# Uniform Computable Categoricity for Fields

## Logic Seminar

## Meeting Details

For more information about this meeting, contact Kristin Berrigan, Jan Reimann, Linda Westrick.

**Speaker:** Russell Miller, CUNY Graduate Center

**Abstract:** Computability theorists have learned not to take isomorphism for granted. It is quite possible
for two computable structures (i.e., structures with decidable atomic diagrams) to be isomorphic
to each other, yet for there to be no isomorphism between them that can be computed
(as a function from one domain onto the other). This is important because, if the structures
fail to be computably isomorphic, then one of them may have very nice computability-theoretic
properties, while the other does not, despite being (non-computably) isomorphic to the first.
For example, two computable fields can be isomorphic, but if they are not computably
isomorphic, then in one it may be undecidable which field elements have square roots,
while in the other the same question may be decidable.
\emph{Computable categoricity} of a structure means that these pathologies cannot happen:
for every pair of computable copies of the structure, there is a computable isomorphism
between them. Fields were the first class of structures for which computable categoricity
was studied, by Fr\"ohlich and Shepherdson in the 1950's. We will discuss
a uniform version of computable categoricity for fields, which will be accessible
with no background at all beyond a very basic knowledge of field theory. Even so, we
will be able to understand several recent results, some by the speaker and (time permitting)
some joint with Hans Schoutens.

## Room Reservation Information

**Room Number:** 315 McAllister

**Date:** 10/01/2019

**Time:** 2:30pm - 4:00pm