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PRODID:-//talks.bham.ac.uk//v3//EN
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CATEGORIES:Theoretical computer science seminar
SUMMARY:When measurable spaces don't have enough points -
Paolo Perrone (University of Oxford)
DTSTART:20221118T133000Z
DTEND:20221118T142000Z
UID:TALK4967AT
URL:/talk/index/4967
DESCRIPTION:*Zoom details*\n\n* Link: https://bham-ac-uk.zoom.
us/j/81873335084?pwd=T1NaUFg2U1l6d0RLL2RlTzFBam1IU
T09\n* Meeting ID: 818 7333 5084\n* Passcode: 217\
n\n*Abstract*\n\nIn topology\, a space is called s
ober if every irreducible closed subset is the clo
sure of a unique point.\nOne can express this conc
ept in terms of a monad (called "lower Vietoris" o
r "Hoare powerdomain")\, which assigns to a topolo
gical space X the space HX of its closed subsets.\
nGiven a space X\, we can form a parallel pair HX
-> HHX using the unit of the monad\, and the equal
izer of this pair is precisely the set of irreduci
ble closed subsets of X. The space X is sober if a
nd only if it is an equalizer for this pair.\nOne
can instance the same construction in different co
ntexts\, and obtain analogous notions of "sobriety
". For the Giry monad on measurable spaces\, the e
quivalent of an irreducible closed set is a so-cal
led "extremal" or "zero-one" measure. Just as an i
rreducible closed set cannot be written as a nontr
ivial union\, an extremal measure cannot be writte
n as a nontrivial convex combination. A measurable
space is then sober if and only if every extremal
measure is a Dirac delta at a unique point.\nSeve
ral measurable spaces used in mathematics fail to
be sober\, and have important nontrivial extremal
measures. Examples are ergodic measures in dynamic
al systems\, as well as measures arising from the
zero-one laws of probability theory. These objects
\, despite being very useful in practice\, are oft
en described as being "singular"\, or even "badly
behaved".\nOur categorical treatment\, which paral
lels the one of topology\, can give systematic and
structural understanding of these seemingly count
erintuitive objects.\n\nJoint work with Sean Moss.
Relevant papers: arXiv:2204.07003 and arXiv:2207.
0735.
LOCATION:LG23\, SoCS and Zoom (see abstract for link)
CONTACT:George Kaye
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