Spring 2022

In Spring 2022, Juvitop was about Goodwillie Calculus and Spectral (Partition) Lie Algebras.

• • Feb 022022
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  Overview

  Adela Zhang

  Notes

  I will give an overview of the seminar and a brief description of what each talk entails, mainly through the lens of Koszul duality. The first half of the seminar concerns Goodwillie calculus and classical computations about the Goodwillie tower of the identity functor. These computations are essential in understanding the spectral Lie operad and power operations on its algebras, which we shall survey next. Finally we will focus on recent results on spectral partition Lie algebras, including a Koszul duality statement, as well as developments in the chromatic version of the story.

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• • Feb 092022
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  Introduction to Goodwillie Calculus

  Leon Liu

  This talk follows Peter Haine's notes from Juvitop 2017.

  References:

  • Kuhn, Goodwillie towers and chromatic homotopy: an overview, sections 1-5
  • Goodwillie, Calculus III: Taylor Series
  • Lurie, Higher Algebra, sections 6.1.1 - 6.1.5

  In this talk we begin the story of Goodwillie calculus. We will motivate Goodwillie calculus by ordinary calculus, then define cubes and n-excisive functors, which are the categorical analogue of polynomials of degree less or equal to n. Then we will define the Taylor tower, analogous to the Taylor approximation. Lastly, we define homogeneous functors, the analogue of polynomials of pure degrees.

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• • Feb 162022
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  Goodwillie Layers and the Chain Rule

  Lucy Yang

  References:

  • Arone-Ching, Operads and chain rules for the calculus of functors
  • Lurie, Higher Algebra, sections 6.1.4, 6.3

  This talk will continue and build on the introduction to Goodwillie calculus from last week. We will review and extend the theory of excisive approximation to functors of several variables (multilinear approximation) and introduce the differential of a functor. We then use this to classify n-homogeneous functors, i.e. the layers appearing in the Taylor tower. Next, we discuss the chain rule and indicate the formalism/setup needed to make the statement precise. Finally, we use the chain rule to sketch an argument for producing a (co)operad structure on the derivatives of the identity functor.

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• • Feb 232022
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  Examples of Goodwilie Calculus

  Cameron Krulewski

  Notes

  Reference: Kuhn, Goodwillie towers and chromatic homotopy: an overview, section 6

  In this talk, we will discuss convergence of Goodwillie towers and see some examples. We'll start by seeing how connectivity conditions are an analog of a radius of convergence for the Goodwillie tower. Then, we'll discuss an example functor for which there is a model, due to Arone, of its nth excisive approximations. Using Arone's model, we will first show that the tower converges, then deduce a result known as Snaith splitting. If there is extra time, we may discuss a connection between Goodwillie towers and the Kahn-Priddy theorem.

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• • Mar 022022
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  The Goodwillie Tower of the Identity Functor

  Nat Pacheco-Tallaj

  References:

  • Arone-Mahowald, The Goodwillie tower of the identity and the unstable periodic homotopy of spheres
  • [Arone-Kankaanrinta, A functorial model for iterated Snaith splitting with applications to calculus of functors]

  We will discuss how the Goodwillie derivatives of the identity can be expressed as the Spanier-Whitehead duals of certain partition complexes. Then we will outline the computation of the stable homology of the goodwillie layers of the identity evaluated at an odd-dimensional sphere.

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• • Mar 092020
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  The Goodwillie Tower of the Identity and Tits Buildings

  Ishan Levy

  This talk follows Mike Hopkins' notes from the Thursday Seminar in Spring 2018.

  Reference:

  • Arone-Dwyer, Partition complexes, Tits buildings, and symmetric products.
  • Mitchell-Priddy, Stable splittings derived from the Steinberg module
  • Arone-Brantner, The action of Young subgroups on the partition complex, section 5.5 and 6

  I will explain the relationship due to Arone-Dwyer between the Goodwillie Layers on odd spheres, Tits buildings, and the Symmetric power filtration on the integers.

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• • Mar 162020
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  Operadic Koszul Duality and the Spectral Lie Operad

  Natalie Stewart

  References:

  • Behrens, The Goodwillie tower and the EHP sequence, section 1.1-1.3
  • Ching, Bar constructions for topological operads and the Goodwillie derivatives of the identity
  • Lurie, Thursday seminar notes, lecture 15,16

  We will begin by discussing Koszul duality for associative algebra objects in a monoidal ∞-category, and sketch the specialization of this to operads in nice enough ∞-categories. This will be computed via a dual to a derived tensor product, which we will describe via a geometric realization of the simplicial bar construction; we will go on to describe an alternative tree-based bar construction for operads due to Ching (with explicit cooperad structure) and prove an equivalence to the simplicial bar construction. With Arone-Mahowald's computation in mind, Ching's bar construction will allow us to recognize the Goodwillie derivatives of the identity as the spectral Lie operad, i.e. as the Koszul dual to the commutative operad in spectra, whose singular homology is the desuspension of the Lie operad in graded vector spaces.

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• • Mar 232020
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  No talk

  Spring break
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• • Mar 302020
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  Differentiating the EHP sequence and the Hilton-Milnor Theorem

  David Lee

  Notes

  References:

  • Arone-Mahowald, section 4.2
  • Arone-Brantner, The action of Young subgroups on the partition complex, section 5 and 7
  • Arone-Kankaanrinta, The homology of certain subgroups of the symmetric group with coefficients in Lie(n)

  In previous talks, we discussed the Goodwillie layers on odd spheres. In this talk, I will explain how to differentiate the EHP sequence to obtain Goodwillie layers on even spheres from layers on odd spheres. Then, I will discuss Goodwillie layers on a wedge of spheres by differentiating the Hilton-Milnor theorem.

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• • Apr 062020
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  Power Operations on Spectral Lie Algebras

  Eunice Sukarto

  Notes

  References:

  • Behrens,The Goodwillie tower and the EHP sequence, section 1.4-1.5
  • Antolin-Camarena, The mod 2 homology of free spectral Lie algebras
  • Kjaer, On the odd primary homology of free algebras over the spectral lie operad
  • Brantner, The Lubin-Tate theory of spectral Lie algebras

  We will construct power operations on the mod 2 homology of spectral Lie algebras as an analog to Dyer-Lashof operations on E_∞ algebras, and look at relations between them. Next, we construct a shifted Lie bracket on homology of spectral Lie algebras and use partition complexes to show that it satisfies the Jacobi identity. This shifted Lie bracket is compatible with the action of the power operations, giving the homology of a spectral Lie algebra the structure of an allowable-bar{R}-sLie-algebra. Finally, we will show that for a simply connected space, the homology of the free sLie-algebra is the free allowable-bar{R}-sLie-algebra on the reduced homology.

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• • Apr 132020
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  Spectral Partition Lie Algebras and a Koszul Duality Theorem

  Arpon Raksit

  Notes

  Reference: Brantner-Mathew, Deformation Theory and Partition Lie Algebras

  We've been discussing the Lie operad, the Koszul dual of the commutative operad. One might guess that there is some kind of Koszul duality at the level of algebras, i.e. between Lie algebras and commutative algebras. This is correct in characteristic zero but not in positive characteristic. Rather, there is a duality between commutative algebras and a variant notion called spectral partition Lie algebras. In this talk, I will try to explain this variant and the duality result.

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• • Apr 202020
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  Spectral Lie Algebras and Unstable Homotopy Theory

  Piotr Pstragowski

  Notes

  References:

  • Heuts, Lie algebras and v_n-periodic spaces,
  • Eldred-Heuts-Mathew-Meier, Monadicity of the Bousfield-Kuhn functor

  In 1969, Quillen showed that the homotopy theory of rational spaces is equivalent to that of rational spectral Lie algebras, the equivalence established by taking the Chevalley-Eilenberg complex of the cochain algebra. We will discuss an analogue of this result for higher chromatic heights, due to Heuts - the homotopy theory of v_n-periodic spaces is equivalent to that of spectral Lie algebras in T(n)-local spectra.

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• • Apr 272020
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  No talk

  Simons Lectures
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• • May 042020
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  Partition Lie algebras and two algebraic classification problems

  Lukas Brantner

  References:

  • Brantner-Mathew, Deformation Theory and Partition Lie Algebras
  • Brantner-Waldron, Purely Inseparable Galois theory I: The Fundamental Theorem

  I will explain how partition Lie algebras can be used to classify formal deformations in characteristic p (B.-Mathew) and to prove the fundamental theorem of purely inseparable Galois theory (B.-Waldron).

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This seminar was organized by Nat Pacheco-Tallaj and Adela Zhang.