Logic and operator algebras

The most recent wave of applications of logic to operator algebras is a young and rapidly developing field. This is a snapshot of the current state of the art.Comment: A minor chang

Expansions, omitting types, and standard systems

Recursive saturation and resplendence are two important notions in models of arithmetic. Kaye, Kossak, and Kotlarski introduced the notion of arithmetic saturation and argued that recursive saturation might not be as rigid as first assumed. In this thesis we give further examples of variations of recursive saturation, all of which are connected with expandability properties similar to resplendence. However, the expandability properties are stronger than resplendence and implies, in one way or another, that the expansion not only satisfies a theory, but also omits a type. We conjecture that a special version of this expandability is in fact equivalent to arithmetic saturation. We prove that another of these properties is equivalent to \beta-saturation. We also introduce a variant on recursive saturation which makes sense in the context of a standard predicate, and which is equivalent to a certain amount of ordinary saturation. The theory of all models which omit a certain type p(x) is also investigated. We define a proof system, which proves a sentence if and only if it is true in all models omitting the type p(x). The complexity of such proof systems are discussed and some explicit examples of theories and types with high complexity, in a special sense, are given. We end the thesis by a small comment on Scott's problem. We prove that, under the assumption of Martin's axiom, every Scott set of cardinality <2^{\aleph_0} closed under arithmetic comprehension which has the countable chain condition is the standard system of some model of PA. However, we do not know if there exists any such uncountable Scott sets.Comment: Doctoral thesi

Normality of nilpotent varieties in E6E_6

We determine which nilpotent orbits in $E_6$ have normal closure and which do not. We also verify a conjecture about small representations in rings of functions on nilpotent orbit covers for type $E_6$.Comment: 13 page

Model Theory for a Compact Cardinal

We would like to develop model theory for T, a complete theory in L_{theta,theta}(tau) when theta is a compact cardinal. We already have bare bones stability theory and it seemed we can go no further. Dealing with ultrapowers (and ultraproducts) naturally we restrict ourselves to "D a theta-complete ultrafilter on I, probably (I,theta)-regular". The basic theorems of model theory work and can be generalized (like Los theorem), but can we generalize deeper parts of model theory? The first section tries to sort out what occurs to the notion of stable T for complete L_{theta,theta}-theories T. We generalize several properties of complete first order T, equivalent to being stable (see [Sh:c]) and find out which implications hold and which fail. In particular, can we generalize stability enough to generalize [Sh:c, Ch. VI]? Let us concentrate on saturation in the local sense (types consisting of instances of one formula). We prove that at least we can characterize the T's (of cardinality < theta for simplicity) which are minimal for appropriate cardinal lambda > 2^kappa +|T| in each of the following two senses. One is generalizing Keisler order which measures how saturated are ultrapowers. Another asks: Is there an L_{theta,theta}-theory T_1 supseteq T of cardinality |T| + 2^theta such that for every model M_1 of T_1 of cardinality > lambda, the tau(T)-reduct M of M_1 is lambda^+-saturated. Moreover, the two versions of stable used in the characterization are different