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SUMMARY:David Mehrle (Cornell University)
DTSTART;VALUE=DATE-TIME:20211027T194500Z
DTEND;VALUE=DATE-TIME:20211027T210000Z
DTSTAMP;VALUE=DATE-TIME:20230529T063446Z
UID:rts/1
DESCRIPTION:Title: When
free algebras are not free modules and the weird world of equivariant alg
ebra\nby David Mehrle (Cornell University) as part of Rochester topolo
gy seminar\n\nLecture held in Hylan 1106A.\n\nAbstract\nIn homological alg
ebra\, we take for granted that the free $\\mathbb{Z}$-algebra $\\mathbb{Z
}$[x] is also free as a $\\mathbb{Z}$-module. This fact is crucial for cer
tain computations in homotopy theory. We want to make some of these same c
omputations in equivariant homotopy theory\, where $\\mathbb{Z}$-algebras
are replaced by incomplete Tambara functors and $\\mathbb{Z}$-modules are
replaced by Mackey functors. However\, a free incomplete Tambara functor i
s almost never (i.e. with probability zero) free as a Mackey functor! In t
his talk\, I will explain this oddity of equivariant homotopy theory and o
ne way to resolve it. This is joint work with Mike Hill and J.D. Quigley.\
n
LOCATION:https://researchseminars.org/talk/rts/1/
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BEGIN:VEVENT
SUMMARY:Carissa Slone (University of Kentucky)
DTSTART;VALUE=DATE-TIME:20211110T204500Z
DTEND;VALUE=DATE-TIME:20211110T220000Z
DTSTAMP;VALUE=DATE-TIME:20230529T063446Z
UID:rts/2
DESCRIPTION:Title: Char
acterizing 2-slices over $C_2$ and $K_4$\nby Carissa Slone (University
of Kentucky) as part of Rochester topology seminar\n\nLecture held in Hyl
an 1106A.\n\nAbstract\nThe slice filtration focuses on producing certain i
rreducible spectra\, called slices\, from a genuine $G$-spectrum $X$ over
a finite group $G$. We have a complete characterization of all 1-\, 0-\, a
nd (-1)-slices for any such $G$. We will characterize 2-slices over $C_2$
and expand this characterization to $K_4 = C_2 \\times C_2$.\n\nZoom meeti
ng ID: 933 8835 4137\nPasscode: 212814\n
LOCATION:https://researchseminars.org/talk/rts/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeff Carlson (Imperial College London)
DTSTART;VALUE=DATE-TIME:20211117T204500Z
DTEND;VALUE=DATE-TIME:20211117T220000Z
DTSTAMP;VALUE=DATE-TIME:20230529T063446Z
UID:rts/3
DESCRIPTION:Title: Prod
ucts on Tor\, homogeneous spaces\, and Borel cohomology\nby Jeff Carls
on (Imperial College London) as part of Rochester topology seminar\n\nLect
ure held in Hylan 1106A.\n\nAbstract\nThe Eilenberg-Moore spectral sequenc
e converges from the classical Tor of a span of cohomology rings to the di
fferential Tor of a span of cochain algebras (which is the cohomology of t
he homotopy pullback). These are both rings\, the first classically and th
e second as a corollary of the Eilenberg-Zilber theorem. \n\nOne might wel
l ask when a more general differential Tor of DGAs admits a ring structure
\, though apparently no one did. We will show that when the DGAs in questi
on admit a certain sort of $E_3$-algebra structure\, Tor is a commutative
graded algebra. \n\nWe have not done this out of an innocent interest in h
omotopy-commutative algebras. In 1960s and '70s there was a flurry of acti
vity developing A-infinity-algebraic techniques with an aim toward computi
ng the Eilenbergâ€“Moore spectral sequence (for example\, of a loop space
or homogeneous space). Arguably the most powerful result this program prod
uced was the 1974 theorem of Munkholm that the sequence collapses when the
three input spaces have polynomial cohomology over a given principal idea
l domain\, which however only gives the story on cohomology groups. Our re
sult shows that Munkholm's map is in fact an isomorphism of rings. \n\nThe
proof hinges on homotopy properties of the (1-)category of augmented DGAs
. This work is all joint with several large commutative diagrams\, who sho
uld be considered the true authors.\n\nZoom meeting ID: 988 2359 9895 pass
code: 553391\n
LOCATION:https://researchseminars.org/talk/rts/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cihan Okay (Bilkent University)
DTSTART;VALUE=DATE-TIME:20211203T193000Z
DTEND;VALUE=DATE-TIME:20211203T210000Z
DTSTAMP;VALUE=DATE-TIME:20230529T063446Z
UID:rts/4
DESCRIPTION:Title: Homo
topy classification of operator solutions of linear systems\nby Cihan
Okay (Bilkent University) as part of Rochester topology seminar\n\nLecture
held in Hylan 1106A.\n\nAbstract\nLinear systems of equations over a fini
te field play an important role in quantum information theory. Instead of
looking for solutions over the base field one can look for solutions (in a
certain sense) over the unitary group\, which are called operator solutio
ns. The data of this system of equations can be expressed using a hypergra
ph and the operator solutions can be studied from a topological point of v
iew by considering certain topological realizations of these hypergraphs.
In this talk I will describe how homotopical methods provide a way to clas
sify operator solutions of linear systems. Our basic approach is to interp
ret operator solutions as maps from a topological realization of the hyper
graph to a certain classifying space first introduced by Adem-Cohen-Torres
-Giese.\n\nZoom Meeting ID: 972 4663 8781 Passcode: 031045\n
LOCATION:https://researchseminars.org/talk/rts/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matt Feller (University of Virginia)
DTSTART;VALUE=DATE-TIME:20211208T204500Z
DTEND;VALUE=DATE-TIME:20211208T220000Z
DTSTAMP;VALUE=DATE-TIME:20230529T063446Z
UID:rts/5
DESCRIPTION:Title: Gene
ralizing quasi-categories via model structures on simplicial sets\nby
Matt Feller (University of Virginia) as part of Rochester topology seminar
\n\nLecture held in Hylan 1106A.\n\nAbstract\nQuasi-categories are particu
lar simplicial sets which behave like categories up to homotopy. Their the
ory has been massively developed in the past two decades\, thanks largely
due to Joyal and Lurie\, and they have become vital tools in many areas of
algebraic topology\, algebraic geometry\, and beyond. Due to the success
of quasi-categories\, it would be nice to extend the theory to up-to-homot
opy versions of objects more general than categories\, such as the 2-Segal
sets of Dyckerhoff-Kapranov and GĂ lvez-Kock-Tonks. Such a generalization
would ideally come with an associated model structure on the category of
simplicial sets\, but finding a model structure with a more general class
of fibrant objects than a given model structure is a nontrivial and open-e
nded task. In this talk\, I will explain how to use Cisinski's machinery t
o construct model structures on the category of simplicial sets whose fibr
ant objects generalize quasi-categories. In particular\, one of these mode
l structures has fibrant objects precisely the simplicial sets that satisf
y a lifting condition which captures the homotopical behavior of quasi-cat
egories without the algebraic aspects.\n\nZoom Meeting ID: 954 8701 7543\n
Passcode: 123708\n
LOCATION:https://researchseminars.org/talk/rts/5/
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