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CATEGORIES:Conferences/Seminars/Lectures
DESCRIPTION:Featured Speaker : Allen Yuan (MIT Mathematics)\n\nTitle : On
the higher Frobenius\n\nAbstract : Algebraic topology is the study of spa
ces via algebraic invariants. Given such an invariant of a space $X$\, one
can ask: how much of $X$ is captured by that invariant? For instance\, can
one recover $X$ itself (up to homotopy)?\nThis question was first addressed
in work of Quillen and Sullivan on rational ho- motopy theory in the 1960’
s and in work of Dwyer-Hopkins and Mandell on $p$-adic homotopy theory in t
he 1990’s. They showed that various algebraic enhancements of the notion of
cohomology allow one to recover various approximations to a space $X$\, su
ch as its \emph{rationalization} or $p$\emph{-completion}.\nIn this thesis\
, we describe how to unify these ideas and recover a space in its entirety\
, rather than up to an approximation\, using deeper invariants. The approac
h is centered around an insight of Nikolaus and Scholze\, who demonstrate t
hat the classical Frobenius endomorphism for rings in characteristic $p$ na
turally generalizes to a phenomenon in higher algebra (more precisely\, for
$_\infty$-ring spectra)\, which we call the \emph{higher Frobenius}. Our m
ain result is that there is an action of the circle group on (a certain sub
category of) $p$-complete $_\infty$-rings whose monodromy is the higher Fro
benius. Using this higher Frobenius action\, we give a fully faithful model
for a simply connected finite complex $X$ in terms of Frobenius-fixed $_\i
nfty$-rings.\n\nFor information\, write: burklund@mit.edu
DTEND:20200309T213000Z
DTSTAMP:20241106T220140Z
DTSTART:20200309T203000Z
GEO:42.358825;-71.090029
LOCATION:Room : 2 - 131
SEQUENCE:0
SUMMARY:Algebraic Topology Seminar
UID:tag:localist.com\,2008:EventInstance_32875789631476
URL:https://calendar.mit.edu/event/algebraic_topology_seminar_20200309
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