José Iovino

A PRIMER OF SIMPLE THEORIES

Abstract. We present a self-contained exposition of the basic aspects of simple theories while developing the fundamentals of forking calculus. We expound also the deeper aspects of S. Shelahs 1980 paper Simple unstable theories. The concept of weak dividing has been replaced with that of forking. The exposition is from a contemporary perspective and takes into account contributions due to S. Buechler, E. Hrushovski, B. Kim, O. Lessmann, S. Shelah and A. Pillay.

Model theoretic forcing in analysis

Abstract We present a framework for model theoretic forcing in a non first order context, and present some applications of this framework to Banach space theory.

Omitting uncountable types and the strength of [0,1]-valued logics

Abstract We study a class of [ 0 , 1 ] -valued logics. The main result of the paper is a maximality theorem that characterizes these logics in terms of a model-theoretic property, namely, an extension of the omitting types theorem to uncountable languages.

The morley rank of a banach space

We introduce the concepts of Morley rank and Morley degree for structures based on Banach spaces. We characterize co-stability in terms of Morley rank, and prove the existence of prime models for co-stable theories. ?

Beyond First Order Model Theory

We consider model-theoretic properties related to the expressive power of three analogues of Lω1,ω for metric structures. We give an example showing that one of these infinitary logics is strictly more expressive than the other two, but also show that all three have the same elementary equivalence relation for complete separable metric structures. We then prove that a continuous function on a complete separable metric structure is automorphism invariant if and only if it is definable in the more expressive logic. Several of our results are related to the existence of Scott sentences for complete separable metric structures.

A public-key threshold cryptosystem based on residue rings

Abstract We present a generalization of Pedersen’s public-key threshold cryptosystem. Pedersen’s protocol relies on the field properties of ℤ p . We generalize the protocol so that the calculations can be performed in residue rings that are not necessarily fields. The protocol presented here is polynomial-time equivalent to Pedersen’s.