ORGANIC MATHEMATICS WORKSHOP SCHEDULE

Tuesday, December 12

Harbour Centre Room 1900

Session I: Overviews & Panel Discussions

Session II: Lectures

The Conference dinner will be at the Beijing Restaurant.
Beijing Restaurant
865 Hornby
Vancouver, B.C.
688-7788

There is no cost for invited participants and OMPers ; others are invited to attend at a cost of $25 each. Please sign up in the conference office HC 136. Jen Chang will be in that office from 12.00-2.00 and 4.30-6.00 Tuesday and Wednesday.

Wednesday, December 13

Harbour Centre Room 1900

Session III: Lectures

Session IV: Lectures

Note that THURSDAY, December 14 Sessions will be held at the Centre for Experimental and Constructive Mathematics (Shrum Science 8495, Phone 291-6517) on SFU's Burnaby Mountain Campus. Vans will take participants for the morning session, picking participants up at 9:00am at the Waterfront Hotel.

The afternoon session participants will be picked up at 12:00 in time to join the others for lunch at the Halpern Centre. After the lunch, the morning session people will be taken back downtown to the hotel.

Thursday, December 14

Session V: Hands-on Morning Session at the CECM

Session VI: Hands-on Afternoon Session at the CECM

ABSTRACTS

Speaker:

George Andrews

Title:

Pfaff's Method (III): Comparison With the WZ Method

Abstract:

In the 1990's, the WZ method has been the method of choice in resolving new conjectures for hypergeometric identities. The object here is to compare the WZ method with Pfaff's method. Such a comparison should (it is hoped) provide some suggestions for the further development of each method.

(*** Taken from the electronic paper ***)

Speaker:

David H. Bailey

Title of talk:

Recognizing Numerical Constants

Abstract:

The advent of inexpensive, high-performance computers and new efficient algorithms have made possible the automatic recognition of numerically computed constants. In other words, techniques now exist for determining, within certain limits, whether a computed real or complex number can be written as a simple expression involving the classical constants of mathematics.

In this presentation, some of the recently discovered techniques for constant recognition, notably integer relation detection algorithms, will be presented. As an application of these methods, the author's recent work in recognizing "Euler sums" will be described in some detail. The latter work was done jointly with Jonathan Borwein and Roland Girgensohn of SFU/CECM.

Speaker:

David Boyd

Title:

Are Salem numbers beta-numbers?

Abstract:

The beta transformation is the mapping on the unit interval defined by multiplication by mod 1. is a beta-number if 1 is an eventually periodic point of this mapping. It is known that all Pisot numbers of any degree and all Salem numbers of degree 4 are beta-numbers. We describe computational and heuristic arguments which suggest that all Salem numbers of degree 6 are beta-numbers but that this is not true for higher degrees.

Speakers:

Joe Buhler and Ron Graham

Title:

Juggling Permutations of the Integers

Abstract:

We will discuss permutations f of the integers Z satisfying , and will give applications to juggling patterns. Demonstrations will be included.

Speaker:

Arjeh Cohen

Title:

A 16-dimensional module in characteristic 3

Abstract:

Let k be an algebraically closed field of characteristic 3. In this paper we show that there are finitely many -orbits in the quotient of the module of all cubic forms by the submodule of cubes of linear forms.

Speaker:

Henri Cohen

Title:

Binary Cubic Forms and Cubic Number Fields

Abstract:

The aim of this paper is to present in a naive manner a small part of the theory of binary cubic forms and in particular its application to cubic number fields. Most of the results are due to Davenport-Heilbronn , but the algorithmic applications seem to be new.

Speaker:

Stan Devitt

Title:

Mathematical Documents -- Live

Abstract:

Interactive computational systems such as Maple can be used in many ways to bring mathematical documents to life. This presentation explores the use of this technology to enhance the access, customization and utilization of general mathematical information.

Speaker:

Frank Garvan

Title:

Approximations to via the Dedekind function.

Abstract:

Arguably the most efficient algorithm currently known for the extended precision calculation of is a quartic iteration due to J.M. and P.B. Borwein. In their paper, the Borwein's show how this iteration and others are intimately connected to the work of Ramanujan. This connection is shown utilizing their alpha-function which is defined in terms of theta-functions. They are able to find p-th order iterations based on this function using modular equations for the theta-functions. In this paper we construct an infinite family of functions . Each gives rise to a p-th order iteration. For p=4 we obtain an iteration related to the Borwein quartic iteration. For p=3 we obtain a cubic iteration due to the Borweins but does not come from the alpha-function. For p=7 we find a septic iteration that is analogous to the cubic iteration. Our method depends on using the computer and a symbolic algebra package to find and solve certain modular equations.

Speaker:

Andrew Granville

Title:

Properties of Binomial Coefficients

Abstract:

Many great mathematicians of the nineteenth century considered problems involving binomial coefficients modulo a prime power (for instance Babbage, Cauchy, Cayley, Gauss, Hensel, Hermite, Kummer, Legendre, Lucas and Stickelberger -- see Dickson). They discovered a variety of elegant and surprising Theorems which are often easy to prove. In this article we shall exhibit most of these results, and extend them in a variety of ways.

(*** Taken from the electronic paper ***)

Speaker:

Jeff Lagarias

Title:

A New View on the Hirsch Conjecture

Abstract:

The Hirsch Conjecture states that any d-dimensioanl bounded polytope with n facets has an edge-path between any two vertices of length at most n*d. The d-step conjecture is the special case n=2d, and is known to be equivalent to the general conjecture. It has long been suspected to be false in high dimensions. In joint work with N. Prabhu and J. Reeds, we discovered striking evidence that it is true in all dimensions, in a strong form. This evidence was base on a connection with Gaussian elimination of a set of (d!)^2 matrices constructed from the d-polytope with 2d facets, and massive computational experiments.

Speaker:

Clement Lam

Title:

The Search for a Finite Projective Plane of Order 10

Abstract:

When I was a graduate student looking for a thesis topic, Herbert Ryser advised me not to work on the projective plane of order 10. Even though he was extremely interested in this subject, he believed that it was too difficult and that I might get nowhere with it. I took his advice and chose another problem. Somehow, this problem has a beauty that fascinates me as well as many other mathematicians. Finally in 1980, I succumbed to the temptation and started working on it with some of my colleagues. We eventually managed to get somewhere, but unfortunately, Dr. Ryser is no longer with us to hear of the final result. This is an expository article describing the evolution of the problem and how computers were used to solve it.

Speaker:

Stan Wagon

Title:

New Visualization Ideas for Differential Equations

Abstract:

Modern software allows us to go farther in the visualization of solutions to differential equations. I will present a Mathematica package called VisualDSolve that does the standard things in terms of presenting solutions to DEs graphically, but also has several new features, such as:

-- the use of curvy fish shapes instead of lines to represent flow fields

-- the use of shading of the regions defined by nullclines to elucidate the phase plane for a system of two autonomous equations.

Speaker:

Beverly West

Title:

Differential Equations: Quantitative Information from Qualitative Analysis

Abstract:

Airy's equation x'' + xt = 0 has solutions that traditionally are found just as infinite series, giving no good information on the behavior of solutions as t approaches infinity, or minus infinity. Guided by PICTURES of the solutions, we can (rigorously) answer these questions, WITHOUT any analytic solving.

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