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Abstracts for Workshop on "Unstable Homotopy Theory"

Alternatively have a look at the program.

talk 1: Perfect $\mathbb{F}_p$-algebras in homotopy theory

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Speaker: 
Robert Burklund
Affiliation: 
University of Copenhagen
Date: 
Mon, 11/11/2024 - 09:30 - 10:30
Location: 
MPIM Lecture Hall

I will discuss some places in homotopy theory where one encounters perfect $\mathbb{F}_p$-algebras. The focus will be illustrating the ways in which the rigidity of this category allows us to prove several a priori surprising theorems.

 

A chromatic Whitehead theorem

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Speaker: 
Shaul Barkan
Affiliation: 
Hebrew University Jerusalem
Date: 
Mon, 11/11/2024 - 11:00 - 12:00
Location: 
MPIM Lecture Hall

Universal differentials in the bar spectral sequence

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Speaker: 
Adela Zhang
Affiliation: 
University of Copenhagen
Date: 
Mon, 11/11/2024 - 14:00 - 15:00
Location: 
MPIM Lecture Hall

The synthetic analogue of the bar comonad controls the universal differentials in the bar spectral sequence of algebras over spectral operads. This can be viewed as a deformation of Koszul duality for such algebras. I will explain ongoing work with Burklund and Senger on identifying the universal differentials in the bar spectral sequence for spectral Lie algebras over $F_p$. This will also shed light on the mod p homology of labeled configuration spaces and $E_n$-Dyer-Lashof operations for Lubin-Tate theories via a theorem of Knudsen.

 

On the chain rule in Goodwillie calculus

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Speaker: 
Thomas Blom
Affiliation: 
University of Copenhagen / MPIM Bonn
Date: 
Mon, 11/11/2024 - 15:00 - 16:00
Location: 
MPIM Lecture Hall

The chain rule of Arone-Ching is a celebrated result in Goodwillie calculus, describing how the derivatives of a composite of two functors between spaces or spectra can be reconstructed from the derivatives of the individual functors. Based on this result, Lurie conjectured that such a chain rule should more generally hold for a large class of $\infty$-categories. In this talk, I will discuss recent joint work with Max Blans in which we give an affirmative answer to this conjecture.

 

talk 1: Hopf algebras and periodic localizations of homotopy theory

Posted in
Speaker: 
Gijs Heuts
Affiliation: 
Utrecht University
Date: 
Tue, 12/11/2024 - 09:30 - 10:30
Location: 
MPIM Lecture Hall

I will discuss various localizations of unstable homotopy theory arising from the chromatic perspective, namely the v${}_n$-periodic localizations and the localizations with respect to T(n)-homology. There are many ways to approach these through "algebraic models". For example, the v${}_n$-periodic localizations can be studied via the theory of spectral Lie algebras. Quite orthogonally, p-finite spaces can be studied through their cochain algebras with values in Morava E-theory.

talk 2: Perfect $\mathbb{F}_p$-algebras in homotopy theory

Posted in
Speaker: 
Robert Burklund
Affiliation: 
University of Copenhagen
Date: 
Tue, 12/11/2024 - 11:00 - 12:00
Location: 
MPIM Lecture Hall

I will discuss some places in homotopy theory where one encounters perfect $\mathbb{F}_p$-algebras. The focus will be illustrating the ways in which the rigidity of this category allows us to prove several a priori surprising theorems.

$E_2$ formality via obstruction theory

Posted in
Speaker: 
Geoffroy Horel
Affiliation: 
Université Sorbonne Paris Nord
Date: 
Tue, 12/11/2024 - 14:00 - 15:00
Location: 
MPIM Lecture Hall

I will explain how to approach the question of $E_2$-formality of differential graded algebras over a prime field via obstruction theory. In particular, $E_2$-algebras whose cohomology ring is a polynomial algebra on even degree classes are intrinsically formal. As a consequence we can prove $E_2$-formality of the classifying space of some compact Lie group or of Davis-Januszkiewicz spaces.

 

The Dehn invariant and spherical scissors congruence as spectral Hopf algebra

Posted in
Speaker: 
Josefien Kuijper
Affiliation: 
Utrecht University
Date: 
Tue, 12/11/2024 - 15:00 - 16:00
Location: 
MPIM Lecture Hall

The Dehn invariant is known to many as the satisfying solution to Hilbert’s 3rd problem: a polyhedron P can be cut into pieces and reassembled into a polyhedron Q if and only if Q and P have not only the same volume, but also the same Dehn invariant. Generalised versions of Hilbert’s 3rd problem concern the so-called scissors congruence groups of euclidean, hyperbolic and spherical geometry in varying dimensions, and in these contexts one can define a generalised Dehn invariant.

Polynomial functors from groups to spectra

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Speaker: 
Gregory Arone
Affiliation: 
Stockholm University
Date: 
Tue, 12/11/2024 - 16:30 - 17:30
Location: 
MPIM Lecture Hall

Let Fr be the category of finitely generated free groups, and D a stable infinity category. We consider polynomial functors from Fr to D. We show that there are equivalences between the following infinity categories

[Polynomial functors from Fr to D] = [Excisive functors from Top${}_*$ to D] = [Truncated right modules in D over the the commutative operad] = [Right modules in D over the Lie operad, whose Koszul dual is truncated]

talk 2: Hopf algebras and periodic localizations of homotopy theory

Posted in
Speaker: 
Gijs Heuts
Affiliation: 
Utrecht University
Date: 
Wed, 13/11/2024 - 09:30 - 10:30
Location: 
MPIM Lecture Hall

I will discuss various localizations of unstable homotopy theory arising from the chromatic perspective, namely the v${}_n$-periodic localizations and the localizations with respect to T(n)-homology. There are many ways to approach these through "algebraic models". For example, the v${}_n$-periodic localizations can be studied via the theory of spectral Lie algebras. Quite orthogonally, p-finite spaces can be studied through their cochain algebras with values in Morava E-theory.

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