An old question about first-order topological structures is: If every set defined 
by a unary formula is a boolean combination of definable open sets, is the same 
true of every definable set (of any arity)? The question is open (as far as I know) 
even under the assumption that, in every elementarily equivalent structure, every 
set defined by a unary formula is a boolean combination of definable open sets. 
I will cast doubt on a positive answer (even under the stronger assumption) but 
show that a natural further strengthening of the hypotheses does yield a positive
answer for expansions of the real line.

It will be proved that the pure-injective envelope of such a domain (as a
module over itself) carries a canonical ring structure compatible with that
of the underlying domain.

Abstract

Abstract

Abstract

Tight closure theory is a fairly new development in commutative algebra 
initiated by Hochster and Huneke at the end of the '80's. It uses 
properties of the Frobenius in positive characteristic and then lifts these 
results to characteristic zero using reduction techniques and Artin 
Approximation. This yields an extremely powerful method which allows 
one to prove many deep theorems with relative ease, at least in 
positive characteristic; the zero characteristic case requires typically 
some additional work. I will explain how one can often circumvent the 
latter complications by using the non-standard Frobenius in characteristic 
zero--essentially this is obtained by taking an ultraproduct of rings of 
positive characteristic in conjunction with certain definability and 
uniformity results. This works especially well for finitely generated 
algebras over the complex numbers as I will illustrate with some 
examples: the Briancon-Skoda Theorem, the Hochster-Roberts Theorem 
and Boutot's Theorem on rational quotient singularities.

Abstract

I will try to sketch some recent developments in the theory of Presburger
groups, more specifically the ordered additive group of integers and elementary
equivalent structures. I will present a dimension theory for Presburger sets and 
a cell decomposition theorem for Z-groups. 
With this tool of cell decomposition we can classify the Presburger sets up to
Presburger-definable bijection: there exists a definable bijection between two
infinite Presburger sets if and only if their dimension is equal. Furthermore I
will talk about quotients built up with Presburger sets: Z-groups eliminate
imaginaries within the Presburger language, without extra sorts.

A new concept of elementary relation between structures is introduced, which
generalizes that of elementary embedding (or monomorphism). When specified
to the case where the relation is the graph of a surjective map, this leads
to elementary epimorphisms, a concept dual to that of elementary
monomorphism. (This dualism is similar to that of pure monomorphism and pure
epimorphism in module theory.) I will prove a dual elementary chain lemma
(or an elementary co-chain lemma).