**LATTICE PROJECT**

[under construction!]

Martin Guest, Richard Palais, and Takashi Sakai

**INTRODUCTION**

This project has both research and educational aspects. The aim is to study the motion of "lattices" of particles connected by springs, both as a dynamical system in its own right and as a way of understanding fundamental concepts in geometry and the theory of integrable systems. If the tension forces in the springs are linear, the equations of motion are a linear system of o.d.e. with constant coefficients, but if the forces are nonlinear the problem quickly becomes nontrivial, and computer simulations are helpful.

The problem was first addressed in the famous experiments of Fermi-Pasta-Ulam in 1954/5, which was one of the earliest examples of experimental mathematics or mathematical visualization using computers (and which was carried out using one of the earliest computers). The surprising results of those experiments led to far-reaching mathematical developments - to soliton theory and beyond.

Fifty years later, the role of the computer as an experimental tool in mathematics is just starting to find acceptance amongst pure mathematicians. The wide availability of fast computers and (reasonably) affordable software is a fairly recent phenomenon in this community, which partly accounts for the delay. The lack of a good "interface" between researchers/teachers and software is another reason: software is often inconvenient to use, and there is a lack of good documentation which takes the mathematics as seriously as the computer. This project is a contribution to filling the gap.

**BACKGROUND**

The starting point of the project was Richard Palais's "Lattice Category" in the mathematical software 3D-XplorMath.

In 2000 (when 3D-XplorMath was called 3D-Filmstrip), Martin Guest gave a series of lectures for a general audience for Tomin College, at the Tokyo Forum exhibition centre. (Tomin College, the "citizens' college" associated to Tokyo Metropolitan University, was abolished a couple of years later, as part of the university reform plan.) As the audience consisted of people with a variety of backgrounds, images and simulations from 3D-XplorMath were used as a substitute for detailed mathematical calculations.

This format was used again with mathematics student audiences, at Ryukyu University (Okinawa) in 2001 and at the National Center for Theoretical Sciences (Taiwan) in 2003. Even though the mathematical preparedness of these audiences was higher, the demonstrations from 3D-XplorMath were extremely useful as a substitute for the complicated calculations which arise when dealing with lattices of more than a handful of particles.

Informal lecture notes from the NCTS lectures are available here. References to 3DXM are given in the form [3DXM :ODE :ODE (2D)-2nd Order: USER]. This means "In 3D-XplorMath, choose the ODE Category, then the ODE (2D)-2nd Order Object, then look at the USER (user-defined) menu item".

**EDUCATION**

In 2003 Martin Guest and Takashi Sakai gave a course for second year undergraduate students at Tokyo Metropolitan University. The title of the course was "Introduction to Maple". As an alternative to a dull litany of techniques, they based the course on the theory of differential equations and lattices (this required virtually all the basic aspects of Maple that the course was supposed to cover).

As a result they decided to write a textbook for undergraduate students in the mathematical sciences, expanding the course into an introduction to a wide area of mathematics, from elementary differential equations to basic ideas of symplectic geometry. The provisional title of the (English version of the) book is

"Exploring Lattices"

Some provisional chapters (lecture notes handed out to the course participants) will be available here shortly.

**RESEARCH**

Richard Palais continued to develop the Lattice Category of 3D-XplorMath, to the extent that it has become capable of rather sophisticated experiments (which would certainly be possible with other software, but would cost an experienced programmer some time and effort).

In the near future a Java version of 3D-XplorMath will become available, but for the time being Macintosh users are invited to download the program and look at the Lattice Category (choose the ODE Category, then Lattices). Martin Guest wrote corresponding documentation which is contained in the download package, but which can be obtained directly here:

ATC (About This Category): Lattice Models

ATO (About This Object): The Fermi-Pasta-Ulam Lattice and The Toda Lattice

A report on some (old and new) experiments performed with 3D-XplorMath will be made available in due course.

**LINKS**

There are many books and papers on lattice theory, and many interesting web sites. The following list [under construction] will contain links to sites that make use of computer experiments.