Zygotop seminar

A proper abstract will be forthcoming, but I'll be talking about Morse (co)homology, in particular the Morse \(A_\infty\)-category of a closed manifold equipped with a Riemannian metric, and how this is basically realised as (part of) a particular Fukaya category. I hope to give an interesting introduction to the basic ideas to an audience not particularly invested in the smooth world.

I will give a brief review of THH and prismatic cohomology, then explain how to recover prismatization and related stacks from THH, with an eye towards recent work by Hahn-Levy-Senger and others.

I will begin by explaining the title of this talk, i.e., I'll provide some background on the usual telescope conjecture and explain how one might be led to ask whether or not various tensor triangulated categories satisfy the "tensor telescope conjecture". Then, following Hall and Rydh, I'll sketch a proof of the fact that for a large class of algebraic stacks X, \(D_{qc}(X)\) satisfies the triangular telescope conjecture.

We will explain the role of formal groups in p-Hodge theory and p-adic Galois representations. We will provide some background on p-adic Hodge theory.

We will introduce power operations. Then, we will consider the case of Morava E-theory and relate these to isogenies of formal groups.

In 2018, Pstrągowski and Gheorghe-Isaksen-Krause-Ricka independently introduced categories of what have come to be known as synthetic spectra. These can be thought of as a categorification of the Adams-Novikov spectral sequence, and further have the surprising feature that they are equivalent (up to \(p\)-completion) with a certain cellular subcategory of the \(\mathbb{C}\)-motivic stable homotopy category. This makes synthetic spectra an interesting middle ground between topology and algebraic geometry, the existence of which has powered much novel work on both sides.

In this talk, I will give an introduction to synthetic spectra, highlighting some examples, applications and the role complex cobordism MU and its motivic counterpart MGL play in the story. Time permitting, I will then discuss some ongoing work, joint with Lucas Piessevaux, attempting to generalize this story to the equivariant setting.

A classical heuristic is that the Hochschild cohomology of a space is closely related to the geometry of its free loop space. (E.g., if X is simply connected, then the HH(CX) is equivalent to H(LX).) This slogan, like most other things, becomes more powerful and unwieldy if we instead consider THH. I will talk about a "baby" example and its role in the disproof of the telescope conjecture [BHLS23]. To be precise, I will analyze a certain fiber measuring the difference between the THH of cochains and the cochains of the free loop space, acknowledging (in order, time permitting) the higher commutative algebra, T-equivariant, and cyclotomic structures.

The goal of this talk is to give a proof of the Segal conjecture for \(BP\langle n \rangle\), focusing on \(THH(\mathbb{Z})/p\) and \(THH(\ell)/(p, v_1)\). Concretely, we're trying to understand how the cyclotomic Frobenius acts on \(THH(BP\langle n \rangle)\). To do so, we exploit the Adams filtration on \(BP\langle n \rangle\) to reduce to the case of graded polynomial \(\mathbb{F}_p\)-algebras, and analyse how the cyclotomic Frobenius behaves there. This is a practice talk in advance of the Talbot workshop next week.

This talk was held at 12 noon in SC 530.

Let G be a connected compact Lie group. The geometric Langlands program relates the topology of various objects built from G to the algebraic geometry of various objects built from the Langlands dual group \(G^\vee\) of G. When applied to classical G, this generalizes many classical and beautiful results having to do with the homotopy theory of based loop spaces. I will give an introduction to these ideas by means of explicit examples, and in particular discuss how simple phenomena on one side of Langlands duality amount to deep phenomena on the other side.

This talk took place 2-4pm on Thursday in SC 232.

For an \(H_\infty\) ring spectrum E, we show that p-typical complex orientations on E are not \(H_\infty\) unless E is K(1)-acyclic. By a result of Hahn, this means that E is K(n)-acyclic for all \(n \geq 1\). The proof uses a result of Ando-Hopkins-Rezk characterizing \(\mathbb{E}_\infty\)-maps from MU to K-theory and a bunch of combinatorics and number theory. This is joint work with Andy Senger.

Let \(n>1\) be an integer, and consider the following questions: Can the sphere \(S^{n-1}\) be made into an H-space? Does \(S^{n-1}\) have a parallelisable differentiable structure? Can \(\mathbb{R}^n\) be made into a real division algebra? Is there an element of Hopf invariant one in \(\pi_{2n-1}(S^n)\)? The Hopf invariant one problem more or less asks for which n the answer to each of these questions is yes. In fact, all of these questions are equivalent, and the resolution (due to Adams in 1958) is that these things are only possible when n = 2, 4, or 8. In this talk, we will first discuss some of the more classical implications relating these four properties, and then present Adams-Atiyah's extraordinarily simple proof via complex topological K-theory that only n = 2, 4, 8 work.

This talk will cover polynomial functoriality of the K-theory space as introduced by Barwick, Glasman, Mathew, and Nikolaus, and will use this additional functoriality to deduce Bökstedt's result on the topological hochschild homology of \(\mathbb{F}_p\). From this we will recover the calculation of the dual Steenrod algebra together with its Dyer-Lashof operations, "without doing topology."

I will introduce a notion of a \(-n\) connective heart category which allows one to extend many tools for studying module categories of connective ring spectra to the nonconnective setting. Examples of \(-n\) heart categories include coherent sheaves on a quasi-affine qc derived scheme, perfect modules over certain fixed points of connective ring spectra by group actions, and perfect module categories of bounded below localizations of connective ring spectra.

I will explain why computing a localizing invariant of a \(-n\) connective heart category is naturally equivalent to computing it for the perfect module category of a connective ring. In particular, this makes the algebraic K-theory accessible via trace methods, and allows for other tools to be imported from the connective setting. This contains work joint with Vova Sosnilo.

This talk was held at the special time and place of 2-4pm in SC 530.

Formal groups are purely algebraic objects, which historically arose out of considerations in number theory and algebraic geometry. However, in the 1960s, Quillen proved a surprising theorem, proving a link between the theory of formal groups and the topological theory of complex orientations. This was the starting point of what is known today as chromatic homotopy theory, which has proven to be a very fruitful way to understanding the homotopy theory of spaces.

Given how successful this line of study has been, it is natural to ask whether we can emulate these techniques to study other 'homotopy theories.' In my talk, I would like to focus on the case of equivariant homotopy theory, which studies spaces with a group action. My goal is to provide some background motivating the equivariant analogue of formal groups, and mention some results (old and new) which prove that these are indeed closely linked to an equivariant theory of complex orientations.

I hope to keep the talk relatively beginner-friendly; it will help to be somewhat familiar with the notion of a spectrum, but I will try to avoid anything too technical and provide background as needed. All are welcome!

We define the category of \(G\)-operads and the hierarchy of generalized \(\mathcal{N}_\infty\)-operads, which are \(G\)-suboperads of \(\text{Comm}^\otimes_G\). We exhibit an isomorphism between the category of generalized \(\mathcal{N}_\infty\)-operads and the self-join poset $$\text{Op}_G^{GN\infty} \simeq \text{Ind}-\text{Sys}_G \star \text{Ind}-\text{Sys}_G, $$ where \(\text{Ind}-\text{Sys}_G\) is the poset of indexing systems in \(G\). This recognizes generalized \(\mathcal{N}_\infty\)-operads as parametrizing some commutative multiplicative transfers and possibly a commutative multiplication. Indeed, their algebras in semiadditive Cartesian categories are incomplete Mackey functors and their algebras in Mackey functors recover incomplete Tambara functors when they are \(\mathcal{N}_\infty\)-operads, i.e. when they contain \(\mathbb{E}_\infty\).

After this, we discuss some in-progress research. Namely, we construct a Boardman-Vogt tensor product of \(G\)-operads and demonstrate that tensor products of generalized \(\mathcal{N}_\infty\)-operads correspond with joins in \(\text{Ind}-\text{Sys}_G \star \text{Ind}-\text{Sys}_G\), i.e. there is an \(\mathcal{N}_{(I \vee J)\infty}\)-monoidal equivalence $$ \textbf{Alg}_{\mathcal{N}_{I\infty}} \textbf{Alg}_{\mathcal{N}_{J\infty}} \mathcal{C} \simeq \textbf{Alg}_{\mathcal{N}_{(I \vee J)\infty}} \mathcal{C} $$ for all \(\mathcal{N}_{(I \vee J)\infty}\)-monoidal categories \(\mathcal{C}\), allowing \(G\)-commutative structures to be constructed “one norm at a time.”

In this talk, we will go over the recent result by Ramzi-Sosnilo-Winges that every spectrum admits a categorification, i.e. that there is a section to the functor \(K : Cat^{st}_\infty \to Sp\) sending a category to its nonconnective \(K\)-theory spectrum. If time permits, we will see an application of this result to disprove the conjectured "nonconnective theorem of the heart."

We introduce the category of \(G\)-Mackey functors, along with a functor \(\Sigma_{\underline{\rho}}^{\infty}:\mathscr{C}_G(\mathcal{C}) \rightarrow \mathscr{M}_G(\mathcal{C}) \) from coefficient systems to Mackey functors. Using isotropy separation, we construct an equivalence $$ \left(\Sigma_{\overline{\rho}}^\infty X \right)^G \simeq \bigoplus_{H \in \mathrm{Conj}(G)} \Sigma^{\infty} X^H_{hW_G H} $$ and derive as a corollary the classical computation $$ A(G) \simeq \pi_0^G(\mathbb{S}),$$ where \(A(G) \simeq \mathbb{Z}[\mathrm{Conj}(G)]\) is the Burnside ring.

The Hecke algebras for all symmetric groups taken together form a braided monoidal category that controls all quantum link invariants of type A and, by extension, the standard canon of topological quantum field theories in dimension 3 and 4. Here we provide the first categorification of this Hecke braided monoidal category, which takes the form of an \(\mathbb{E}_2\)-monoidal \((\infty,2)\)-category whose hom-\((\infty,1)\)-categories are \(k\)-linear, stable, idempotent-complete, and equipped with \(\mathbb{Z}\)-actions. This categorification is designed to control homotopy-coherent link homology theories and to-be-constructed topological quantum field theories in dimension 4 and 5.

Our construction is based on chain complexes of Soergel bimodules, with monoidal structure given by parabolic induction and braiding implemented by Rouquier complexes, all modelled homotopy-coherently. This is part of a framework which allows to transfer the toolkit of the categorification literature into the realm of \(\infty\)-categories and higher algebra. Along the way, we develop families of factorization systems for \((\infty,n)\)-categories, enriched \(\infty\)-categories, and \(\infty\)-operads, which may be of independent interest.

Over a fixed field \(\mathbb{F}\), a classical problem asks for what values of \(r\), \(s\), and \(n\) do there exist identities of the form $$\left(\sum_{i = 1}^r x_{i}^2 \right) \cdot \left(\sum_{i = 1}^s y_i^2 \right) = \sum_{i = 1}^n z_i^2,$$ where the \(z_i\)'s are bilinear expressions in \(x_i\) and \(y_i\). A simple necessary criterion was given by Hopf for \(F = \mathbb{R}\) via a clever little argument using singular cohomology, but for a while the question remained of whether this criterion holds over fields of characteristic \(p > 0\). This question is answered in the affirmative, using an analogous argument in motivic cohomology. In my talk, we will take a stroll through the landscape of this problem, highlighting some features of motivic cohomology along the way, and see how they lead us to the resolution of this problem.

We will define Hochschild homology for ordinary rings and show how to generalize this to THH for ring spectra. We will say some things about THH, following the first few sections of Krause-Nikolaus

I will sketch the physical POV and the mathematical definition of the map \(\mathrm{TMF} \rightarrow \mathrm{KO}((q))\), and thereby attempt to convince you that ideas from physics can predict interesting mathematical results. I'll introduce the necessary physics and assume familiarity with TMF only up to the current point in Stephen's topics course.

Goodwillie originally proved that for an associative \(\mathbb{Q}\)-algebra \(\)R and nilpotent ideal \(I\) that the relative theory \(K(R,I)\) is equivalent to \(HC^-(R,I)\) via the Goodwillie-Jones trace map. The Dundas-Goodwillie-McCarthy theorem allows for general associative rings, but at the cost of replacing cyclic homology with topological cyclic homology. We will discuss a variant, originally due to Beilinson and refined by Antieau-Mathew-Morrow-Nikolaus, which works for commutative rings \(R\) henselian along \((p)\) and the ideal \(I=(p)\). This still allows us to compute in terms of cyclic homology.

We'll then talk about an application of this result towards the \(p\)-adic variational Hodge conjecture. This generalizes a result of Bloch–Esnault–Kerz.

We give an introduction to algebraic K-theory, beginning with definitions and building up basic and not-so-basic properties. We will touch on connective vs nonconnective K-theory, explain why it is difficult to compute, and survey some recent advances. We will be motivated by the computation of \(K(\mathbb{F}_q)\), but do some sightseeing along the way.

A lot is known about additive splittings of (p-local) Thom spectra associated to connective covers of \(BO\times\mathbb{Z}\). After giving an overview of those results I want to showcase what is known about multiplicative splittings, focusing on the (p=2, ht=1)-local spin bordism spectrum. Expect a lot of fun algebra and geometry! Time permitting, I might speculate about (p=2, ht=2)-local string bordisms.

The Dundas-Goodwillie-McCarthy theorem states that for connective \(\mathbb{E}_1\)-algebras, the fiber of the cyclotomic trace (between K-theory and topological cyclic homology) is an equivalence on nilpotent extensions. This is one of the most important tools in the theory of trace methods.

In this talk, we will see a proof of this theorem using modern language, following notes of Sam Raskin. We will reduce to showing that the Goodwillie derivatives of K-theory and TC are equivalent via the cyclotomic trace. We will then show this equivalence by finding that each coincides with THH (up to a shift). This will involve some soft arguments about K-theory and some explicit computations of TC of square-zero extensions.

We recall the definition of the cotangent complex for augmented \(\mathcal{O}\)-algebras and develop the \(\mathcal{O}\)-algebra lift for the postnikov tower obstruction theory. From there, we define TAQ cohomology and note that Postnikov obstructions for \(\mathcal{O}\)-algebras land in \(\mathcal{O}\)-TAQ cohomology groups.

We use this obstruction theory to construct a unique \(\mathbb{E}_4\)-structure on Brown Peterson spectra; this comes down to inductively computing \(\mathbb{E}_4\)-TAQ cohomology in a range for the Postnikov truncations \(\mathrm{BP}[n]\), so we demonstrate how perform this computation using four layers of the bar spectral sequence, in which we leverage Steinberger's computation of Dyer-Lashof operations on \(\mathbb{F}_p\). Time-permitting, we'll show that this \(\mathbb{E}_4\) structure is unique, that it splits via an \(\mathbb{E}_4\) \(\mathrm{MU}\)-idempotent, and that the underlying unique \(\mathbb{E}_2\) structure splits via a lift of the Quillen idempotent.

The Geometric Langlands conjecture predicts that we have an equivalence of DG Categories $$D-\mathrm{Mod}(\mathrm{Bun}_G) = \mathrm{IndCoh}_\mathrm{Nilp}\left(\mathrm{LocSys}_{G^{\vee}}\right).$$ Since for reasonable stacks (including \(\mathrm{LocSys}\)), we have \(\mathrm{IndCoh}_\mathrm{Nilp} = \mathrm{Ind}(\mathrm{Coh}_\mathrm{Nilp})\), we should expect that (and it is the case that) \(D-\mathrm{Mod}(\mathrm{Bun}_G)\) is compactly generated. We will discuss Drinfeld and Gaitsgory's proof of this fact.

\(\mathbb{E}_n\) algebras are associative algebras with various levels of commutativity. They play a crucial in many parts of mathematics, from homotopy theory to topological field theories. In this talk, we are motivated in the following question: how to explicitly construct \(\mathbb{E}_n\) algebras in a \(m\)-categories?

While natural, \(\mathbb{E}_n\) have infinite coherence and are hard to construct by hands. However, by Eckmann-Hilton argument, many of the coherences are redundant. In this talk, we will use a generalization of the EH argument to derive a ''minimal'' construction of En algebras. Along the way we will use the interplay of arity-restriction and connectedness for operads, as well as a general Blakers-Massey theorem for operadic algebras.

This talk will give an overview of the Land-Tamme papers "On the \(K\)-theory of pullbacks" and "On the \(K\)-theory of pushouts", focusing on the recent second paper. We will state and prove their main results, namely that we can measure the failure of excision for localising invariants (think \(K\)-theory) via the \(\odot\)-construction. The key result of the second paper is that this \(\odot\)-construction can sometimes be expressed as a pushout in \(\mathrm{Alg}(k)\), making it computable in many situations of interest. The talk will finish with a tour of some applications.

Computation of the mod \((p,v_1,v_2)\) syntomic cohomology of \(\ell\) using the even filtration, and an application to redshift.

We will start by discussing various definitions of Thom spectra and how they relate to one another. We will then show the equivalence of the two statements of Hopkins-Mahowald before discussing the proof of the theorem, which uses Steinberger's computation of the Dyer-Lashof operations on the dual Steenrod algebra. If time permits, we will see how this computation relates to Bökstedt's computation of \(\operatorname{THH}(\mathbb{F}_p)\).

We will begin by computing the \(\mathbb{F}_2\) cohomology of \(\mathrm{ko}\), which turns out to be \(H^*(\mathrm{ko}) \simeq \mathcal{A} \otimes_{\mathcal{A}(1)} \mathbb{F_2} \), for \(\mathcal{A}(1) = \mathcal{A}\langle \mathrm{Sq}^1, \mathrm{Sq}^2\rangle\) a finitely generated Steenrod subalgebra. By the free-forgetful adjunction, this implies that the \(E_2\)-page for the \(\mathrm{ko}\)-ASSeq is \(\mathrm{Ext}^{s,t}_{\mathcal{A}(1)} (\mathbb{F}_2,\mathbb{F}_2)\).

Using this, we will characterize this \(E_2\)-page using a minimal \(\mathcal{A}(1)\)-resolution and a periodic \(\mathcal{A}(1)\)-resolution. After a short argument, we see that the \(\mathrm{ko}\)-ASSeq collapses at the \(E_2\) page. Along the way, we'll perform the computation pictured on the shirt from Paul Goerss's retirement conference.