Titles and Abstracts of Talks

Dr. Vincent Astier

will talk on

Pfister's subform theorem for spaces of orderings

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

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Reduction of elliptic curves in equal characteristic 2 and 3

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

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

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Valuation Theory and Ordered *-Rings

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

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Ramification of Valuations

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

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Algebras over Prüfer domains: Geometrically relevant properties and problems

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

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

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Sums of squares, convex optimization, and the Positivstellensatz

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

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Multiplicative Subgroups of Fdot/Fdot 2 : Additive Properties and Extensions

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

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Multigraded Hilbert Schemes

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

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A structure theorem for differential algebras with applications to real, differential algebra

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.

The Third Annual Colloquiumfest

Last update: March 18, 2002 --------- created and maintained by Franz-Viktor Kuhlmann