Schedule for: 23w5055 - A Convergence of Computable Structure Theory, Analysis, and Randomness

We'll introduce ourselves, describe our work, and say what we're hoping to get out of the workshop

In this talk, we give an introduction to modern continuous model theory. We use the metric ultraproduct construction to motivate the notion of a continuous language and the appropriate notion of structure for such languages. We give some examples of metric structures of interest in this workshop, such as Banach spaces, Hilbert spaces, and C*-algebras. We then move on to discuss compactness and completeness for continuous logic. Time permitting, we discuss some of the more nuanced aspects of continuous model theory, such as generalized formulas, definable sets, and the metric on type spaces.

In the very early days of computable structure theory there were only specific algebraic questions about specific structures that were presumed to be well known (e.g. does every explicitly given field have a unique computable algebraic closure). Later, this line of thought was abstracted into computable model theory, which could perhaps have looked something like model theory --- except that, as Millar famously remarked, there were "too many counterexamples." Over time what took hold was the study of a new category, taking the putative pathologies (e.g. distinctions between classically isomorphic structures) as features of a new and interesting mathematical realm. Apparently niche interests like infinitary logics, admissible set theory, and alpha-jump priority constructions found a natural home in this discipline. This talk will outline the kinds of questions asked in computable structure theory, and the style of arguments used.

I will give an overview of algorithmic randomness; this will cover definitions, central and illustrative results, and applications of randomness to other areas of computability theory. My intention is to give a picture of the field and show some of its potential for crossover with other areas.

I will talk about several recent results that explicitly relate computable algebra with computable topology via various sorts of effective dualities.

Division into groups to work on open problems.

Working on open problems.

Working on open problems.

Working on open problems.

Discussion of progress from the previous day and group reformation (if desired).

Working on open problems.

Working on open problems.

Discussion of progress from the previous day and group reformation (if desired).

Working on open problems.

Working on open problems.

Discussion of progress from the morning and group reformation (if desired).

Working on open problems.

Working on open problems.

Working on open problems.

Wrapping up, discussing the progress that has been made, and planning for the future.