Juvitop Seminar
Spring 2022
This semester Juvitop will be about Goodwillie Calculus and Spectral (Partition) Lie Algebras.
We meet at 4:00 on Wednesday in 2151 unless otherwise noted.

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Adela Zhang
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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 Kuhn, Goodwillie towers and chromatic homotopy: an overview, sections 15
 Goodwillie, Calculus III: Taylor Series
 Lurie, Higher Algebra, sections 6.1.1  6.1.5
Leon Liu
This talk follows Peter Haine's notes from Juvitop 2017.
References:
In this talk we begin the story of Goodwillie calculus. We will motivate Goodwillie calculus by ordinary calculus, then define cubes and nexcisive 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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 AroneChing, Operads and chain rules for the calculus of functors
 Lurie, Higher Algebra, sections 6.1.4, 6.3
Lucy Yang
References:
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 nhomogeneous 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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Cameron Krulewski
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 KahnPriddy theorem.
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 AroneMahowald, The Goodwillie tower of the identity and the unstable periodic homotopy of spheres
 [AroneKankaanrinta, A functorial model for iterated Snaith splitting with applications to calculus of functors]
Nat PachecoTallaj
References:
We will discuss how the Goodwillie derivatives of the identity can be expressed as the SpanierWhitehead 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 odddimensional sphere.
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Ishan Levy
This talk follows Mike Hopkins' notes from the Thursday Seminar in Spring 2018.
Reference:
I will explain the relationship due to AroneDwyer between the Goodwillie Layers on odd spheres, Tits buildings, and the Symmetric power filtration on the integers.
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Natalie Stewart
References:
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 treebased bar construction for operads due to Ching (with explicit cooperad structure) and prove an equivalence to the simplicial bar construction. With AroneMahowald'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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No talk
Spring break
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David Lee
References:
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 HiltonMilnor theorem.
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Eunice Sukarto
References:
We will construct power operations on the mod 2 homology of spectral Lie algebras as an analog to DyerLashof 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 allowablebar{R}sLiealgebra. Finally, we will show that for a simply connected space, the homology of the free sLiealgebra is the free allowablebar{R}sLiealgebra on the reduced homology.
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Arpon Raksit
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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Piotr Pstragowski
References:
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 ChevalleyEilenberg 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_nperiodic spaces is equivalent to that of spectral Lie algebras in T(n)local spectra.
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No talk
Simons Lectures
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Lukas Brantner
References:
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 is organized by Nat PachecoTallaj and Adela Zhang.