Schedule for: 20w5217 - Equivariant Stable Homotopy Theory and p-adic Hodge Theory

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Dylan Wilson has constructed a $C_p$-equivariant space, $\mathbb{CP}^{\infty}_{\mu_p}$, which generalizes the complex conjugation action on projective space. I will explain several different ways of viewing this space, as well as the associated notion of a $\mu_p$-orientation. In particular, I will discuss $\mu_p$-orientations of height $p-1$ Morava $E$-theories, as well as a $\mu_3$-orientation of tmf(2). I will describe how a single element $v_1^{mu_p}$ in the stable homotopy of $CP^{\infty}_{\mu_p}$ determines the $C_p$-action on the homotopy groups of height $p-1$ Morava $E$-theory. This is joint work with Andrew Senger and Dylan Wilson.

An $H_{\infty}$ ring spectrum comes with an $m$th power operation for any positive integer $m$, and this becomes additive only after collapsing a certain transfer ideal. I will discuss the analogous situation equivariantly, both in the case of $G$-spectra and in the global setting. In each case, I will identity precisely the minimal ideal that must be collapsed in order to make the $m$th power operation a map of Mackey functors. I will provide examples, such as the sphere spectrum and complex $K$-theory. This is joint work with Peter Bonventre and Nat Stapleton.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will describe some structural properties of the motivic filtration on topological cyclic homology constructed by Bhatt--Morrow--Scholze. In particular, I will describe an identification of the graded pieces (in low weights integrally and rationally in all weights) with syntomic cohomology, as introduced by Fontaine--Messing and Kato. Joint with Benjamin Antieau, Matthew Morrow, and Thomas Nikolaus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

This talk will be on several related properties of chromatic localizations of algebraic K-theory obtained in joint work with M. Land and G. Tamme. In particular, we give a criterion when a map of ring spectra induces an equivalence in such localizations. This implies several vanishing results and in particular reproves $L_{K(1)}K(Z/p^k) = 0$, a result recently proven by Bhatt--Clausen--Mathew using prismatic cohomology.

The equivariant A-theory G-spectrum for a smooth G-manifold is expected to split off a G-spectrum of h-cobordisms on M analogously to the nonequivariant splitting result of Waldhausen, Rognes and Jahren. I will talk about recent progress toward this conjecture. This is joint work with C. Malkiewich.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

This talk is a report on joint work with Piotr Pstragowski. Our purpose is to argue that, in higher algebra, there exists an intrinsic structure akin to spin that manifest itself through the fact that the homotopy groups of a commutative algebra form a commutative algebra in the symmetric monoidal category of graded abelian groups. The geometry built from such algebras, which we call Dirac geometry, is a natural extension of $\mathbb{G}_m$-equivariant geometry in which half-integer Serre twists exist.

A fundamental problem in 4-dimensional topology is the following geography question: "which simply connected topological 4-manifolds admit a smooth structure?" After the celebrated work of Kirby-Siebenmann, Freedman, and Donaldson, the last uncharted territory of this geography question is the "11/8-Conjecture''. This conjecture, proposed by Matsumoto, states that for any smooth spin 4-manifold, the ratio of its second-Betti number and signature is least 11/8. Furuta proved the ''10/8+2''-Theorem by studying the existence of certain Pin(2)-equivariant stable maps between representation spheres. In this talk, we will present a complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald invariants. In particular, we improve Furuta's result into a ''10/8+4''-Theorem. Furthermore, we show that within the current existing framework, this is the limit. This is joint work with Mike Hopkins, Jianfeng Lin and XiaoLin Danny Shi.

In this talk I will revisit the computation, originally due to Hesselholt and Madsen, of the K-theory of truncated polynomial algebras for perfect fields of positive characteristic. The original proof relied on an understanding of cyclic polytopes in order to determine the genuine equivariant homotopy type of the cyclic bar construction for a suitable monoid. Using the Nikolaus-Scholze framework for topological cyclic homology I achieve the same result using only the homology of said cyclic bar construction, as well as the action of Connes’ operator. Time permitting, I will sketch how to use this method to make new computations of K-theory, in particular for the coordinate axes in affine d-space over perfect fields of positive characteristic. This extends work by Hesselholt in the case d = 2. The analogous results for fields of characteristic zero were found by Geller, Reid and Weibel in 1989. I also extend their computations to base rings which are smooth Q-algebras.

Let $R$ be a perfectoid ring. Hesselholt and Bhatt-Morrow-Scholze have identified the Postnikov filtration on $\mathrm{THH}(R;\mathbb Z_p)$: it is concentrated in even degrees, generated by powers of the Bökstedt generator $\sigma$, generalizing classical Bökstedt periodicity for $R=\mathbb F_p$. We study an equivariant generalization, the \emph{regular slice filtration}, on $\mathrm{THH}(R;\mathbb Z_p)$. The slice filtration is again concentrated in even degrees, generated by $RO(\mathbb T)$-graded classes which can loosely be thought of as \emph{norms} of $\sigma$. The slices themselves are $RO(\mathbb T)$-graded suspensions of certain Mackey functors. When $R$ is $p$-torsionfree, the slice spectral sequence is concentrated in even degrees and collapses on the $E^2$ page.

By work of Voevodsky, the slice filtration in motivic homotopy theory provides a powerful tool for computing homotopy groups of motivic spectra. In this talk, I will describe the motivic slice filtration and the effective slice spectral sequence it produces. Then I will discuss how the motivic effective slice spectral sequence gives rise to a spectral sequence in $C_2$ equivariant homotopy theory. One of the main applications is computing the homotopy groups of $ko_{C_2}$, the connective $C_2$ equivariant $K$ theory. This gives a reproof of work by Guillou-Hill-Isaksen-Ravenel that computes homotopy of $ko_{C_2}$ using equivariant Adams spectral sequence.

Joint with Mike Hill, Kyle Ormsby, Paul Arne Østvær. We perform Hochschild homology calculations in the algebro-geometric setting of mod-2 motivic cohomology in the sense of Suslin and Voevodsky. Via Betti realization this recovers B\”okstedt’s calculation of the topological Hochschild homology of finite prime fields.

The uniqueness of complex $K$-theory as an $E_\infty$ ring spectrum was shown by Baker and Richter in 2005 using obstruction theory. Working rationally, we show for any finite abelian group this extends uniquely to a naive-commutative ring structure for equivariant $K$-theory. The proof involves finding the image of $K$-theory in the algebraic model of Barnes, Greenlees, and Kedziorek given by rational CDGAs with an action of the Weyl group. Despite lacking an explicit description of the CDGAs corresponding to $K$-theory, we compute the homology from the homotopy of the geometric fixed-points and prove formality. This is joint work with Anna Marie Bohmann, Christy Hazel, Jocelyne Ishak, and Magdalena Kedziorek.

I will give a survey of recent work on the C_2-equivariant stable homotopy groups. Topics to be discussed include: the rho-Bockstein spectral sequence, the C_2-equivariant Adams spectral sequence, the equivariant effective spectral sequence, relationships to R-motivic computations, and new methods for computing classical Mahowald invariants. There will be many explicit computational results. I will mention some problems for further study.

We define Witt vectors for non-commutative rings (following Hesselholt) and characteristic polynomials for endomorphism over non-commutative rings. This is the \pi_0-Effekt of a certain refinement of the cyclotomic trace with values in TR. We also explain how to give a 'motivic' picture of TR with coefficients refining work of Lindenstrauss-McCarthy.

In this talk, we will present certain Real-oriented models of Lubin-Tate theories at p=2 and arbitrary heights. For these models, we give explicit formulas for the action of certain finite subgroups of the Morava stabilizer groups on the coefficient rings. This is an input necessary for future computations of the homotopy fixed point spectral sequences for the associated higher real K-theories. The Real orientations will provide information about the differentials. The construction utilizes equivariant formal group laws associated with the norms of the Real bordism theory, and are based on techniques introduced by Hill-Hopkins-Ravenel. This is joint work with Agnes Beaudry, Mike Hill, and Mingcong Zeng.

Recent work of Blumberg-Gerhardt-Hill-Lawson defines Witt vectors for Green functors using an algebraic analogue of topological Hochschild homology relative to a finite subgroup of the circle, described in terms of the Hill-Hopkins-Ravenel norm. In my talk, I will make explicit the perspective that real topological Hochschild homology is the norm from the cyclic group of order two to O(2). It is then natural to construct a theory of Witt vectors for Hermitian Mackey functors. I will then illustrate the computability of this theory with examples. This is based on joint work with T. Gerhardt and M. Hill.

At height $h=2^{n-1}m$, the Morava stabilizer group contains a cyclic group $G$ of order $2^n$. In this talk, I will present equivariant spectra that refine the classical height h Morava $K$-theories. These are obtained from $G$-equivariant models of Lubin-Tate spectra which were constructed in recent joint work with Hill-Shi-Zeng. I will present some preliminary results and conjectures about their slice filtration and equivariant homotopy groups, often focusing on special cases. This is joint work with Hill-Shi-Zeng.

This is joint work with Guchuan Li and Vitaly Lorman. The Johnson--Wilson spectra $E(n)$ play a fundamental role in chromatic homotopy theory. In the late 90's, Ando--Morava--Sadofsky showed that the Tate construction with respect to a trivial $\mathbb{Z}/p$-action on $E(n)$ splits into a wedge of $E(n-1)$'s. I will describe a $C_2$-equivariant lift of this result involving the Real Johnson--Wilson theories $E\mathbb{R}(n)$ studied by Hu--Kriz and Kitchloo--Lorman--Wilson. Our result simultaneously generalizes the work of Ando--Morava--Sadofsky (by taking underlying spectra) and a classical Tate splitting for real topological K-theory proven by Greenlees--May (by taking $C_2$-fixed points). I will outline the proof and highlight an essential tool, the parametrized Tate construction (developed in joint work with Jay Shah), which has other applications relevant to the workshop.

Fundamental work of Segal and Waldhausen gives us two versions of K-theory that produces spectra from certain types of categories. These constructions agree, in the sense that appropriately equivalent categories yield weakly equivalent spectra. In the 2000s, work of Elmendorf--Mandell and Blumberg--Mandell produced more structured versions of Segal and Waldhausen K-theory, respectively. These versions are "multiplicative," in the sense that appropriate notions of pairings of categories yield multiplication-type structure on their resulting spectra. In this talk, I will discuss joint work with Osorno in which we show that these constructions agree as multiplicative versions of K-theory. Consequently, we get comparisons of rings spectra built from these two constructions. Furthermore, the same result also allows for comparisons of related constructions of spectrally-enriched categories.

The topological Hochschild homology $THH(R)$ constitutes a powerful and well-studied invariant of an associative ring $R$. As originally shown by Bokstedt, Hsiang and Madsen, $THH(R)$ admits the elaborate structure of a cyclotomic spectrum, whose formulation depends upon equivariant stable homotopy theory. More recently, inspired by considerations in p-adic Hodge theory, Nikolaus and Scholze demonstrated (under a bounded-below assumption) that the data of a cyclotomic spectrum is entirely captured by a system of circle-equivariant Frobenius maps, one for each prime p. They also give a formula for the topological cyclic homology $TC(R)$ directly from these maps. The purpose of this talk is to extend the work of Nikolaus and Scholze in order to accommodate the study of real topological Hochschild homology $THR$, which is a $C_2$-equivariant refinement of $THH$ defined for an associative ring with an anti-involution, or more generally an $E_\sigma$-algebra in $C_2$-spectra. The key idea is to make use of the $C_2$-parametrized Tate construction. This is joint work with J.D. Quigley and is based on the arXiv preprint 1909.03920.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.