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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Introenumerable sets and the cototal enumeration d
egrees - Joseph Miller (University of Wisconsin-Ma
dison)
DTSTART;TZID=Europe/London:20220606T133000
DTEND;TZID=Europe/London:20220606T143000
UID:TALK174782AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/174782
DESCRIPTION:In 2015\, Emmanuel Jeandel pointed out two interes
ting properties of the language L of a minimal sub
shift. First\, it is enumeration reducible to its
complement\; we say that L is cototal. Second\, th
ere is an enumeration operator that recovers L fro
m any infinite subset\; we say that L is uniformly
introenumerable. The first observation motivated
the study of cototal sets and their enumation degr
ees\, which has been fruitful. Less attention has
been paid to the second observation.\nIn 2018\, Mc
Carthy showed that every cototal enumeration degre
e contains the language of a minimal subshift\, an
d hence contains a uniformly introenumerable set.
This leaves open the question of whether all unifo
rmly introenumerable sets have cototal degree. Goh
\, Jacobsen-Grocott\, Soskova\, and I have answere
d this question in the negative.\nIn this talk\, I
will give the background on cototality and its co
nnection to symbolic dynamics and computable struc
ture theory. I will put the result above in a larg
er conext\, describing other related subclasses of
the enumeration degrees. I will also discuss the
forcing partial order that we use to prove the sep
aration between uniform introenumerability and cot
otality.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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