THE THIRD ANNUAL COLLOQUIUMFEST
Titles and Abstracts of Talks
Dr. Vincent Astier
will talk on
Pfister's subform theorem for spaces of orderings
Abstract:
The classical version of Pfister's subform theorem cannot be formulated
for spaces of orderings because of its use of transcendental field
extensions. After recalling some basic facts, we will present and prove
a reformulation of it for spaces of orderings (or more precisely, for
the reduced theory of quadratic forms) which avoids this difficulty.
Dr. Roland Auer
Saskatoon
will talk on
Reduction of elliptic curves in equal characteristic 2
and 3
Abstract:
The conductor of an elliptic curve is an important ingredient in both,
the Shimura-Taniyama-Weil Conjecture (proved in 1999) and the still
unsolved Birch and Swinnerton-Dyer Conjecture (see
http://www.claymath.org/prizeproblems).
If E is an elliptic curve over a local field K, then the conductor
exponent f = f (E/K) usually equals 0, 1 or 2 according to whether E has
good, multiplicative or additive reduction. Only if the residue
characteristic is 2 or 3, can it occur that f > 2.
This talk on joint work with Jaap Top (Groningen, NL) deals with the
case of K having characteristic 2 or 3. It turns out that then f can
exceed any bound. We also give a full classification of the reduction
type and minimal discriminant of E/K in terms of (Artin-Schreier reduced
valuations of) the coefficients of a Weierstrass equation for E, thereby
circumventing Tate's Algorithm, which had still been necessary in this
case. The results can also be used to formulate statements on the
possible global conductors.
Professor Tom Craven
Hawaii
will talk on
Valuation Theory and Ordered *-Rings
Abstract:
After a brief introduction to the idea of ordering the symmetric
elements of a skew field with involution, the recent work of Marshall,
Craven and Smith will be discussed for extending these ideas to rings.
To make the set of positive elements multiplicatively closed, one must
order more than just the symmetric elements (which are not closed under
multiplication); at this point valuation theory becomes crucial.
Professor Dale Cutkosky
Missouri
will talk on
Ramification of Valuations
Abstract:
We consider ramification theory of general valuations, and show that in
algebraic function fields of characteristic zero, it is suprisingly
similar to that of local Dedekind domains.
Dr. Hagen Knaf
Kaiserslautern, Germany
will talk on
Algebras over Prüfer domains: Geometrically relevant
properties and problems
Abstract:
In the talk an overview of results concerning the dimension and ideal
theory of flat finitely generated algebras over a Prüfer domain is
given: Equidimensionality, catenarity, altitude formula, reflexivity.
All results are related to the study of integral schemes of finite type
over a Prüfer domain, a class of schemes that provides a geometric
approach to valued function fields.
A collection of open (?) problems arising from the attempt to weaken
finiteness conditions is presented.
Professor Edward Mosteig
Tulane University
will talk on
An Overview of Generalized Gröbner Bases Via
Valuations
Abstract:
Classical computations of Gröbner bases rely upon initially setting
a term order to be used in the division algorithm. Alternatively,
Sweedler suggested in a paper (1986) another method of computing
Gröbner bases by using valuations in place of term orders. This
talk will address the use of valuations, and the connections with
filtrations and graded structures. Examples (of different types) will be
included that demonstrate the existence of valuations that don't come
about from term orders, but share many of the same properties.
Professor Pablo A. Parrilo
ETH Zürich
will talk on
Sums of squares, convex optimization, and the
Positivstellensatz
Abstract:
We present an overview of a recently introduced convex optimization
framework for semialgebraic problems. Along the way, we'll learn how to
compute sum of squares decompositions for polynomials using semidefinite
programming, and the computational shortcuts that are possible whenever
additional properties (such as sparsity, symmetries, or an ideal
structure)
are present.
The results are used to develop hierarchies of progressively stronger
convex tests, based on the Positivstellensatz, to prove emptiness of
semialgebraic sets. The developed techniques unify and generalize many
well-known existing results. The ideas and algorithms will be
illustrated with examples from a broad range of application domains,
such as continuous and combinatorial optimization and systems and
control theory.
Professor Tara L. Smith
Cincinnati
will talk on
Multiplicative Subgroups of Fdot/Fdot 2 : Additive
Properties and Extensions
Abstract:
We consider multiplicative subgroups of Fdot/Fdot 2,
classify them according to certain additive properties (such as
"rigidity") and consider extensions of F to which the subgroups
extend in a natural way that preserves the additive structure. In
particular we consider the existence of "closures," maximal extensions
of the subgroups, which still preserve the additive properties. This
generalizes the classical theory of orderings on fields, which are
maximal proper subgroups closed under addition, and real closures or
Euclidean closures of those fields.
Professor Bernd Sturmfels
Berkeley
will talk on
Multigraded Hilbert Schemes
Abstract:
How can one give a computer-friendly parametrization of all ideals
in a polynomial ring? To answer this question, we assume that the
polynomial ring is graded by some abelian group and the Hilbert function
is fixed. We construct the Hilbert scheme which parametrizes all such
ideals. Our construction is widely applicable, it provides explicit
equations, and it allows us to prove a range of new results, including
Bayer's conjecture on equations defining Grothendieck's classical
Hilbert scheme and the construction of a Chow morphism in toric
geometry. This is joint work with Mark Haiman (math.AG/0201271).
Dr. Markus Tressl
Regensburg
will talk on
A structure theorem for differential algebras
with applications to real, differential algebra
Abstract:
We show that each differentially finitely generated
domain over a differential ring can be locally decomposed
in an affine part and a differentially free part.
As a first application we reduce the solvability
of an implicit system of ordinary differential
equations to an affine problem about ystem.
Moreover our structure theorem transfers
quantifier elimination results from
(real) algebra to (real) differential algebra.
Last update: March 18, 2002
--------- created and maintained by Franz-Viktor Kuhlmann