Computable analysis on the space of marked groups

Abstract: I will present some results that concern the study of Markovian computable analysis on the topological space of marked groups. While this work was originally motivated by questions in group theory, it turned out to be also very interesting from the point of view of computable analysis alone, providing the first naturally arising Polish space that is not effectively Polish. Indeed, while it is effectively complete and separable, the space of marked groups is not effectively separable: any sequence that is dense in it is non-computable. I will also present a phenomenon of failure of an effective axiom of choice: there is no algorithm that can, given a non-empty basic clopen set, produce a group that belongs to this set. Because of this, none of the well known results of Kreisel-Lacombe-Schoenfield, Ceitin and Moschovakis, and that establish the effective continuity of computable functions in different settings, can be applied to the space of marked groups. My talk will focus on presenting some classical theorems of the theory of decision problems for groups (Boone-Novikov, Boone-Rogers, etc), in order to show how those theorems apply to the space of marked groups.

logic

Audience: researchers in the topic

Computability theory and applications

Series comments: Description: Computability theory, logic

The goal of this endeavor is to run a seminar on the platform Zoom on a weekly basis, perhaps with alternating time slots each of which covers at least three out of four of Europe, North America, Asia, and New Zealand/Australia. While the meetings are always scheduled for Tuesdays, the timezone varies, so please refer to the calendar on the website for details about individual seminars.