Probability and Measure

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Classical Descriptive Set Theory

A surprise announcement by then prime minister Tony Blair in 1998 led to the creation of a new senior clinical role - the nurse consultant.

Work in progress: The evolving nurse consultant role has all-round benefits and should be embraced, finds Frances Pickersgill

Classification theory and the number of non-isomorphic models

An outline Definitions and technicalities Turbulence Classifying homeomorphisms Infinite dimensional group representations A generalized Scott analysis GE groups The dark side Beyond Borel Looking ahead Ordinals Notation Bibliography Index.

Classification and Orbit Equivalence Relations

Introduction Baire-measurable homomorphisms of analytic quotients Open coloring axiom and uniformization Homomorphisms of analytic quotients under OCA Weak extension principle Gaps and limits in analytic quotients Bibliography Index Index of special symbols.

Analytic Quotients: Theory of Liftings for Quotients over Analytic Ideals on the Integers

We introduce a version of logic for metric structures suitable for applications to C*-algebras and tracial von Neumann algebras. We also prove a purely model-theoretic result to the effect that the theory of a separable metric structure is stable if and only if all of its ultrapowers associated with nonprincipal ultrafilters on ℕ are isomorphic even when the Continuum Hypothesis fails.

http://www.math.yorku.ca/~ifarah/Ftp/2010d-stable-th.pdf

Model theory of operator algebras II: model theory

We introduce a version of logic for metric structures suitable for applications to C*-algebras and tracial von Neumann algebras. We also prove a purely model-theoretic result to the effect that the theory of a separable metric structure is stable if and only if all of its ultrapowers associated with nonprincipal ultrafilters on N are isomorphic even when the Continuum Hypothesis fails.

Model theory of operator algebras II: Model theory

We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II_1 factors and their ultrapowers. Among other things, we show that for any II_1 factor M, there are continuum many nonisomorphic separable II_1 factors that have an ultrapower isomorphic to an ultrapower of M. We also give a poor man's resolution of the Connes Embedding Problem: there exists a separable II_1 factor such that all II_1 factors embed into one of its ultrapowers.

Model theory of operator algebras III: elementary equivalence and II1 factors

Several authors have considered whether the ultrapower and the relative commutant of a C*-algebra or II_1 factor depend on the choice of the ultrafilter. We settle each of these questions, extending results of Ge-Hadwin and the first author.

Ilijas Farah

Papers

Analytic Quotients: Theory of Liftings for Quotients over Analytic Ideals on the Integers

Model theory of operator algebras II: model theory

Model theory of operator algebras II: Model theory

Model theory of operator algebras III: elementary equivalence and II1 factors

Model theory of operator algebras I: Stability