Ultraproducts and metastability

Kiddie Talk: Ultraproducts and Szeméredi’s Theorem

As promised, this talk will not assume any knowledge of ultrafilters, ultraproducts, or any model theory. The plan is as follows: (1) Ultrafilters (2) Ultraproducts (3) Joke (4) What are ultraproducts good for? (5) Szeméredi’s theorem via ultraproducts References: Mostly Terence Tao’s various blog notes on ultraproducts. For model theory, see David Marker’s Model Theory: An Introduction for a v...

Nonstandard Analysis and Ultraproducts in Banach Spaces and Functional Analysis Lecture 1: Nonstandard Analysis and Ultraproducts of Structures

Ultraproducts and possible worlds semantics in institutions

We develop possible worlds (Kripke) semantics at the categorical abstract model theoretic level provided by the so-called ‘institutions’. Our general abstract modal logic framework provides a method for systematic Kripke semantics extensions of logical systems from computing science and logic. We also extend the institution-independent method of ultraproducts of [R. Diaconescu, Institution-inde...

Ultraproducts of measure preserving actions and graph combinatorics

Ultraproducts of measure preserving actions of countable groups are used to study the graph combinatorics associated with such actions, including chromatic, independence and matching numbers. Applications are also given to the theory of random colorings of Cayley graphs and sofic actions and equivalence relations.

Institution-independent Ultraproducts

We generalise the ultraproducts method from conventional model theory to an institutionindependent (i.e. independent of the details of the actual logic formalised as an institution) framework based on a novel very general treatment of the semantics of some important concepts in logic, such as quantification, logical connectives, and ground atomic sentences. Unlike previous abstract model theore...