resolution of this problem developed by West, Enquist and Brown at Los Alamos/Santa Fe. The basic idea is that the structure of the networks that supply nutrients to a body is the crucial factor. Links to river networks will be babbled about and if time allows I will mention a few absurd little problems of my own.

hy du tea-leefes eggregete-a et zee center ooff zee coop oonce-a ve-a stup sturreeng? Es un undergred in Svedee I ves ooffffered un ixpluneshun fur thees in terms ooff zee B\"odewadt buoondery leyer fur a ruteteeng flooeed oofer a steshunery flet plete-a. Bork Bork Bork! I veell present thees sulooshun tugezeer veet a durect noomereecel seemooleshun ooff speen-doon ooff a soospenseeun in a teecoop-leeke-a geumetry. Zee relefunce-a ooff but zeese-a epprueches tu zee reel prublem veell be-a deescoossed. Bork Bork Bork

(Why do tea-leaves aggregate at the center of the cup once we stop stirring? As an undergrad in Sweden I was offered an explanation for this in terms of the B\"odewadt boundary layer for a rotating fluid over a stationary flat plate. I will present this solution together with a direct numerical simulation of spin-down of a suspension in a teacup-like geometry. The relevance of both these approaches to the real problem will be discussed.)