Isaac Goldbring

THE PSEUDOARC IS A CO-EXISTENTIALLY CLOSED CONTINUUM

Abstract Answering a question of P. Bankston, we show that the pseudoarc is a co-existentially closed continuum. We also show that C ( X ) , for X a nondegenerate continuum, can never have quantifier elimination, answering a question of the first and third named authors and Farah and Kirchberg.

High density piecewise syndeticity of sumsets

Abstract Renling Jin proved that if A and B are two subsets of the natural numbers with positive Banach density, then A + B is piecewise syndetic. In this paper, we prove that, under various assumptions on positive lower or upper densities of A and B , there is a high density set of witnesses to the piecewise syndeticity of A + B . Most of the results are shown to hold more generally for subsets of Z d . The key technical tool is a Lebesgue density theorem for measure spaces induced by cuts in the nonstandard integers.

Model-theoretic aspects of the Gurarij operator system

We establish some of the basic model theoretic facts about the Gurarij operator system GS recently constructed by the second-named author. In particular, we show: (1) GS is the unique separable 1-exact existentially closed operator system; (2) GS is the unique separable nuclear model of its theory; (3) every embedding of GS into its ultrapower is elementary; (4) GS is the prime model of its theory; and (5) GS does not have quantifier-elimination, whence the theory of operator systems does not have a model companion. We also show that, for any q ∈ ℕ, the theories of Mq-spaces and Mq-systems do have a model companion, namely the Fra¨ıssé limit of the class of finite-dimensional Mq-spaces and Mq-systems respectively; moreover, we show that the model companion is separably categorical. We conclude the paper by showing that no C* algebra can be existentially closed as an operator system.

Robinson forcing and the quasidiagonality problem

We introduce weakenings of two of the more prominent open problems in the classification of

Pseudofinite and Pseudocompact Metric Structures

\mathrm{C}^*

Hilbert’s 5th Problem

-algebras, namely the quasidiagonality problem and the UCT problem. We show that the a positive solution of the conjunction of the two weaker problems implies a positive solution of the original quasidiagonality problem as well as allows us to give a local, finitary criteria for the MF problem, which asks whether every stably finite

An approximate Herbrand’s theorem and definable functions in metric structures

\mathrm{C}^*

Bezout Domains and Elliptic Curves

-algebra is MF.

Hindman's theorem and idempotent types

We initiate the study of pseudofiniteness in continuous logic. We introduce a related concept, namely that of pseudocompactness, and investigate the relationship between the two concepts. We establish some basic properties of pseudofiniteness and pseudocompactness and provide many examples. We also investigate the injective-surjective phenomenon for definable endofunctions in pseudofinite structures.

A monad measure space for logarithmic density

Algebra × Topology = Analysis Important Lie groups are the vector groups Rn, their compact quotients Rn/Zn, the general linear groups GLn(R), and the orthogonal groups On(R). For each of these the group structure and the real analytic manifold structure is the obvious one; for example, GLn(R) is open as a subset of Rn 2 , and thus an open submanifold of the analytic manifold Rn2 . Hilbert’s 5th problem asks for a characterization of Lie groups that is free of smoothness or analyticity requirements. A topological group is said to be locally euclidean if some neighborhood of its identity is homeomorphic to some Rn. A Lie group is obviously locally euclidean, and the most common version of Hilbert’s 5th problem (H5) can be stated as follows: