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SUMMARY:Robert Burklund (University of Copenhagen)
DTSTART;VALUE=DATE-TIME:20230911T200000Z
DTEND;VALUE=DATE-TIME:20230911T213000Z
DTSTAMP;VALUE=DATE-TIME:20240522T202831Z
UID:MITAlgebraicTopologySeminar/1
DESCRIPTION:Title: Beyond the telescope conjecture\nby Robert Burklun
d (University of Copenhagen) as part of MIT Algebraic Topology Seminar\n\n
Lecture held in 2-131.\n\nAbstract\nThere is a natural dichotomy between t
elescopic (T(n)-local) and chromatic (K(n)-local) homotopy theory. Telesco
pic homotopy theory is more closely tied to the stable homotopy groups of
spheres and through them to geometric questions\, but is generally computa
tionally intractable. Chromatic homotopy theory is more closely tied to ar
ithmetic geometry and powerful computational tools exist in this setting.
Ravenel’s telescope conjecture asserted that these two sides coincide. I
will present a family of counterexamples to this conjecture based on usin
g trace methods to analyze the algebraic K-theory of a family of K(n)-loca
l ring spectra beginning with the K(1)-local sphere. As a consequence of t
his we obtain a new lower bound on the average rank of the stable homotopy
groups of spheres. Time permitting\, I will then describe the galois grou
p of the T(n)-local sphere and how this informs our understanding of teles
copic homotopy theory. This talk is based on projects joint with Carmeli\,
Clausen\, Hahn\, Levy\, Schlank and Yanovski.\n
LOCATION:https://researchseminars.org/talk/MITAlgebraicTopologySeminar/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ben Antieau (Northwestern University)
DTSTART;VALUE=DATE-TIME:20230918T203000Z
DTEND;VALUE=DATE-TIME:20230918T220000Z
DTSTAMP;VALUE=DATE-TIME:20240522T202831Z
UID:MITAlgebraicTopologySeminar/2
DESCRIPTION:Title: Integral models for spaces\nby Ben Antieau (Northw
estern University) as part of MIT Algebraic Topology Seminar\n\nLecture he
ld in 2-131.\n\nAbstract\nGeneralizing and building on the work of Kriz\,
Ekedahl\, Goerss\, Lurie\, Mandell\, Mathew\, Mondal\, Quillen\, Sullivan\
, Toën and Yuan\, I will describe an integral cochain model for nilpotent
spacees of finite type. A binomial ring is a lambda-ring in which all Ada
ms operations act as the identity. A derived binomial ring is a derived Λ
-ring equipped with simultaneous trivializations of the commuting Adams op
erations. For example\, if X is a space\, then ZX\, the integral cochains
on X\, is naturally a derived binomial ring. The induced contravariant fun
ctor from spaces to derived binomial rings is fully faithful when restrict
ed to nilpotent spaces of finite type. This is related\, closely\, to rece
nt work of Horel and of Kubrak—Shuklin—Zakharov.\n
LOCATION:https://researchseminars.org/talk/MITAlgebraicTopologySeminar/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hana Jia Kong (Harvard University)
DTSTART;VALUE=DATE-TIME:20230925T203000Z
DTEND;VALUE=DATE-TIME:20230925T220000Z
DTSTAMP;VALUE=DATE-TIME:20240522T202831Z
UID:MITAlgebraicTopologySeminar/3
DESCRIPTION:Title: A deformation of Borel equivariant homotopy\nby Ha
na Jia Kong (Harvard University) as part of MIT Algebraic Topology Seminar
\n\nLecture held in 2-131.\n\nAbstract\nThe real motivic stable homotopy c
ategory has a close connection\nto the $C_2$-equivariant stable homotopy c
ategory. From a computational perspective\, the real motivic computation c
an be viewed\nas a simpler version which “removes the negative cone” i
n the $C_2$-\nequivariant stable homotopy groups. On the other hand\, by w
ork of\nBurklund–Hahn–Senger\, one can build the completed Artin–Tat
e real\nmotivic category from the completed $C_2$-equivariant category usi
ng\nthe deformation construction associated to the $C_2$-effective filtrat
ion.\nIn work with Gabriel Angelini-Knoll\, Mark Behrens\, and Eva Belmont
\,\nwe try to build an analog of this deformation story for a general fini
te\ngroup $G$. We give a new interpretation of the $C_2$-effective filtrat
ion\nin the Borel equivariant category which generalizes for $G$. Using th
is\nnew interpretation\, the deformation construction gives a deformation\
nof the Borel equivariant stable homotopy category for general finite\ngro
ups.\n
LOCATION:https://researchseminars.org/talk/MITAlgebraicTopologySeminar/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dan Berwick-Evans (University of Illinois Urbana-Champaign)
DTSTART;VALUE=DATE-TIME:20231002T203000Z
DTEND;VALUE=DATE-TIME:20231002T220000Z
DTSTAMP;VALUE=DATE-TIME:20240522T202831Z
UID:MITAlgebraicTopologySeminar/4
DESCRIPTION:Title: Supersymmetric field theories and elliptic cohomology<
/a>\nby Dan Berwick-Evans (University of Illinois Urbana-Champaign) as par
t of MIT Algebraic Topology Seminar\n\nLecture held in 2-131.\n\nAbstract\
nSince the mid 1980s\, there have been hints of a deep connection between
2-dimensional field theories and elliptic cohomology. This lead to Stolz a
nd Teichner's conjectured geometric model for the universal elliptic cohom
ology theory of topological modular forms (TMF) in which cocycles are 2-di
mensional supersymmetric field theories. Basic properties of these field t
heories lead to expected integrality and modularity properties\, but the a
bundant torsion in TMF has always been mysterious. In this talk\, I will d
escribe deformation invariants of 2-dimensional field theories that realiz
e some of the torsion in TMF.\n
LOCATION:https://researchseminars.org/talk/MITAlgebraicTopologySeminar/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Brazelton (Harvard University)
DTSTART;VALUE=DATE-TIME:20231016T203000Z
DTEND;VALUE=DATE-TIME:20231016T220000Z
DTSTAMP;VALUE=DATE-TIME:20240522T202831Z
UID:MITAlgebraicTopologySeminar/5
DESCRIPTION:by Thomas Brazelton (Harvard University) as part of MIT Algebr
aic Topology Seminar\n\nLecture held in 2-131.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/MITAlgebraicTopologySeminar/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Piotr Pstrągowski (Harvard University)
DTSTART;VALUE=DATE-TIME:20231023T203000Z
DTEND;VALUE=DATE-TIME:20231023T220000Z
DTSTAMP;VALUE=DATE-TIME:20240522T202831Z
UID:MITAlgebraicTopologySeminar/6
DESCRIPTION:Title: The even filtration and prismatic cohomology\nby P
iotr Pstrągowski (Harvard University) as part of MIT Algebraic Topology S
eminar\n\nLecture held in 2-131.\n\nAbstract\nThe even filtration\, introd
uced by Hahn-Raksit-Wilson\, is a canonical filtration attached to a commu
tative ring spectrum which measures its failure to be even. Despite its si
mple definition\, the even filtration recovers many arithmetically importa
nt constructions\, such as the Adams-Novikov filtration of the sphere or t
he Bhatt-Morrow-Scholze filtration on topological Hochschild homology\, sh
owing that they are all invariants of the commutative ring spectrum alone.
I will describe a linear variant of the even filtration which is naturall
y defined on associative rings and can be effectively calculated through r
esolutions of modules\, as well as joint work with Raksit on the resulting
extension of prismatic cohomology to the context of $E_2$-rings.\n
LOCATION:https://researchseminars.org/talk/MITAlgebraicTopologySeminar/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tomer Schlank (Hebrew University of Jeruselam)
DTSTART;VALUE=DATE-TIME:20231030T203000Z
DTEND;VALUE=DATE-TIME:20231030T220000Z
DTSTAMP;VALUE=DATE-TIME:20240522T202831Z
UID:MITAlgebraicTopologySeminar/7
DESCRIPTION:Title: Higher Semi-additivity and Chromatically localized $K$
-theory\nby Tomer Schlank (Hebrew University of Jeruselam) as part of
MIT Algebraic Topology Seminar\n\nLecture held in 2-131.\n\nAbstract\nWe s
hall consider the functor $L_{T(n)}K $ of chromatically localized algebrai
c $K$-theory. We shall discuss its interaction with pi-finite colimits. Th
is will lead to a possible alternative characterization of this functor as
well as results about it's interaction with cyclotomic hyper-descent. Thi
s is a key input to the proof of the telescope conjecture. This talk is ba
sed on joint works with Shay Ben-Moshe\, Shachar Carmeli\, and Lior Yanovs
ki\, as well as with Robert Burklund\, Jeremy Hahn\, and Ishan Levy.\n
LOCATION:https://researchseminars.org/talk/MITAlgebraicTopologySeminar/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Weinstein (Boston University)
DTSTART;VALUE=DATE-TIME:20231106T213000Z
DTEND;VALUE=DATE-TIME:20231106T223000Z
DTSTAMP;VALUE=DATE-TIME:20240522T202831Z
UID:MITAlgebraicTopologySeminar/8
DESCRIPTION:Title: The rational homotopy groups of the $K(n)$-local spher
e\nby Jared Weinstein (Boston University) as part of MIT Algebraic Top
ology Seminar\n\nLecture held in 2-131.\n\nAbstract\nWe compute the ration
al homotopy groups of the $K(n)$-local sphere for all heights $n$ and all
primes $p$\, verifying a prediction that goes back to Morava in the early
1970s. The key ingredients are (1) the Devinatz-Hopkins spectral sequence
(2) the isomorphism between the Lubin–Tate tower and the Drinfeld tower
at the level of perfectoid spaces (3) integral $p$-adic Hodge theory\, and
(4) an integral refinement of a theorem of Tate on the Galois cohomology
of nonarchimedean fields. This is joint work with Tobias Barthel\, Tomer S
chlank\, and Nathaniel Stapleton.\n
LOCATION:https://researchseminars.org/talk/MITAlgebraicTopologySeminar/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Craig Westerland (University of Minnesota)
DTSTART;VALUE=DATE-TIME:20231113T200000Z
DTEND;VALUE=DATE-TIME:20231113T210000Z
DTSTAMP;VALUE=DATE-TIME:20240522T202831Z
UID:MITAlgebraicTopologySeminar/9
DESCRIPTION:Title: Moments of L-functions via the homology of braid group
s.\nby Craig Westerland (University of Minnesota) as part of MIT Algeb
raic Topology Seminar\n\nLecture held in The seminar will meet at 3:00 PM
in Harvard Science Center SC507 at Arithmetic Statistics seminar..\n\nAbst
ract\nIn 2005\, Conrey\, Farmer\, Keating\, Rubinstein\, and Snaith posed
a conjecture on the asymptotics of moments of quadratic L-functions. While
this conjecture originates as a question about number fields\, it has a m
ore geometric version when posed over function fields in positive characte
ristic. I’ll talk about how one can reinterpret the central object in th
is conjecture in terms of the action of the Galois group of a finite field
on the cohomology of braid groups with certain coefficients coming from t
he braid group’s interpretation as the hyperelliptic mapping class group
. We will see the “arithmetic factor” in this conjecture appear in the
part of this cohomology that is accessible through tools of homological s
tability. This is joint work with Jonas Bergström\, Adrian Diaconu\, and
Dan Petersen.\n
LOCATION:https://researchseminars.org/talk/MITAlgebraicTopologySeminar/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akhil Mathew (University of Chicago)
DTSTART;VALUE=DATE-TIME:20231204T213000Z
DTEND;VALUE=DATE-TIME:20231204T223000Z
DTSTAMP;VALUE=DATE-TIME:20240522T202831Z
UID:MITAlgebraicTopologySeminar/10
DESCRIPTION:by Akhil Mathew (University of Chicago) as part of MIT Algebra
ic Topology Seminar\n\nLecture held in 2-131.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/MITAlgebraicTopologySeminar/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kathryn Lesh (Union College)
DTSTART;VALUE=DATE-TIME:20231127T213000Z
DTEND;VALUE=DATE-TIME:20231127T223000Z
DTSTAMP;VALUE=DATE-TIME:20240522T202831Z
UID:MITAlgebraicTopologySeminar/11
DESCRIPTION:Title: Normalizer decompositions of p-local compact groups\nby Kathryn Lesh (Union College) as part of MIT Algebraic Topology Semi
nar\n\nLecture held in 2-131.\n\nAbstract\nI will talk about a 'normalizer
decomposition' for the classifying space of a p-local compact group. The
decomposition generalizes those of Dwyer for finite groups and of Libman f
or p-local finite groups and (separately) for compact Lie groups. I'll sho
w how the decomposition gives a homotopy pushout square for the exotic p-c
ompact groups of Aguade and Zabrodsky by building on the example of SU(p).
This is joint work with Belmont\, Castellana\, Grbic\, and Strumila.\n
LOCATION:https://researchseminars.org/talk/MITAlgebraicTopologySeminar/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Efimov (Steklov Mathematical Institute of Russian Academ
y of Sciences and National Research University Higher School of Economics)
DTSTART;VALUE=DATE-TIME:20240129T213000Z
DTEND;VALUE=DATE-TIME:20240129T223000Z
DTSTAMP;VALUE=DATE-TIME:20240522T202831Z
UID:MITAlgebraicTopologySeminar/12
DESCRIPTION:Title: Localizing motives and corepresentability of $TR$ and
$TC$\nby Alexander Efimov (Steklov Mathematical Institute of Russian
Academy of Sciences and National Research University Higher School of Econ
omics) as part of MIT Algebraic Topology Seminar\n\nLecture held in 2-131.
\n\nAbstract\nI will explain some of my recent results on the category of
localizing motives -- the target of the universal localizing invariant com
muting with filtered colimits. The main surprising result about this categ
ory is that it is rigid as a symmetric monoidal category (in the sense of
Gaitsgory and Rozenblyum).\n \n\nAs an application of the proof of rigidit
y\, we will deduce that the functors $TR$ (topological restriction) and $T
C$ (topological cyclic homology) are corepresentable in this category\, if
we restrict to connective $E_1-rings$.\n\nIf time permits\, I will explai
n how rigidity of $Mot^loc$ allows to construct refined versions of (topol
ogical) Hochschild homology and its variants\, which contain much more inf
ormation about the $E_1-algebra$ than the usual variants of $(T)HH$.\n
LOCATION:https://researchseminars.org/talk/MITAlgebraicTopologySeminar/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eva Belmont (Case Western Reserve University)
DTSTART;VALUE=DATE-TIME:20240205T213000Z
DTEND;VALUE=DATE-TIME:20240205T223000Z
DTSTAMP;VALUE=DATE-TIME:20240522T202831Z
UID:MITAlgebraicTopologySeminar/13
DESCRIPTION:Title: Computations with the modified Adams spectral sequenc
e\nby Eva Belmont (Case Western Reserve University) as part of MIT Alg
ebraic Topology Seminar\n\nLecture held in 2-131.\n\nAbstract\nThe modifie
d Adams spectral sequence\, which computes the homotopy groups of a Borel-
equivariant spectrum\, is a combination of the homotopy fixed points spect
ral sequence and the nonequivariant Adams spectral sequence. One can also
use it to read off (completed) $\\mathbb{R}$-motivic homotopy groups\, via
a synthetic spectra construction. We will explain how to compute it using
examples coming from $ko_{C_2}$ and $kq$. This is joint work with Gabriel
Angelini-Knoll\, Mark Behrens\, and Hana Kong.\n
LOCATION:https://researchseminars.org/talk/MITAlgebraicTopologySeminar/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gabriel Angelini-Knoll (Sorbonne Paris Nord)
DTSTART;VALUE=DATE-TIME:20240304T213000Z
DTEND;VALUE=DATE-TIME:20240304T223000Z
DTSTAMP;VALUE=DATE-TIME:20240522T202831Z
UID:MITAlgebraicTopologySeminar/14
DESCRIPTION:Title: Syntomic cohomology of real topological $K$-theory\nby Gabriel Angelini-Knoll (Sorbonne Paris Nord) as part of MIT Algebrai
c Topology Seminar\n\nLecture held in 2-131.\n\nAbstract\nWork of Hahn-Rak
sit-Wilson extended the Bhatt-Morrow-Scholze filtration on topological cyc
lic homology and topological periodic cyclic homology to sufficiently nice
ring spectra. This allows one to define syntomic cohomology and prismatic
cohomology at this level of generality. One example of such a nice ring s
pectrum is real topological $K$-theory. In joint work with Christian Auson
i and John Rognes\, we compute the syntomic cohomology of real topological
$K$-theory modulo $(2\,η\,v_1)$. This computation produces a new example
of pure redshift and arithmetic duality. As an application\, we compute t
he algebraic $K$-theory of real topological $K$-theory modulo $(2\,η\,v_1
)$ and show that it satisfies a higher chromatic complexity version of the
Lichtenbaum-Quillen conjecture.\n
LOCATION:https://researchseminars.org/talk/MITAlgebraicTopologySeminar/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stephen McKean (Harvard University)
DTSTART;VALUE=DATE-TIME:20240311T203000Z
DTEND;VALUE=DATE-TIME:20240311T213000Z
DTSTAMP;VALUE=DATE-TIME:20240522T202831Z
UID:MITAlgebraicTopologySeminar/15
DESCRIPTION:Title: Motivic Euler characteristics and power structures\nby Stephen McKean (Harvard University) as part of MIT Algebraic Topolog
y Seminar\n\nLecture held in 2-131.\n\nAbstract\nThere is a quadratic form
-valued version of the compactly supported Euler characteristic coming fro
m motivic homotopy. A feature of this Euler characteristic is that it desc
ends to a ring homomorphism out of the Grothendieck ring of varieties. In
characteristic 0\, this Euler characteristic was constructed by Röndigs a
nd later Arcila-Maya—Bethea—Opie—Wickelgren—Zakharevich\, who used
Bittner’s blow up presentation of $K_0(Var)$. In characteristic not 2\,
Azouri gave a characterization in terms of the six functor formalism. I w
ill discuss a hybrid approach using a sort of universal property of $K_0(V
ar)$. I will then discuss power structures on $K_0(Var)$ and the Grothendi
eck—Witt ring of quadratic forms\, and conclude with a conjecture relati
ng these two power structures. This is joint work in progress with Dori Be
jleri\n
LOCATION:https://researchseminars.org/talk/MITAlgebraicTopologySeminar/15/
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