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Seminar

A Computable Functor from Groups to Fields August 02, 2022 (02:15 PM PDT - 03:00 PM PDT)
Parent Program:
Location: SLMath: Eisenbud Auditorium, Online/Virtual
Speaker(s) Meng-Che Ho (California State University, Northridge)
Description No Description
Keywords and Mathematics Subject Classification (MSC)
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

A Computable Functor From Groups To Fields

Abstract/Media

In descriptive set theory, the complexities of classes of countable structures are studied. A classical example is the isomorphism problem $\cong_r$ on the class of torsion-free abelian groups of a fixed rank $r$.  Baer gave a simple invariant for $r = 1$. However, jorth showed that $\cong_1 <_B \cong_2$ and Thomas generalized this to show that $\cong_n <_B \cong_{n+1}$. Recently, Paolini and Shelah, and independently Laskowski and Ulrich, showed that the class of torsion-free abelian group with domain $\omega$ is Borel complete.

We compare the class of torsion-free abelian groups and the class of fields in the computability setting. In particular, we show that there is a computable functor from the class of torsion-free abelian groups of rank $r$ to the class of fields of characteristic $0$ with transcendence degree $r$. We will also discuss the (im)possibility of a computable functor that goes in the other direction.

93842?type=thumb A Computable Functor from Groups to Fields 553 KB application/pdf

A Computable Functor From Groups To Fields