## Spring 2022

This semester Juvitop will be about Goodwillie Calculus and Spectral (Partition) Lie Algebras.

We meet at 4:00 on Wednesday in 2-151 unless otherwise noted.

• Feb 022022
$\begingroup$

#### Overview

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.

$\endgroup$
• Feb 092022
$\begingroup$

### 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 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.

$\endgroup$
• Feb 162022
$\begingroup$

### 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 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.

$\endgroup$
• Feb 232022
$\begingroup$

### 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 Kahn-Priddy theorem.

$\endgroup$
• Mar 022022
$\begingroup$

### References:

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.

$\endgroup$
• Mar 092020
$\begingroup$

### Ishan Levy

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

### Reference:

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.

$\endgroup$
• Mar 162020
$\begingroup$

### Natalie Stewart

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.

$\endgroup$
• Mar 232020
$\begingroup$

### Spring break

$\endgroup$
• Mar 302020
$\begingroup$

### David Lee

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.

$\endgroup$
• Apr 062020
$\begingroup$

### Eunice Sukarto

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.

$\endgroup$
• Apr 132020
$\begingroup$

### 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.

$\endgroup$
• Apr 202020
$\begingroup$

Notes

### 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 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.

$\endgroup$
• Apr 272020
$\begingroup$

### Simons Lectures

$\endgroup$
• May 042020
$\begingroup$

### Lukas Brantner

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).

$\endgroup$

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