COLLOQUIUM
of the
Department of Mathematics
and Statistics
University of Saskatchewan
106 Wiggins Road
Saskatoon, SK, S7N 5E6
Canada
Phone: (306) 966-6081
Fax: (306) 966-6086
Professor M.R. Gonzalez Dorrego
Universidad Autonoma de Madrid
gave a talk on
Wednesday, September 9, 1998, 4:00 p.m.
Professor Mark Spivakovsky
University of Toronto
gave a talk on
Friday, October 2, 1998, 4:00 p.m.
Professor Alexander Prestel
Universitaet Konstanz, Germany
gave a talk on
5:00 p.m.
Professor Eberhard Becker
Universitaet Dortmund, Germany
gave a talk on
Monday, October 5, 1998, 4:00 p.m.
Professor Bernard Teissier
Ecole Normale Superieure, Paris
gave a talk on
Friday, October 16, 1998, 4:00 p.m.
Professor George Patrick
University of Saskatchewan
gave a talk on
Friday, November 13, 1998, 4:00 p.m.
Professor Juliana Erlijman
University of Regina
gave a talk on
Friday, November 27, 1998, 4:00 p.m.
Professor Salma Kuhlmann
University of Saskatchewan
gave a talk on
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
University of Saskatchewan
gave a talk on
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
University of Saskatchewan
gave the talk
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
University of Saskatchewan
gave a talk on
Friday, March 5, 1999, 3:30 p.m.
Professor Erhard Neher
University of Ottawa
gave a talk on
4:30 p.m.
Professor Jun Morita
Tsukuba University, Japan
gave a talk on
Friday, March 12, 1999, 4:00 p.m.
Professor J. Tavakoli
University of Saskatchewan
gave a talk on
Friday, March 19, 1999, 4:00 p.m.
Professor Soek-Jin Kang
Seoul National University
gave a talk on the
Thursday, May 27, 1999, 2:00 p.m.
Dr. Enrique G. Reyes
McGill University, Montreal
gave a talk on
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
University of Saskatchewan
gave a talk on
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
University of Saskatchewan
gave a talk on
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
University of Saskatchewan
gave a talk on
Monday, June 28, 1999, 4:00 p.m.
Dr. Saeid Azam
gave a talk:
Last update: January 26, 2008 --------- created and maintained by Franz-Viktor Kuhlmann