COLLOQUIUM
of the
Department of Mathematics and Statistics
Phone: (306) 966-6081
Fax: (306) 966-6086

## Colloquium Talks 1998/99

#### Colloquium chair: Salma Kuhlmann

Friday, September 4, 1998, 4:00 p.m.

Professor M.R. Gonzalez Dorrego

gave a talk on

## Curves on a Kummer Surface

Abstract:
Let K be an algebraically closed field of characteristic different from 2. A Kummer surface in P^3_K is a quartic surface with 16 ordinary double points. We shall discuss the geometry of a Kummer surface in P^3_K and (16,6) configurations. We shall describe curves on a general Kummer surface.

Wednesday, September 9, 1998, 4:00 p.m.

Professor Mark Spivakovsky
University of Toronto

gave a talk on

## Resolution of Singularities

Abstract:
The problem of resolution of singularities and related questions account for a sizable portion of all the research in algebraic geometry in this century. The purpose of this talk is to introduce the audience to the subject of resolution of singularities. We will state the desingularization problem and outline the methods used in its solution. We will give a proof of resolution of singularities of curves, but with a view to the general case: as much as possible, we will point out how to extend all the concepts and results to higher dimensions.

Friday, October 2, 1998, 4:00 p.m.

Professor Alexander Prestel
Universitaet Konstanz, Germany

gave a talk on

## Positive definite polynomials and the Moment Problem

Abstract:
The Moment Problem is asking for conditions on a linear form on the algebra of real polynomials in n variables to be induced by some positive Borel measure with compact support. This problem is closely related to certain representations for real polynomials being positive on a compact semi-algebraic set. We shall be talking about this conection.

5:00 p.m.

Professor Eberhard Becker
Universitaet Dortmund, Germany

gave a talk on

## Symbolic solving of systems of polynomial equations with finitely many solutions

Abstract:
The talk will report on recent developments in symbolic computation which allows solution of systems of polynomial equations and inequalities in several variables provided the system has only finitely many solutions over the complex field C. The solutions in C^n are encoded as the roots of a univariante polynomial in T, and the coordinates of the solutions are given as rational functions in T. One of the main methods is the Buchberger Algorithm for computing Gröbner bases. The talk will also present some examples.

Monday, October 5, 1998, 4:00 p.m.

Professor Bernard Teissier
Ecole Normale Superieure, Paris

gave a talk on

## Valuations and Binomial Ideals

Abstract:
The topic of the lecture is resolution of singularities; I will show how to resolve singularities in the very special case of irreducible algebraic varieties defined by binomial equations, and give some ideas of a strategy to reduce to this case the proof of local uniformization, which is a local form of resolution of singularities.

Friday, October 16, 1998, 4:00 p.m.

Professor George Patrick

gave a talk on

## The Lie theoretic structure of transversal relative equilibria

Abstract:
A relative equilibrium is a solution of a (Hamiltonian) dynamical system with symmetry which is also the action of a one-parameter subgroup of the symmetry group. The circular orbit of a geostationary satellite and the rotation of a flywheel about its axis of symmetry are everyday examples of relative equilibria.
Evolutions of such dynamical systems are in one-one correspondence with points of a manifold called phase space, and so the relative equilibria may be considered as a subset of phase space. As such, under some mild assumptions, this set is a symplectic submanifold of phase space with dimension the sum of the dimension and the rank of the symmetry group. However, singularities in this set can and often do occur.
I will indicate why the singularities of this set are, generically, locally diffeomorphic to the commuting pairs of the Lie algebra of a certain subgroup of the symmetry group. It is the story of the search for a theorem that became the finding of a definition.

Friday, November 13, 1998, 4:00 p.m.

Professor Juliana Erlijman
University of Regina

gave a talk on

## Von Neumann algebras and braid representations

Abstract:
We give a brief introduction to Jones' theory of subfactors. We discuss various constructions of natural examples related to braid group representations.

Friday, November 27, 1998, 4:00 p.m.

Professor Salma Kuhlmann

gave a talk on

## The Exponential-Logarithmic Power Series Fields

Abstract:
In my last Colloquium talk, I showed that NO power series field can admit a logarithm. In this talk, I will show that EVERY power series field admits a PRELOGARITHM, that is, a non-surjective logarithm. I will give an explicit formula for this prelogarithm.

I then use this prelogarithm to give a canonical construction of non-archimedean real closed fields that admit logarithmic functions. These canonically constructed real closed fields are called the Exponential Logarithmic Power Series Fields. They have the following amazing properties:

1) They admit "bad" logarithms, that is, logarithms that DO NOT satisfy the elementary properties that the real logarithm satisfies.

2) On the other hand, they admit infinitely many pairwise distinct logarithmic functions, such that each of these logarithmic functions DOES satisfy all of the elementary properties of the real logarithm. These loagarithms are distinct in a very essential manner: indeed they pairwise distinct GROWTH RATES.

3) The Exponential Logarithmic Power Series Fields have the following Universal Property:
Any real closed field endowed with a logarithm that satisfies the elementary properties of the real logarithm embeds in an appropriate Exponential-Logarithmic Power Series Field (the embedding is an embedding of ordered fields which moreover commutes with the given logarithm).

Furthermore, the embedding may be chosen in such a way that it preserves the EXPONENTIAL RANK.

Friday, December 4, 1998, 4:00 p.m.

Professor Franz-Viktor Kuhlmann

gave a talk on

## Large Fields

Abstract:
A field K is called "large" if every smooth curve over K has infinitely many K-rational points, provided it has at least one. This notion was introduced by Florian Pop in an Annals paper in 1996. There he deals with problems of inverse Galois theory (a short description will be given in the talk). Among other results, Pop proves a theorem which "approximates" the Shafarevich Conjecture (which states that the absolute Galois group of the maximal cyclotomic extension of the field of rationals is profinite free).

There are several properties of fields which are equivalent to "large". Some of them are tightly connected to my own work of the early 1990's. I state these properties and give an idea of how the equivalence is proved. It turns out that there is a nice relation to properties that origin from model theoretic algebra. These express that K is "existentially closed" in suitable extensions L, that is, every elementary sentence asserting the existence of certain elements will hold in K, provided it holds in L. We show the connection of this notion with the existence of rational points and rational places.

There are many large fields. Basic examples are the algebraically closed, real closed and p-adically closed fields (and then PAC, PRC, PpC fields and fields with universal local-global principles). I explain what "existentially closed" means for the first three examples, by the Nullstellensatz framework.

Finally, I describe a new result about large fields which can be derived in two different ways, either from my results about local uniformization, or from my theory of the space of all (rational) places of an algebraic function field.

Friday, January 22, 1999, 4:00 p.m.

Professor Murray Bremner

gave the talk

## On Free Partially Associative Triple Systems

Abstract:
A triple system is partially associative (by definition) if it satisfies the identity

(abc)de + a(bcd)e + ab(cde) = 0.

This talk presents a computational study of the free partially associative triple system on one generator with coefficients in the ring $\Bbb Z$ of integers. In particular, the $\Bbb Z$-module structure of the homogeneous submodules of (odd) degrees $\le 11$ is determined, together with explicit generators for the free and torsion components in degrees $\le 9$. Elements of additive order 2 exist in degrees $\ge 7$, and elements of additive order 6 exist in degrees $\ge 9$. The most difficult case (degree 11) requires finding the row-reduced form over $\Bbb Z$ of a matrix of size $364 \times 273$. These computations were done with Maple V.4 on Symmetry and Hammer.

Friday, January 29, 1999, 4:00 p.m.

Professor Mik Bickis

gave a talk on

## Conditional Probability; Algebras and Trees

Abstract:
Some elementary examples in conditional probability lead to paradoxes if knowledge of an event is taken to imply conditioning on that event. Correct interpretation of these examples depends on incorporating the knowledge acquisition in the probability model. The confusion lies in failing to recognize the temporal nature of probability arguments. The classical theory of probability is based on Kolmogorov's axiomatization which, while putting probability on the solid mathematical foundation of measure theory, neglects the dynamic aspect of the subject. An alternative approach has recently been developed by Glen Shafer, based on the concept of an event tree. Some aspects of Shafer's theory will be presented in this expository talk.

Friday, March 5, 1999, 3:30 p.m.

Professor Erhard Neher
University of Ottawa

gave a talk on

## Lie and Jordan Algebras

Abstract:
The general theme of the talk is the interplay of Lie and Jordan algebras. In particular, I will explain in some detail and with examples, how to construct a Lie algebra from any Jordan algebra. In the second part of my talk I will indicate how this construction can be used to describe Lie algebras graded by a root system.

4:30 p.m.

Professor Jun Morita
Tsukuba University, Japan

gave a talk on

## Quasicrystals and Shelling

Abstract:
Starting from an introduction to quasicrystals with several examples, we will give an algebraic approach to the study of their structures. Especially, we will obtain a certain elementary formula for shelling.

Friday, March 12, 1999, 4:00 p.m.

Professor J. Tavakoli

gave a talk on

## Infinitesimal Objects in Synthetic Differential Geometry

Abstract:
In the category of smooth presheaves Sets Lop where L is the opposite of the category of all finitely generated C*-rings and C*-homomorphisms, the object D of first-order infinitesimals is tiny, in the sense that the exponential functor (-)D has a right adjoint. In this talk we establish an explicit description of the automorphism group object of D in terms of the object of units of the smooth real line R. As a consequence we prove that the automorphism group of D has two connected components.

Friday, March 19, 1999, 4:00 p.m.

Professor Soek-Jin Kang
Seoul National University

gave a talk on the

## Peterson-type trace formula for generalized Kac-Moody superalgebras and monstrous moonshine

Abstract:
In this talk, a recursive root multiplicity formula and a trace formula for generalized Kac-Moody superalgebras will be presented. These formulas are generalizations of Peterson's root multiplicity formula for Kac-Moody algebras. We will discuss some of the applications of these formulas to various class of Lie algebras and Lie superalgebras. The applications to the Monster Lie algebra and Thompson series will also be discussed.

Thursday, May 27, 1999, 2:00 p.m.

Dr. Enrique G. Reyes
McGill University, Montreal

gave a talk on

## Pseudo-spherical surfaces and integrability of evolution equations

Abstract:
This lecture will be about some geometric aspects of the theory of integrable partial differential equations. A crucial problem in the theory is precisely to decide what "integrability" should mean. Two notions which have been proposed for evolution equations are "formal" and "kinematic" integrability. I will introduce them (roughly speaking, formal integrability refers to the existence of a formal pseudo-differential operator determined by the equation at hand; kinematic integrability refers to the existence of linear problems for which the given equation is the integrability condition) and then I will show how one can compare them by using classical tools from the differential geometry of surfaces.

Actually, in broad outline, what one does is this. First, one defines the class of differential equations which "describe pseudo-spherical surfaces". Second, one classifies the evolution equations belonging to this class. Third, one uses this classification to prove comparison theorems. Two results have been obtained up to now:
(a) Every second order autonomous evolution equation which is formally integrable is kinematically integrable.
(b) This implication cannot be extended to third order equations.

Friday, May 28, 1999, 4:00 p.m.

Professor Doug MacLean

gave a talk on

## Optimization of Dynamic Programming Functions

Abstract:
A search algorithm for the maximum of a function expressed as the sum of a monotone increasing and a monotone decreasing function.

In the practical calculation of solutions to Dynamic Programming problems it necessary to estimate the maximum value of functions of a single control variable a great number of times. The functions are defined recursively, and it is seldom possible to assume that they have the nicer properties one is accustomed to in the older branches of Applied Mathematics. In particular, assumptions about the existence and continuity of derivatives are often not justified in sufficiently realistic models of the phenomena being optimized, so the standard methods of differential calculus fail us: finding the points of discontinuity of the derivative may be very difficult. There are very few results known regarding the computation of maxima of non-differentiable functions on an interval. Most of the functions which arise in Dynamic Programming can, in very natural way, be expressed as the sum of a monotone increasing and a monotone decreasing function. It is, of course, known that any function of bounded variation can be expressed in this way, but the usual method of constructing such monotone summands is not amenable to computation. Examples drawn from Financial Planning (including personal pension planning), Agricultural Economics (livestock sale), Forestry (optimal harvest rotation), and Sawmill Process Control will be given.

Friday, June 11, 1999, 4:00 p.m.

Professor Salma Kuhlmann

gave a talk on

## On Shelling E_8 Quasicrystals (after Robert Moody and Al Weiss)

Abstract:
The classical crystallographic condition now has to be taken with a grain of borax'' was Donald Coxeter's comment after the discovery of quasicrystals: (quoting from Marjorie Senechal's book) "Fifteen years ago, the world of solid state science was startled by the announcement of the discovery of a metallic phase with long-range orientational order and no translational symmetry'' (Shechtman, Blech, Gratias und Cahn 1984).

The material in question was an alloy of aluminium-manganese, produced by Shechtman from a melt by a rapid cooling technique. Its diffraction images showed icosahedral symmetry, long believed to be impossible for matter in the crystalline state." Since this discovery a whole team of scientists study quasicrystals, trying to understand their mathematics, and to develop methods to construct models. In [1], using the cut and project method, the authors obtain a quasicrystal from the E_8-lattice. This quasicrystal has the symmetries of the non-crystallographic Coxeter group H_4. In [4] a method for shelling the E_8-quasicrystal is described and a conjecture concerning the number of points in every shell'' is formulated. This conjecture is proved in a corrected form in [3].

A crucial point in the proof is the algebraic description of the quasicrystal through the "icosian ring'', a maximal order in the quaternion algebra over \Q[\sqrt{5}]. With this description, the conjecture above turns into an arithmetic Problem. In the talk, we will lecture about the proof of the conjecture.

References:
[1] V. Elser and N. J. A. Sloane: A highly symmetric four dimensional quasicrystal, J. Phys. A: Math. Gen. 20 (1987), 6161-6167
[2] R. V. Moody und J. Patera: Quasicrystals and icosians, J. Phys. A: Math. Gen. 26 (1993), 2829-2853
[3] R. V. Moody und A. Weiss: On Shelling E_8 Quasicrystals, J. Number Theory 47 (1994), 405-412
[4] J.-F. Sadoc und R. Mosseri: The E_8 lattice and quasicrystals: geometry, number theory and quasicrystals, J. Phys. A: Math. Gen. 26 (1993), 1789-1809

Friday, June 18, 1999, 4:00 p.m.

Professor John Martin

gave a talk on

## Salma's Lecture and Poincare's Homology Sphere

Abstract: Surprise.

Monday, June 28, 1999, 4:00 p.m.

Dr. Saeid Azam

gave a talk:

## On the relation of extended affine Weyl groups and definite Weyl groups

Abstract: We show that any (reduced) extended affine Weyl group is homomorphic image of some indefinite Weyl group where the homomorphism and its kernel are given explicitly.

### Centre for Algebra, Logic and Computation

Last update: January 26, 2008 --------- created and maintained by Franz-Viktor Kuhlmann