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SUMMARY:Introenumerable sets and the cototal enumeration degrees - Joseph 
 Miller (University of Wisconsin-Madison)
DTSTART:20220606T123000Z
DTEND:20220606T133000Z
UID:TALK174782@talks.cam.ac.uk
DESCRIPTION:In 2015\, Emmanuel Jeandel pointed out two interesting propert
 ies of the language L of a minimal subshift. First\, it is enumeration red
 ucible to its complement\; we say that L is cototal. Second\, there is an 
 enumeration operator that recovers L from any infinite subset\; we say tha
 t L is uniformly introenumerable. The first observation motivated the stud
 y of cototal sets and their enumation degrees\, which has been fruitful. L
 ess attention has been paid to the second observation.\nIn 2018\, McCarthy
  showed that every cototal enumeration degree contains the language of a m
 inimal subshift\, and hence contains a uniformly introenumerable set. This
  leaves open the question of whether all uniformly introenumerable sets ha
 ve cototal degree. Goh\, Jacobsen-Grocott\, Soskova\, and I have answered 
 this question in the negative.\nIn this talk\, I will give the background 
 on cototality and its connection to symbolic dynamics and computable struc
 ture theory. I will put the result above in a larger conext\, describing o
 ther related subclasses of the enumeration degrees. I will also discuss th
 e forcing partial order that we use to prove the separation between unifor
 m introenumerability and cototality.
LOCATION:Seminar Room 1\, Newton Institute
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