Department of Mathematics & Statistics

Saskatchewan Algebra & Number Theory Mini-meeting

September 20 & 21, 2002

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ABSTRACTS

Talk 1: Roland Auer (U of S)
Ray class fields of global funcion fields, and curves with many points.
Curves over finite fields with many rational points have their applications in Coding Theory and to (Quasi-)Monte Carlo Methods.

While there are theoretical upper bounds for the number of rational points on a (smooth, projective, absolutely irreducible, algebraic) curve of a given genus over a fixed finite field, establishing meaningful lower bounds would actually involve constructing curves with many points. This can be done in various ways.

In the present talk, we will consider abelian covers of explicitly given low genus curves in which many of the rational points are forced to split. The existence of these covers, their degree and the genus of the covering curve is derived from (the function field case) of global Class Field Theory, a classical topic in Algebraic Number Theory.

By construction, the covering curves are expected to have many rational points, and some of them actually beat previously known records.
Talk 2: Martin Argerami (U of R)
Talk 3: Murray Marshall (U of S)
Optimizing polynomial functions using semidefinite programming.

Recently progress has been made (by N.Z. Shor, P.A. Parrilo, P. Parrilo & B. Sturmfels and by J.B. Lasserre) in the development of fast algorithms for optimizing polynomials. The main idea is to relax the problem to a simpler problem (involving sums of squares and moment sequences) which can be solved by semidefinite programming. The method has potential application in control theory, for example. In most cases the method produces exact results and dramatically outperforms existing algebraic methods. The talk, which is a survey of this work, involves a nice mixture of real algebraic geometry, functional analysis, and optimization.
Talk 4: Murray Bremner, Department of Mathematics and Statistics, University of Saskatchewan
Cohomology of infinite dimensional Lie algebras isn't as hard as it sounds
The derivation algebra of the ring of complex Laurent polynomials is one of the most important examples of a simple infinite dimensional Lie algebra. It is called the Witt algebra, and is isomorphic to the Lie algebra of complex vector fields on the circle. For this reason it is closely related to affine Kac-Moody algebras and string theory. This talk will give an elementary proof of the simplicity of the Witt algebra, and then discuss its central extensions. The universal central extension is called the Virasoro algebra, which has a very rich and fascinating representation theory related to modular functions. This talk will give an elementary determination of the cocycle which defines the Virasoro algebra (originally discovered by Block in characteristic p, and Gelfand-Fuchs in characteristic 0). This computation also shows that the second cohomology group of the Witt algebra (with coefficients in the base field) is one-dimensional.
Talk 5: Richard McIntosh (U of R)
On the Largest Prime Factor of a Number

Abstract: PDF File
Talk 6: Mikhail Kotchetov (U of S)
Polynomial Identities in Hopf Algebras.

The notion of a polynomial identity for algebras is classical. We will start by introducing the dual notion for coalgebras and discussing its general properties. In particular, we will study what classes of coalgebras are varieties, i.e. can be defined by a set of identities. In the case of algebras, varieties are characterized by a classical theorem of Birkhoff. Somewhat unexpectedly, the dual statement for coalgebras does not hold. Further, we will give two realizations of a relatively (co)free coalgebra of a variety: one via the so called finite dual of a relatively free algebra and the other a direct construction using some kind of symmetric functions.

Given a Hopf algebra, one can consider two kinds of polynomial identities: for the underlying algebra or for the underlying coalgebra. It is convenient to refer to the former as simply "identities" and the latter - "coidentities". We concentrate on the problem when a Hopf algebra has a nontrivial identity or coidentity. It appears that these two kinds of identities are totally independent of each other: a free algebra can be endowed with the structure of a cocommutative Hopf algebra, and, dually, a free coalgebra - with the structure of a commutative Hopf algebra. This suggests two natural special cases of the above problem (which are formal duals of each other). Namely, we ask: 1) when a cocommutative Hopf algebra satisfies an identity and 2) when a commutative Hopf algebra satisfies a coidentity?
Talk 7: Francesco Barioli (U of R)
Maximal Cp-rank

An n-by-n symmetric nonnegative matrix A is called completely positive if A=V^T V for some t-by-n nonnegative matrix V. The minimal t for which such a matrix V exists is called the Cp-rank of A.

While at the moment no definitive test is known that computes the Cp-rank of a matrix, a sharp upper bound for the Cp-rank of matrices with a prescribed rank is presented. For matrices with a prescribed order a sharp bound is not yet determined. However we present a counterexample to the Drew-Johnson-Loewy conjecture, that was Cp-rank(A)<= n² / 4.
Talk 8: Olivier Piltant, Universite de Versailles, St Quentin, France
Local Parametrizations of Algebraic Curves and Surfaces

An algebraic variety is usually locally described in terms of equations, parametrization or in terms of its ring of regular functions. Using equations, it is viewed as embedded in affine space while using a parametrization it is viewed as a covering of the affine space.

For example, a local parametrization of a complex algebraic curve is of the form
u = f(x),
with f a series in x vanishing at zero, and which is algebraic over the field of rational functions C(x). In this case, the order of vanishing of f at zero expresses the ramification of the induced covering of the complex affine line.

The purpose of this lecture is to present recent results obtained in a joint work with Dale Cutkosky on the ramification of coverings of the affine plane. I will mainly focus on the case of ground fields of positive characteristic, where the situation is much more subtle than for the field of complex numbers.
Talk 9: Franz-Viktor Kuhlmann (U of S)
Places of algebraic function fields in arbitrary characteristic.

We consider the Zariski space of all places of an algebraic function field F|K of arbitrary characteristic and investigate its structure by means of its patch topology. We show that certain sets of places with nice properties (e.g., prime divisors, places of maximal rank, zero-dimensional discrete places) lie dense in this topology.

Further, we give several equivalent characterizations of fields that are large, in the sense of Florian Pop's Annals paper Embedding problems over large fields. We also consider the question whether a field K is existentially closed in an extension field L if L admits a K-rational place. Finally, we sketch a proof of the fact that the Zariski space with the Zariski topology is quasi-compact and that it is a spectral space.
Talk 10: Salma Kuhlmann (U of S)
An Exponential Integer Part for the Exponential-Logarithmic Power Series Field.
An integer part Z of an ordered field F is a discretely ordered subring, with 1 as least positive element, and such that for every field element x, there is a ring element z so that x is between z and z+1.

Sheperdson showed that integer parts of real closed fields are precisely the models of Open Induction (OI is a fragment of Peano Arithmetic).

He used this observation to investigate the arithmetic properties of these rings; for example, he constructed an integer part of the field of Puiseux series (with coefficients in the field of real algebraic numbers, and exponents in the group of rational numbers) in which the set of primes is not cofinal.

On the other hand, subsequent to his work, several authors constructed such rings with unboundedly many irreducubles and primes.

Thus the "infinity of primes" is not provable from OI.
In this talk, we will first review these notions.
Then we shall consider the exponential analogue to the above:
we study Exponential Integer Parts of ordered exponential fields (that is, integer parts that are moreover satisfy some closure conditions under the exponential function).

It is an open problem whether such rings necessarily contain unboundedly many primes.

We construct an exponential integer part of the Exponential- Logarithmic power series fields. These fields are universal for models of real exponentiation.
We end the talk by a discussion of a possible approach to tackle the open problem mentioned above.