ALGEBRA AND LOGIC GROUP
of the
Mathematical Sciences Group
University of Saskatchewan
106 Wiggins Road
Saskatoon, SK, S7N 5E6, Canada
Phone: (306) 966-6081 - Fax: (306) 966-6086
Past Talks of Our Guests
Friday, November 19, 1997, 4:00 p.m.
Professor Sibylla Priess-Crampe
Ludwig Maximillian Universität München, Germany
gave a talk in the Department Colloquium on
Fixed Point and Coincidence Theorems for Ultrametric
Spaces
Abstract:
An ultrametric space (X,d,G) is a set X with an ultrametric distance
functions d from X to G , where G is a partially ordered set with a
smallest element 0. d has the same properties as a metric but instead of
the triangle inequality the following one:
For all g of G , if d(x,y) and d(y,z) are at most g then also d(x,z) is
at most g .
A special role for ultrametric spaces play spherically complete
ultrametric spaces. "Sperically complete" corresponds to the property
"maximal valued" for valued fields.
For spherically complete ultrametric spaces there holds a fixed point
theorem which looks like Banach's fixed point theorem for metric spaces.
One has furthermore a generalization of this singlevalued fixed point
theorem to multivalued mappings (again as it is the case in the metric
situation). Some hints to applications of the theorems will be given.
Friday, February 13, 1998, 4:00 p.m.
Professor Hans Brungs
University of Alberta
gave a talk in the Department Colloquium on
Extending Valuation Rings
Abstract:
For any field extension K/F every valuation ring of F can be extended to
K. There are three competing definitions for noncommutative valuation
rings and their extension properties will be discussed; in particular
extensions of valuation rings in the center of a finite dimensional
division algebra. These results can be used to show that every rooted
tree can be realized as the graph of all valuation rings of a finite
dimensional division algebra.
Friday, March 6, 1998, 4:00 p.m.
Professor Niels Schwartz
University of Passau, Germany
gave a talk in the Department Colloquium on
From rings of continuous functions to real closed rings
Abstract:
Rings of continuous functions into the real numbers are obviously an
important tool in topology. Therefore they have been studied intensely
from many points of view. Their algebraic analysis is a difficult
problem, partly because, as a category, rings of continuous functions
have very poor properties. There are only very few ring theoretic
constructions that produce rings of continuous functions when applied to
such rings. A related fact is that there is no axiomatization in first
order model theory. One major topic in real geometry is the
investigation of semi-algebraic spaces. A semi-algebraic space is always
defined with reference to a real closed field. Similar to rings of
continuous functions on topological spaces, the functions that are most
useful for the analysis of semi-algebraic spaces are the rings of
continuous semi-algebraic functions into the real closed fields
associated with the spaces. These rings have the same poor category
theoretic and model theoretic properties as rings of continuous
functions. But there is a larger class of rings, called real closed
rings, which is the smallest axiomatizable class of rings containing the
rings of continuous semi-algebraic functions. Thus, real closed rings
can be studied by model theoretic methods. The category theoretic
properties are also very favorable; the class is closed with respect to
o large number of the usual ring theoretic constructions. All rings of
continuous functions are real closed. In fact, currently the real closed
rings are the smallest known class of rings that contains the rings of
continuous functions, is axiomatizable and is closed under so many ring
theoretic constructions. Thus, although they were first introduced as a
tool for real geometry, the real closed rings also have a potential for
applications in topology.
Friday, September 4, 1998, 4:00 p.m.
Professor M.R. Gonzalez Dorrego
Universidad Autonoma de Madrid
gave a talk in the Department Colloquium 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 in the Department Colloquium 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
Universität Konstanz, Germany
gave a talk in the Department Colloquium 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
Universität Dortmund, Germany
gave a talk in the Department Colloquium 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 in the Department Colloquium 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, July 16, 1999, 4:00 p.m.
Professor Sudesh Kaur Khanduja
Punjab University, Chandigarh, India
gave a talk in the Department Colloquium on the
Generalized Hensel's Lemma
Abstract:
We intend to talk about a generalization of the classical Hensel's Lemma
and some of its applications using prolongations of a valuation v,
defined on a field K to a simple transcendental extension K(x) of K.
Friday, July 23, 1999, 4:00 p.m.
Professor Peter Roquette
Universität Heidelberg
gave a talk in the Department Colloquium on
The Grunwald and Wang Story
Abstract:
(1) Explain the statement of the famous
Grunwald existence theorem.
(2) Report on its origin and its significance within Number Theory.
(3) Tell about the shock to the mathematical community when Wang found a
counter example, and about the attempts to mend matters.
(4) Reinterpretation of the (corrected) theorem within valuation theory.
Friday, July 30, 1999, 7:30 p.m.
Professor Paulo Ribenboim
Queens University, Kingston
gave a public lecture on
These marvelous prime numbers
This lecture was supported by the
Special
Lectureship Fund of the University of Saskatchewan.
Friday, August 6, 1999, 4:00 p.m.
Professor Hans Schoutens
Wesleyan University, Middletown
gave a talk in the Department Colloquium on
The image of an analytic map; the ultrametric case
Abstract:
Consider an analytic map f:Y->X (either complex, real or
rigid analytic). We want to describe the image I=Im f of f
viewed as a subset of X. Unfortunately, I is not necessarily a
semi-analytic set. With a semi-analytic set we mean a subset given by (a
Boolean combination of) norm-inequalities |p(x)|<|q(x)| between
analytic functions p and q. We [S. + Gardener] show that in the rigid
case, I can be described though by norm-inequalities if we allow for
the functions p,q,... in this description to be (iterated) quotients
of analytic functions. Moreover, after a well-choosen blowing up
process \pi:\tilde X -> X, we can arrange for \pi^{-1}(I) to become
semi-analytic. In fact, we show how to flatten the original map f by
blow ups, after which we apply the Theorem of Raynaud which asserts that
the image of a flat (affinoid) map is always semi-analytic. To give an
idea of the proof, which is a rigid analytic adaptation of Hironaka's
proof for the real case, I will work out the Osgood example, where
f(s,t)=(s, st, s exp(t)).
Friday, September 10, 1999, 4:00 p.m.
Professor Michael Baake
University of Tübingen
gave a talk in the Department Colloquium on
Which distributions of matter diffract
Abstract:
Mathematical diffraction theory is concerned with the spectral analysis
of unbounded complex measures, and has important applications in
crystallography and diffraction physics, and to the understanding of
quasicrystals in particular.
After introducing the basic questions and methods, I will briefly survey
the present state of affairs by means of characteristic examples from
lattices, quasicrystals, and stochastic structures such as lattice
gases. My emphasis will be on cases that can be treated explicitly, but
without loss of rigour, and I'll also point out some open questions.
Friday, February 18, 2000, 4:00 p.m.
Professor Alfred Weiss
University of Alberta, Edmonton
gave a talk in the Department Colloquium on
L-values and multiplicative Galois module structure
Abstract:
Zeta functions 'know' much about the arithmetic of their algebraic
number field. Examples of this phenomenon are the analytic class
number formula and the Main Conjecture of Iwasawa theory. This appears
to persist also 'relatively', i.e., for the Galois action on an
extension of number fields. An example here is the Frohlich-Taylor
theory of Galois structure of the ring of integers when the
extension is tame. The multiplicative analogue of this, the Galois
structure of units, seems to continue the pattern in a suitable sense.
Proofs of this seem to require strengthening aspects of all of the
examples above. Perhaps this is not surprising as the Galois structure
of units is at the core of class field theory.
Friday, March 24, 2000, 4:00 p.m.
Professor Albrecht Pfister
Universität Mainz, Germany
gave a talk in the First Colloquiumfest on
On the Milnor Conjectures: History, Influence, Applications
(in particular, among the applications, Marshall's signature conjecture
was emphasized)
Abstract:
In the first part of my talk I introduce some preliminary statements
about quadratic forms, Galois cohomology and algebraic K-theory which
are necessary to formulate the Milnor Conjectures. Then there will be
some metamathematical remarks about the impact of these conjectures.
The second part will outline the various attempts (from 1970 till
now) to prove the conjectures, it also contains several applications.
5:00 p.m.
Professor Konrad Schmüdgen
Universität Leipzig, Germany
gave a talk in the First Colloquiumfest on
The Classical Multidimensional Moment Problem
(and its relations and analogies to semialgebraic geometry, in particular
to the Positivstellensatz, and Marshall's recent generalizations)
Abstract:
Let K be a closed subset of Rd. The K-moment problem
asks under what conditions for a given multisequence s=(sn ;
n \in N0d) there exists a positive Borel
measure \mu on Rd such that the support of s is
contained in K and s is the moment sequence of the measure \mu, that is,
s_n = \int tn d\mu(t) for all n \in
N0d.
After a brief excurse to the historical roots two approaches to this
problem are explained. Particular emphasis is placed on the case when K
is a semialgebraic set. Then there is a close interrelation between the
K-moment problem and the archimedean Positivstellensatz for K. For a
compact semialgebraic set K, a solution of the K-moment problem can be
given by using the Positivstellensatz of G. Stengle and conversely the
archimedean Positivstellensatz can be proved by means of the K-moment
problem. Two recent variants of the archimedean Positivstellensatz (due
to M. Marshall and due to T. Jacobi and A. Prestel) are discussed. Some
results for non-compact sets K and some open problems are mentioned.
Saturday, March 25, 2000, 10:15
Professor Ludwig Bröcker
Universität Münster, Germany
gave a talk in the First Colloquiumfest on
From Murrays miraculous lemma to real algebraic geometry
Abstract:
The talk describes the development from the study of
quadratic forms over formally real fields in the seventies to some
modern aspects of real algebraic geometry. In particiular it includes
some remarks on Marshalls work and beyond.
11:15
Professor Victoria Powers
Emory University, USA
gave a talk in the First Colloquiumfest on
A new bound for Polya's Theorem with applications to polynomials
positive on polyhedra
Abstract:
This is joint work with Bruce Reznick.
Let R[X] := R[x1,...,xn]. Polya's
Theorem says that if f \in R[X] is homogeneous and positive on
the simplex
{(x1,..., xn) | xi
\geq 0, \sumi xi = 1},
then for sufficiently large N \in N all the
coefficients of
(x1 +...+ xn)N
f(x1,...,xn)
are positive. We give an explicit bound for N, improving a previous
bound by de Loera and Santos, and give an application to some special
representations of polynomials positive on polyhedra.
2:00 p.m.
Professor Max Dickmann
Universite Paris VII, France
gave a talk in the First Colloquiumfest on
Proof of Murray's signature conjecture and generalizations
Abstract:
In this talk I will outline and compare the proofs, by F. Miraglia
(Sao Paulo, Brazil) and myself, of:
(1) Marshall's signature conjecture for quadratic forms over Pythagorean
fields (Inventiones Math., 1998).
(2) Lam's generalization of (1) to arbitrary formally real fields
(proved in February 1999, unpublished).
I will point out, as well, a consequence of (1) concerning the
representation of forms of a given degree by linear combinations of
Pfister forms of a given degree.
3:00 p.m.
Dr. Jonathon Funk
Saskatoon
gave a talk in the First Colloquiumfest on
Branched covers and orderings of braid groups
Abstract:
The concept of a branched cover can be used to obtain orderings of a
braid group. The orderings obtained in this way are precisely the ones
of ``finite type'', as described by B. Wiest and H. Short, ``Orderings
of mapping class groups after Thurston''.
4:00 p.m.
Professor Alexander Lichtman
University of Wisconsin-Parkside
gave a talk in the First Colloquiumfest on
Valuation methods in group rings and skew fields
Abstract:
We construct a family of discrete valuations in group rings of
residually torsion free nilpotent groups and extend these valuations to
the Malcev-Neumann power series skew fields of these group rings. We
apply our results and methods for study of the universal fields of
fractions of free algebras and the universal fields of fractions of the
Magnus power series ring; we give a description of the centralizer of a
non-central element in this skew field. We obtain new methods for
constructing the universal fields of fractions for free algebras.
Friday, September 15, 2000, Room 206 ARTS
4:00 p.m.
Herve Perdry
Universite de Franche-Comte, Besancon, France
gave a talk in the Department Colloquium on
A constructive approach to Henselian fields
Abstract:
We briefly describe what constructive mathematics is. We outline the
construction of the real closure of an ordered field. Then we give a
classical background about valued fields and in particular, Henselian
fields. After that, we give a construction for the Henselisation of a
valued field, as well as constructive proofs of classical theorems about
valued fields.
Friday, September 22, 2000, 4:00 p.m.
Professor Niels Schwartz
University of Passau, Germany
gave a talk in the Department Colloquium on
The role of partially ordered rings in real algebraic
geometry
Abstract:
Real algebraic geometry is an old topic, but has grown only recently
into being a major special branch of algebraic geometry. Therefore many
of its basic tools and techniques still need to be identified and
developed. One algebraic structure that is going to be fundamental in
real geometry are partially ordered rings. It will be shown in the
lecture why this is the case. The whole range of future applications is
not foreseeable at present, but some substantial uses that have already
been recognized will be pointed out.
Thursday, September 28, 2000, 2:30 pm
Professor Alexander Prestel
University of Konstanz, Germany
gave a talk in the Algebra Seminar on
Positive Polynomials over non-archimedean Fields
In September and October of 2000
Herve Perdry
Besancon, France
gave talks in the Algebra Seminar on
Explicit Construction of the Henselization of a Valued
Field
November 22, 2000
Professor Danielle Gondard
Universite Paris VI, France
gave a talk in the Algebra Seminar on
From Hilbert's 17th problem to valuation fans
In January of 2001
Dr. Jonathan Funk
Saskatoon
gave talks in the Algebra Seminar on
Inverse Semigroups and Order Etendue
Monday, March 5, 2001, 10:30 a.m.
Matthias Aschenbrenner
Urbana, Illinois, USA
gave a talk in the Algebra Seminar on
Asymptotic Couples, H-Fields and their Liouville
Extensions, I
Thursday, March 8, 2001, 2:30 p.m.
Matthias Aschenbrenner
Urbana, Illinois, USA
gave a talk in the Algebra Seminar on
Asymptotic Couples, H-Fields and their Liouville
Extensions, II
Friday, March 9, 2001, 10:30 a.m.
Matthias Aschenbrenner
Urbana, Illinois, USA
gave a talk in the Algebra Seminar on
Asymptotic Couples, H-Fields and their Liouville
Extensions, III
Wednesday March 14, 2001, 10:30 a.m.
Markus Schweighofer
Konstanz, Germany
gave a talk in the Algebra Seminar on
Extending Schmüdgen's Theorem to non-compact varieties, I
Friday March 16, 2001, 10:30 a.m.
Markus Schweighofer
Konstanz, Germany
gave a talk in the Algebra Seminar on
Extending Schmüdgen's Theorem to non-compact varieties,
II
Friday, March 16, 2001, 4:00 p.m.
Professor
Eberhard
Becker
University of Dortmund, Germany
gave a talk in the Department Colloquium on
The cone of positive polynomials - from an optimization point
of view
Abstract:
Researchers in optimization are interested in the cone of positive
polynomials (up to a certain degree). In particular, the search for a
"concrete" barrier function for the entire cone or for subcones is one
of the main tasks. On the other hand, algebraists develop an interest in
the concepts of modern optimization to study the above cone from the
point of view of real algebra. The talk outlines some of the few results
together with an application to finding minima of polynomials on compact
basic closed semi-algebraic sets.
Monday, March 19, 2001, 10:30 a.m.
Professor
Jaka Cimpric
Ljubljana, Slowenia
gave a talk in the Algebra Seminar on
Artin-Schreier theory for semigroups
Tuesday, March 20, 2001, 1:00 p.m.
Professor
Claus
Scheiderer
University of Duisburg, Germany
gave a talk in the Algebra Seminar on
The moment problem for non-compact semialgebraic sets, I
Wednesday, March 21, 2001, 10:30 a.m.
Professor
Claus
Scheiderer
University of Duisburg, Germany
gave a talk in the Algebra Seminar on
The moment problem for non-compact semialgebraic sets, II
Thursday, March 22, 2001, 2:30 p.m.
Professor
Victoria
Powers
Emory University, Atlanta, USA
gave a talk in the Algebra Seminar on
Constructive approaches to Hilbert's Theorem on ternary
quartics
Friday, March 23, 2001, 10:30 a.m.
Professor Alexander Prestel
Konstanz, Germany
gave a talk in the Algebra Seminar on
Representation theorems for `archimedian' rings
11:30 a.m.
Professor
Deirdre
Haskell
McMaster University
gave a talk in the Algebra Seminar on
Elimination of imaginaries in Algebraically closed
valued fields
Friday, March 23, 2001, 4:00 p.m.
Professor
Victoria
Powers
Emory University, Atlanta, USA
gave a talk in the Second Colloquiumfest on
Real algebraic geometry and convex optimization
Abstract:
Semidefinite programming is an important tool for solving many problems
in applied math and engineering, for example in systems and control
theory. In this talk we will give an overview of the interaction of
concepts in real algebraic geometry and semidefinite programming. In
particular, we will talk about applications to convex optimization
problems. Much of the talk will be based on recent work of Pablo
Parrilo, who has developed practical methods for studying semidefinite
programming using ideas from real algebraic geometry. No prior knowledge
of semidefinite programming or convex optimization will be assumed.
5:00 p.m.
Professor
Claus
Scheiderer
University of Duisburg, Germany
gave a talk in the Second Colloquiumfest on
Sums of squares and the moment problem
Abstract:
The question whether a non-negative polynomial is always a sum of
squares of polynomials was raised in the 1880s by Minkowski and
answered by Hilbert. I'll first discuss the generalization of this
question to polynomial functions on affine real algebraic sets.
The hardest case is that of compact curves and surfaces. These
questions are directly related to the (multi-dimensional) moment
problem from analysis. The latter asks for a characterization of
the possible moment (multi-) sequences of positive Borel measures
with support in a given closed subset K of Rn.
The case when K is compact is solved completely by a theorem of
Schmüdgen. I will present these facts, and in the end try to
discuss a few recent results for non-compact K.
Saturday, March 24, 2001, 10:15 a.m.
Professor Max Dickmann
Universite Paris 7, France
gave a talk in the Second Colloquiumfest on
Bounds for the representation of quadratic forms
Abstract:
The (affirmative) solution to Marshall's signature conjecture for
Pythagorean fields implies that, for fixed integers n,m >= 1,
there is a uniform bound on the number of Pfister forms of degree n
over any Pythagorean field F necessary to represent (in the Witt ring
of F) any form of dimension m as a linear combination of such forms
with non-zero coefficients in F. "Uniform" means that the bound does
not depend either on the form nor on the field F; it is given by a
recursive function f of n and m. We single out a large class of
Pythagorean fields and, more generally, of reduced special groups for
which f has a simply exponential bound of the form cmn-1
(c a constant). Such a class is closed under certain - possibly
infinitary - operations which preserve Marshall's signature conjecture.
In the case of groups of finite stability index s, we obtain an upper
bound for f which is quadratic on [m/2n], where the
coefficient c depends on s.
11:15 a.m.
Markus Schweighofer
Konstanz, Germany
gave a talk in the Second Colloquiumfest on
Extension of Schmüdgen's Positivstellensatz to algebras of finite
transcendence degree
Abstract:
We investigate the iterated real holomorphy ring of rings as introduced
by Becker and Powers. First we give a new and simple proof for their
stationarity result. Then we prove the conjecture of Monnier saying that
Schmüdgen's Positivstellensatz holds true not only for affine algebras
but also for algebras of finite transcendence degree. From this it
follows that the stationary object of Becker and Powers is exactly the
archimedean hull of the subsemiring of sums of squares. As a corollary
we obtain a new proof of Marshall's generalization of Schmüdgen's
result to the non-compact case.
2:30 p.m. --- This talk was supported by the University
of Saskatchewan Role Model Speaker Fund
Professor Isabelle Bonnard
Angers, France
gave a talk in the Second Colloquiumfest on
Nash constructible functions
Abstract:
A Nash constructible function on a real algebraic set is defined as a
linear combination (with integer coefficients) of Euler caracteristic of
fibres of regular proper morphisms intersected with connected components
of algebraic sets. The aim of the talk is to prove that Nash
constructible functions on a compact set coincide with sums of signs of
semialgebraic arc-analytic functions.
3:30 p.m.
Raf Cluckers
Kathlieke Universiteit Leuven, Belgium
gave a talk in the Second Colloquiumfest on
Semi-algebraic p-adic geometry
Abstract:
Semi-algebraic p-adic geometry is the p-adic counterpart of real
semi-algebraic geometry. In both cases semi-algebraic sets have a
well-defined dimension which is invariant under semi-algebraic
isomorphisms and which corresponds to the algebro-geometric dimension of
the Zariski-closure. In the real case there is also an Euler
characteristic to the integers; this Euler characteristic together with
the dimension leads to a classification of the real semi-algebraic sets
up to semi-algebraic isomorphism. In the p-adic case, D. Haskell and R.
Cluckers proved that every (abtstract) Euler characteristic on the
p-adic semi-algebraic sets is trivial. Nevertheless, it was possible to
give a classification of p-adic semi-algebraic sets up to semi-algebraic
isomorphism.
4:30 p.m.
Matthias Aschenbrenner
Urbana, Illinois, USA
gave a talk in the Second Colloquiumfest on
Ideal membership in polynomial rings over the integers:
Kronecker's Problem
Abstract:
Given polynomials
f0(X), f1(X),..., fn(X) in Z[X],
X = (X1,..., XN), are there g1(X),...,
gn(X) in Z[X] such that f0 =
g1f1 +...+ gnfn? This is the
ideal membership problem for polynomial rings over the integers. It
constitutes a key problem in Kronecker's "finite type" mathematics. A
decision procedure has been known for about 40 years. More recently, the
method of Gröbner bases has led to a procedure whose number of
steps
could be explicitly bounded in terms of the size of the coefficients,
degrees of the fj's, and the number N of variables. While
for fixed N this upper bound is primitive recursive, as a function of
N it involves the notorious Ackermann function (and thus is not
primitive recursive).
In this talk, we will present a novel approach to this problem. We will
discuss the following three aspects:
(1) existence of bounds for the degrees and coefficients of
g1,..., gn (in terms of the degrees and
coefficients of f0,..., fn);
(2) decidability of ideal membership
by a primitive recursive algorithm;
(3) definability:
dependence on parameters, from an arithmetic-logical viewpoint. In
particular, our method yields bounds which drastically improve the
previously known ones.
Saturday, June 2, 2001
Professor Michael Makkai
McGill University, Montreal
gave a talk in the Special Session on Model Theoretic Algebra at
the CMS Summer 2001 Meeting:
Logic and D. G. Quillen's homotopical algebra
Abstract:
In [Logic Colloquium '95, Lecture Notes of Logic 11, Springer,
1998; 153-190] the author introduced First Order Logic with Dependent
Sorts (FOLDS) for certain foundational purposes. The main semantical
feature of FOLDS is the presence, associated with any given FOLDS
signature L, of a concept called L-equivalence,
taking the place of the usual notion of isomorphism of L-structures. In
this talk, modeltheoretical results for FOLDS, of both of a general and
of an applied nature, will be presented. In particular, connections with
Quillen's classical approach to homotopy theory via the so-called model
categories will be developed. An example of the connections is the fact
that L-equivalence, with a suitable L, for simplicial
Kan-complexes is the same as homotopy equivalence.
Professor Ross Willard
Department of Pure Mathematics, Waterloo
gave a talk in the Special Session on Model Theoretic Algebra at
the CMS Summer 2001 Meeting:
Palyutin's h-formulas and a problem from
universal algebra
Abstract:
In this talk we describe a nice syntactic characterization of a
structural property of general algebras (the strict refinement
property) in terms of certain formulas (h-formulas)
defined by E. A. Palyutin in Categorical Horn classes,
I. Algebra and Logic 19(1980), 377-400.
Professor Toniann Pitassi
University of Toronto
gave a talk in the Special Session on Model Theoretic Algebra at
the CMS Summer 2001 Meeting:
Topics in propositional proof complexity
Professor Bradd Hart
McMaster University and Fields Institute
gave a talk in the Special Session on Model Theoretic Algebra at
the CMS Summer 2001 Meeting:
The stable forking conjecture
Ziv Shami
McMaster University and Fields Institute
gave a talk in the Special Session on Model Theoretic Algebra at
the CMS Summer 2001 Meeting:
Towards a binding group theorem for simple theories
Abstract:
Let T be simple. Let p Î S(Æ) be internal in Q Î L and
suppose the global algebraic closure is weakly transitive on
QC, that is
DCL(aclR(Q)) Í ACL(QCÈA) for every one-to-finite definable relation
R defined over A (where ACL(S) (DCL(S)) for
a set S denotes those elements (in Ceq) which
have finite orbit (an orbit of size 1 resp.) with respect to the action
of Aut(Ceq/QC).
aclR(Q)={b |
| R(b,[`(c)]) for some tuple [`(c)] from QC}). Then
Aut(p/Q)={s| pC | s Î Aut(C/QC) } with its action on
pC is type-definable in Ceq over
Æ. If p is Q-internal via p-regular generic parameter
(i.e. every forking extension of the type of the generic
parameter is orthogonal to p) then Aut(p/Q) is
direct-limit-definable in Ceq over Æ and its action on pC is
definable. This is a joint work with B. Hart.
Sunday, June 3, 2001
Professor Robert E. Woodrow
University of Calgary
gave a talk in the Special Session on Model Theoretic Algebra at
the CMS Summer 2001 Meeting:
Absorbing sets in arc-coloured tournaments
Abstract:
Let T be a tournament whose arcs are coloured with k
colours. Call a subset X of the vertices of T absorbing if
from each vertex of T no in X there is a monochromatic
directed path to some vertex in X. We consider the question of
the minimum size of absorbing sets, extending known results and using
new approaches based on notions from the Theory of Relations. Most of
the work deals with finite tournaments, but extensions to the infinite
are also discussed.
Professor Jim Loveys
McGill University, Montreal
gave a talk in the Special Session on Model Theoretic Algebra at
the CMS Summer 2001 Meeting:
Tiny models of strongly minimal theories
Monday, June 4, 2001
Professor Zoe Chatzidakis
CNRS, Universit‚ Paris 7, France
gave a plenary talk and a session talk in the CMS Summer 2001 Meeting:
Model theory of generic difference fields
Abstract:
A difference field is a field with a distinguished automorphism. A
generic difference field is a difference field such that every system of
difference equations which has a solution in a difference field
extension, has a solution in the field.
In the first part of the talk I will state the main model-theoretic
results obtained on these fields and explain their significance and
importance. In the second part of the talk, I will mention some
applications obtained by Hrushovski to the solution of diophantine
problems (e.g., the Manin-Mumford conjecture and the Jacobi conjecture
for difference fields). I will also mention some intriguing questions,
which lie at the boundary of model theory and diophantine geometry.
Professor Chris Miller
Ohio State University, Columbus, Ohio
gave a talk in the Special Session on Model Theoretic Algebra at the CMS
Summer 2001 Meeting:
Hausdorff dimension, analytic sets and transcendence
Abstract:
Every analytic (in the sense of descriptive set theory) set of real
numbers having positive Hausdorff dimension contains a transcendence
base. Equivalently, every analytic proper real-closed subfield of the
reals has Hausdorff dimension zero. (Joint work with
G. A. Edgar.)
Reed Solomon
University of Wisconsin-Madison
gave a talk in the Special Session on Model Theoretic Algebra at the
CMS Summer 2001 Meeting:
The effective content of spaces of orders on groups
and fields
Abstract:
For a formally real field F, the space of orders
X(F) has a natural topology which makes it a Boolean space
(compact, Hausdorff, and totally disconnected). Craven proved that given
any Boolean space B, there is a formally real field F such
that B is homeomorphic to X(F). Metakides and
Nerode examined the computable content of this correspondence by
considering effectively closed subspaces of the Cantor space (called
P01 classes) in
place of Boolean spaces. They showed that for every P01 class C, there is
a computable field F such that C is homeomorphic to
X(F) by a Turing degree preserving homeomorphism.
Downey and Kurtz asked whether a similar correspondence holds for
computable orderable groups. In this talk, we will discuss these
results as well as give a negative answer to the Downey and Kurtz
question for the classes of torsion-free abelian and nilpotent groups.
Professor Hans Schoutens
Rutgers University, New Brunswick
gave a talk in the Special Session on Model Theoretic Algebra at
the CMS Summer 2001 Meeting:
Determining the number of equations of an affine curve
Abstract:
It is in general a very hard problem to find the minimal number of
generators of an ideal in a finitely generated algebra over a field. In
fact, there can be no computer algebra system that calculates this
number exactly (I will briefly discuss a counterexample due to
SCHMIDT). The obstruction lies in the possible
unboundedness of the degrees of a minimal set of generators and is also
related to the presence of non-trivial line bundles.
I will show that for the defining ideal of an affine curve
C Ì
AKn, such a uniform bound
does exist, except possibly when C is locally generated by the
least possible minimal number of generators (namely n-1). This answers a question raised by
VAN
DEN DRIES for affine
curves. Consequently, there exists an algorithm calculating the exact
number of generators of the ideal of an affine reduced curve C
(barring the exceptional case) provided we take the arithmetic of the
field as an oracle. In the exceptional case (which includes the smooth
case!), we know at least that the ideal of C requires either
n-1 or n generators.
The proof uses a non-standard argument together with the Forster-Swan
Theorem and (the positive solution of) the EE-Conjecture.
Tuesday, June 5, 2001, 4:00 p.m.
Dr. Hagen Knaf
Institute for Industrial Mathematics, Kaiserslautern, Germany
gave a talk in the Department Colloquium:
On uniformization of Abhyankar places over base rings of small
dimension
Abstract:
In the year 1995 A.J. de Jong proved that given a geometrically integral
variety X over the field K there exists a finite extension L|K and
a regula} alteration Y\rightarrow X xK L. In particular,
every K-trivial place P of F=K(X) posesses a prolongation Q to a
finite extension E=K(Y) of FL such that the valuation ring OQ
dominates a regular local ring O\subset E essentially of finite type
over L (local uniformization of P after finite extension of
F). The latter was independently proved by means of purely
valuation-theoretic methods in 1998 by F.-V. Kuhlmann. More recently it
became clear that for a so-called Abhyankar place P local
uniformization is possible without extending F. The method used to
verify this result can also be applied when working over a base ring R
instead of the field K, but does not directly yield local
uniformization in this case.
The aim of the talk is to show how one can combine this method with
Abhyankar's results on uniformization in dimension \leq 2 to prove:
(A) Let P be an Abhyankar place of the function field F|K and R\subseteq
K a regular local ring of dimension dim(R)\leq 2 that is dominated by
OP\cap K. Assume that R is Nagata if \dim(R)=2, that the
group vP(F) /vP(K) is torsion-free, that the
extension FP|KP is separable, and that (K,vP) is defectless.
Then there exists an R-scheme X of finite type such that P is centered
in a regular point of X, and K(X)=F holds.
For a discrete valuation ring R de Jong has proved similiar results as
in the case of a base field, so that (A) for dim(R)=1 can be deduced
from his results at least up to finite extensions.
Some higher dimensional cases in which (A) is true as well as a more
general version of this statement will also be discussed in the talk.
This talk was supported by the
Colloquium Fund of the Department of Mathematics and Statistics.
Friday, July 20, 2001, 4:00 p.m.
Professor
Edward Mosteig
Tulane University, New Orleans
gave a talk in the Department Colloquium on
Applications of Gröbner Bases
Abstract:
Gröbner bases are a computational tool used in solving systems of
polynomial equations by exact means. Currently, they are employed in
many fields of mathematics including commutative algebra, algebraic
geometry, algebraic combinatorics, statistics, linear programming,
numerical analysis, and differential equations. Although they were
developed in the 1960's, they have only recently appeared at the
forefront of computational mathematics. The advent of the personal
computer has permitted computations that were previously
impossible to perform by hand.
Some immediate applications that have elegant expositions via
Gröbner bases include the following.
- Solving the Three Color Problem
- Determining Dependency within Networks
- Computing Hilbert series
- Constructing Minimal Polynomials of Algebraic Numbers
- Geometric Theorem Proving
- Multi-dimensional Reed-Solomon Codes
- Robotics
My goal is to introduce Gröbner bases from an elementary standpoint
and examine their development as given by Bruno Buchberger. From there,
I will highlight a few key results and demonstrate their importance.
Along the way, I will demonstrate how certain examples from a few
different fields of mathematics can be solved using Gröbner bases.
This talk was supported by the University of Saskatchewan Visiting
Lecturers' Fund.
Friday, September 7, 2001, 4:00 p.m.
Dr. Alexander Nenashev
University of Saskatchewan
gave a talk in the Colloquium on
Invariants of quadratic forms over exact categories with
duality
Abstract:
We define an infinite series e0, e1, ... of
invariants of the Witt classes of symmetric bilinear forms over an exact
category with duality. These invariants take their values in suitable
subquotients of Quillen K-groups of the category in question. In the
lower degrees, we recover the generalized dimension index and
generalized discriminant. In the classical case of quadratic forms over
a field of characteristic different from 2 we expect to recover the
invariants related to the Milnor conjecture. Our definition of these
invariants is explicit and is based on the use of a self-dual model for
the K-theory space of the category.
This talk was supported by the Colloquium Fund of the
Department of Mathematics and Statistics.
Tuesday, September 25, 2001, 10:00 am
Professor Niels Schwartz
University of Passau, Germany
gave a talk in the Algebra and Logic Seminar on
Positive polynomials in the plane
Last update: February 5, 2008
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