The electronic Computational Homotopy Theory Seminar is an online international research seminar. Topics include any part of homotopy theory that has a computational flavor, including but not limited to stable homotopy theory, unstable homotopy theory, chromatic homotopy theory, equivariant homotopy theory, motivic homotopy theory, and K-theory.

In the 2020-2021 academic year, the seminar will meet about once per month. The format is two 45-minute talks separated by a 30-minute coffee break. The seminar meets on Thursdays at 10:15am-12:15pm in Detroit (Eastern Time).

Here is the meeting link for the 2020-2021 academic year. The meeting ID is 914 2383 0339, and the password, if needed, is “eCHT”.

The current organizers are Dan Isaksen (Wayne State University), J.D. Quigley (Cornell University), and Hana Kong (University of Chicago). Contact any of us for more information. Guchuan Li (Northwestern University) was an organizer in 2018-2019.

Some of the previous talks are available on the eCHT Youtube channel.

Please subscribe to our mailing list to be kept informed of all eCHT news and events.

See below for the schedule of talks, in reverse chronological order.

**Next meeting: 22 October
**

Speakers: Tom Bachmann (LMU Munich) and Irina Bobkova (Texas A&M)

**19 November 2020 at 11:30
**

Speaker: Foling Zou, University of Michigan

**19 November 2020 at 10:30
**

Speaker: Paul Arne Østvær, University of Oslo

**22 October 2020 at 11:30
**

Speaker: Irina Bobkova, Texas A&M University

Title: Invertible E_2^{hC_4}-modules

Abstract: For Morava E-theory E_n and a finite subgroup F of the Morava stabilizer group, the homotopy fixed points spectrum E_n^hF is periodic and the Picard group of the category of modules over the ring spectrum E_n^hF contains the cyclic subgroup generated by Sigma E_n^hF. In most known examples, the Picard group is found to be precisely this cyclic group. However, at n = p = 2, the Picard group of the category of E_2^{hC_4}-modules is not cyclic and contains an extra element of order 2. I will describe the tools we use to compute this Picard group: a group homomorphism from RO(C_4) to it and the Picard spectral sequence. This talk is based on joint work with Agnes Beaudry, Mike Hill and Vesna Stojanoska.

**22 October 2020 at 10:15
**

Speaker: Tom Bachmann, LMU Munich

Title: The eta-periodic motivic image of j spectrum over fields

Abstract: (joint work with Mike Hopkins) Let ko and ksp denote the very effective motivic spectra corresponding to orthogonal and symplectic K-theory, respectively. One may show that the Fasel-Haution Adams operation Psi^3: ko_(2) -> ko_(2) lifts to Sigma^4,2 ksp_(2); we denote by jo its fiber. In this talk I will explain how to show that after inverting the motivic Hopf element eta, jo just becomes the (periodized, 2-local) sphere spectrum. This determines in particular all the eta-periodic motivic stable stems.

**24 September 2020 at 11:30
**

Speaker: Hood Chatham, UCLA

Title: Snaith Constructions on Wilson Spaces

Abstract:

Snaith showed that complex K theory can be constructed from CP^\infty by taking the suspension spectrum and inverting a “Bott class”. We generalize this construction and build a new Landweber exact complex oriented E_\infty ring spectrum R_h for each prime p and height h. Unlike prior arbitrary height ring spectra, R_h is both simple to construct and has simple homotopy groups. We relate R_h to work of Peterson and Westerland constructing a fixed point spectrum of Morava E theory from K(Z, h+1). We use this connection to investigate when there exist structured ring maps from R_h to Morava E theories. This depends on the arithmetic of the formal group. For instance, there are structured ring maps from R_2 to a Morava E theory associated to a supersingular elliptic curve but not to general Morava E theories. It is our hope to identify some collection of Morava E theories oriented by R_3 as “geometric” E theories, analogous to complex K theory at height 1 and Elliptic E theories at height 2. This work is joint with Jeremy Hahn and Allen Yuan.

**24 September 2020 at 10:15
**

Speaker: Stefan Schwede, Universität Bonn

Title: Splittings of global Mackey functors and regularity of equivariant Euler classes

Reference: arxiv.org/abs/2006.0943

Abstract: In 1962, Dold published an elegant proof of Nakaoka’s splitting of the cohomology of symmetric groups. Dold’s proof only uses formal aspects of group cohomology, and I’ll explain that his argument is way more general and provides a splitting of the values of global Mackey functors at symmetric groups. The relevant kind of global Mackey functor is also known as `inflation functors’, or `biset functors with restriction, transfers and inflations’ (but possibly no deflations).

Equivariant homotopy theory provides an even more general kind of global Mackey functor with values at all compact Lie groups. The main result of the talk is an analog of Dold’s splitting for the values of these `global functors’ at orthogonal, unitary and symplectic groups. As a consequence of these splittings, certain long exact sequences of equivariant homotopy groups decompose into short exact seqeunces. This in turn implies that the Euler class of the tautological U(n)-representation in homotopical equivariant bordism is a non-zero divisor.

**16 April 2020
**

Speaker: Mark Behrens, University of Notre Dame

Title: C_{2}-equivariant stable stems

Abstract: I will explain how to compute C_{2}-equivariant stable stems by using Mahowald’s “metastable homotopy theory” charts and Mahowald invariants.

**26 March 2020**

Speaker: Tilman Bauer, Kungliga Tekniska Högskolan (KTH)

Title: Homotopy representations of Lie groups and 2-compact groups

Abstract: Abstract: Let G be a compact Lie group and p a prime. A (p-complete, complex, n-dimensional) homotopy representation of G is a homotopy class BG → BU(n)^{^}_{p}. Every genuine representation G → U(n) gives rise to a homotopy representation, but this association is neither surjective nor injective unless G is an extension of a finite p-group by a torus (a so-called p-toral group). In fact, homotopy representations are controlled by representations of such p-toral subgroups of G. Every compact Lie group has a maximal p-toral subgroup, which is a generalization of a Sylow subgroup of a finite group, and a representation of this Sylow subgroup extends to one of G if certain invariance conditions are fulfilled and a lifting problem for diagrams in the homotopy category is solvable.

In my talk, I will show how to construct homotopy representations computationally. Particular focus will be on the Dwyer-Wilkerson 2-compact group G_{3}, which is not quite a compact Lie group but looks like a 2-completion of one. The smallest currently known nontrivial homotopy representation of G_{3} has dimension 2^{46}.

I will not assume any knowledge about p-compact groups, p-local finite groups or fusion systems from the audience.

**12 March 2020
**

Speaker: Bjørn Dundas, University of Bergen

Title: Motivic Hochschild homology

Abstract: 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*ö*kstedt’s calculation of the topological Hochschild homology of finite prime fields.

**27 February 2020
**

Speaker: Inbar Klang, Columbia University

Title: Computing C_{n}-relative topological Hochschild homology

Abstract: Let C_{n} denote the cyclic group of order n. Given a C_{n}-ring spectrum, Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell defined its C_{n}-relative topological Hochschild homology. Just as Hochschild homology is an algebraic approximation to topological Hochschild homology, this has an algebraic approximation in the form of Hochschild homology for Green functors, defined by Blumberg, Gerhardt, Hill, and Lawson. I will introduce these concepts and discuss joint work with Adamyk, Gerhardt, Hess, and Kong in which we develop computational tools for C_{n}-relative topological Hochschild homology.

**13 February 2020**

Speaker: Guchuan Li, University of Copenhagen

Title: Blue shift for real oriented cohomology theories

Abstract: This is joint work with Vitaly Lorman and James D. Quigley. The ℤ/p-Tate cohomology spectrum of the n’th Johnson–Wilson theory splits as a wedge of (n-1)’st Johnson–Wilson theories (after completion). We construct a C2-equivariant lifting of this splitting for Real Johnson–Wilson theories. The C2-fixed points of this splitting is a higher height analogue to Davis and Mahowald’s splitting of the Tate cohomology spectrum of ko as a wedge of Hℤ.

**30 January 2020**

Speaker: Prasit Bhattacharya, University of Virginia

Title: On the -orientation order of complex vector bundles

**16 January 2020**

Speaker: Oscar Randal-Williams, University of Cambridge

Title: E_k-algebras and homological stability

**31 October 2019**

Speaker: XiaoLin Danny Shi, University of Chicago

Title: Real Orientations of Lubin-Tate Spectra and the Slice Spectral Sequences of Height 4 Lubin-Tate theories

Abstract: We show that Lubin-Tate spectra at the prime 2 are Real oriented and Real Landweber exact. The proof is an application of the Goerss-Hopkins-Miller theorem to algebras with involution. For each height n, we compute the entire homotopy fixed point spectral sequence for E_{n} with its C_{2}-action by the formal inverse. Then, I will talk about the slice spectral sequence of a C_{4}-equivariant spectrum. This spectrum is a variant of the detection spectrum of Hill-Hopkins-Ravenel. After periodization and K(4)-localization, this spectrum is equivalent to a height 4 Lubin-Tate theory E_{4} with C_{4}-action induced from the Goerss-Hopkins-Miller theorem. In particular, our computation shows that E_{4}^{hC4} is 128-periodic, and E_{4}^{hC12} is 384-periodic. This talk contains joint work with Agnès Beaudry, Hood Chatham, Jeremy Hahn, Mike Hill, Hana Jia Kong, Guozhen Wang, Zhouli Xu, and Mingcong Zeng.

**3 October 2019**

Speaker: Doug Ravenel, University of Rochester

Title: The eight fold way: how to build the right model structure on orthogonal G-spectra

Abstract: This is joint work with Mike Hill and Mike Hopkins. First I will recall what the category orthogonal G-spectra is. Then I will list three attributes that a model structure on it needs to be the right one, meaning one that enables us to prove the Kervaire invariant theorem. The most naive model structure, the projective one, has none of these attributes. For each of them I will describe a construction that will give a new model structure that has it. These commute with each other and can thus be done in any order, yielding seven new model structures including the one we need.

**19 September 2019**

Speaker: Achim Krause, Universität Münster

Title: Bökstedt periodicity and quotients of DVRs

Abstract: Bökstedt periodicity refers to Bökstedt’s classical computation of topological Hochschild homology of finite fields. I will discuss a recent project with Nikolaus on computations of THH based on a relative version of Bökstedt periodicity. Our main applications are quotients of discrete valuation rings, generalizing Brun’s results for Z/p^n.

**5 September 2019**

Speaker: Jay Shah, University of Notre Dame

Title: C_2-equivariant stable homotopy from real motivic stable homotopy

Lecture notes

Abstract: In this talk, I will describe how the infinity-category of p-complete C_2-equivariant spectra embeds fully faithfully into cellular real motivic spectra via the right adjoint to C_2-Betti realization. `Decategorifying’ this statement, I will then describe a procedure that computes C_2-equivariant stable homotopy groups given knowledge of related real motivic stable homotopy groups, which tend to be simpler algebraically. This is joint work with Mark Behrens and is based on the arXiv preprint 1908.08378.

**25 April 2019**

Speaker: Mingcong Zeng, Utrecht University

Title: Real cobordism, its norms and the dual Steenrod algebra

Abstract: The real cobordism spectrum MU_R and its norms play a central role in the proof of the non-existence of classes of Kervaire invariant one by Hill, Hopkins and Ravenel. However, these spectra are still very mysterious and their equivariant homotopy groups are difficult to compute.

In this talk I will focus on the norm of real cobordism into C_4, and draw a connection between it and the dual Steenrod algebra spectrum HF_2 \smash HF_2 with C_2-action by conjugation. Then I will discuss how computations on both sides help each other.

This is joint work with Lennart Meier.

**11 April 2019**

Speaker: Eva Belmont, Northwestern University

Title: The R-motivic Adams spectral sequence and the Mahowald invariant

Abstract: Motivic homotopy theory over R is interesting in part because of its connection to ordinary stable homotopy theory and to C2-equivariant homotopy theory. In this talk I will review some of these connections, and discuss work in progress with Dan Isaksen to compute R-motivic stable homotopy groups of spheres using an Adams spectral sequence. One of our main applications is to a variant of the Mahowald invariant which can be computed using knowledge of the R-motivic Adams spectral sequence.

**28 March 2019**

Speaker: Nick Kuhn, University of Virginia

Title: How to use the representation theory of the symmetric groups to handcraft finite spectra

Abstract: We review and expand upon Jeff Smith’s use of the modular representation theory of the symmetric groups to construct useful functors from finite spectra to finite spectra. Here `useful’ means that when the input is `pretty good’, the output might be `very good’. Examples include the construction of a finite spectrum whose mod 2 cohomology is free over A(3) on a single generator. (A(3) is the 1024 dimensional subalgebra of the Steenrod algebra generated by Sq^1, Sq^2, Sq^4, and Sq^8.)

**14 March 2019**

Speaker: Thomas Nikolaus, Universität Münster

Title: Topological periodic homology as non-commutative crystalline cohomology

Abstract: We explain the basic setup of non-commutative geometry (following Kontsevich) and how topological periodic homology can be considered as a Weil cohomology theory in this setting. The specific results that we present are joint work with A. Mathew and B. Antieau.

**28 February 2019**

Speaker: Martin Frankland, University of Regina

Title: The DG-category of secondary cohomology operations

Abstract: In joint work with Hans-Joachim Baues, we study track categories (i.e., groupoid-enriched categories) endowed with additive structure similar to that of a 1-truncated DG-category, except that composition is not assumed right linear. We show that if such a track category is right linear up to suitably coherent correction tracks, then it is weakly equivalent to a 1-truncated DG-category. This generalizes work of Baues on the strictification of secondary cohomology operations. As an application, we show that the secondary integral Steenrod algebra is strictifiable.

In the talk, I will present this structural result and how it relates to some computational problems.

**31 January 2019**

Speaker: Gabe Angelini-Knoll, Michigan State University

Title: Iterated algebraic K-theory of the integers and Higher Lichtenbaum-Quillen conjectures

Abstract: The Lichtenbaum-Quillen conjecture (LQC) suggests a relationship between special values of zeta functions and algebraic K-theory groups. For example, the algebraic K-theory of the integers encodes special values of the Riemann zeta function. These special values are known to correspond to the Hurewicz image of the alpha family in the homotopy groups of spheres. Inspired by the red-shift conjectures of Ausoni-Rognes, which generalize the LQC to higher chromatic heights, I conjecture that the n-th Greek letter family is detected in the Hurewicz image of the n-th iteration of algebraic K-theory of the integers. In my talk, I will sketch a proof of this conjecture in the case n=2 using the theory of trace methods. Consequently, by work of Behrens, iterated algebraic K-theory detects information about modular forms.

**17 January 2019**

Speaker: Lukas Brantner, University of Oxford

Title: On the E-theory of Configuration Spaces

Abstract: Given natural numbers n and h, one can investigate the Morava K- and E-theory of n-fold loop spaces at height h. Partial computations have been carried out by Langsetmo, Ravenel, Tamaki, and Yamaguchi, but their techniques either rely on phenomena specific to height h=1 or become increasingly intractable as the number n of loops grows large.

In joint work with Knudsen and Hahn, we introduce a new computational technique whose difficulty is uniform in n. More precisely, we exhibit a spectral sequence converging to the E-theory of configuration spaces in n-manifolds and, in good cases, identify its E_2 page as the purely algebraic Chevalley-Eilenberg complex of a Hecke Lie algebra. We illustrate the tractability of our approach by performing several new computations.

**29 November 2018**

Speaker: Tom Bachmann, MIT

Title: Power operations in normed motivic spectra

Abstract: In joint work with M. Hoyois, we established (the beginnings of) a theory of “normed motivic spectra”. These are motivic spectra with some extra structure, enhancing the standard notion of a motivic E_oo-ring spectrum (this is similar to the notion of G-commutative ring spectra in equivariant stable homotopy theory). It was clear from the beginning that the homotopy groups of such normed spectra afford interesting *power operations*. In ongoing joint work with E. Elmanto and J. Heller, we attempt to establish a theory of these operations and exploit them calculationally. I will report on this, and more specifically on our proof of a weak motivic analog of the following classical result of Würgler: any (homotopy) ring spectrum with 2=0 is generalized Eilenberg-MacLane.

**15 November 2018**

Speaker: Clover May, UCLA

Title: Some structure theorems for RO(G)-graded cohomology

**1 November 2018**

Speaker: Zhouli Xu, MIT

Title: The intersection form of spin 4-manifolds and Pin(2)-equivariant Mahowald invariants

Abstract: 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 of powers of certain Euler classes in the RO(Pin(2))-graded equivariant stable homotopy groups of spheres. 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. For the proof, we use the technique of cell-diagrams, known results on the stable homotopy groups of spheres, and the j-based Atiyah-Hirzebruch spectral sequence.

**4 October 2018**

Speaker: Craig Westerland, University of Minnesota

Title: Structure theory for braided Hopf algebras and their cohomology

Abstract: Braided Hopf algebras are Hopf algebra objects in a braided monodical category (e.g., the category of Yetter-Drinfeld modules). Computation of their cohomology can be closely related to computations of the cohomology of the braid groups with certain families of coefficients. When working in the category of graded vector spaces, particularly over a field of characteristic zero, the Milnor-Moore and Poincare-Birkhoff-Witt theorems yield a characterization of primitively generated Hopf algebras which are particularly amenable to cohomology computations (e.g. via Lie algebra cohomology and various May-type spectral sequences). In the genuinely braided (and not symmetric) setting, very little of this structure theory carries over. The purpose of this work is to develop some of that machinery, which will be phrased in the language of braided operads. While still very much in progress, this is already elucidating some cohomology computations.

**20 September 2018**

Speaker: Vigleik Angeltveit, Australian National University

Title: Picard groups and the algebraic K-theory of cuspidal singularities.

**6 September 2018**

Speaker: Kristen Wickelgren, Georgia Institute of Technology

Title: An arithmetic count of the lines through 4 lines in 3-space

Abstract: Given four general lines in 3-dimensional space, it is a classical result that the number of lines intersecting all four is two, provided you allow the coefficients of the lines to be complex numbers. Over a general field k, say with characteristic not 2, and for example the real numbers, the two lines may be a conjugate pair over a quadratic extension of the field. We give a count of the lines weighted by their field of definition and arithmetic-geometric information about the configuration, by using an Euler class in A1-homotopy theory. Because the target of Morel’s degree homomorphism is the Grothendieck-Witt group GW(k) of quadratic forms over the field, this count takes the form of an equality in GW(k). More generally, we give such a count for the lines intersecting 2n-2 codimension 2 hyperplanes in P^n for n odd. This is joint work with Padmavathi Srinivasan, building on joint work with Jesse Kass.

**3 May 2018
**

Speaker: Justin Noel, Universitaet Regensburg

Title: Nilpotence and periodicity in equivariant stable homotopy theory

**19 April 2018
**

Speaker: Dominic Culver, University of Illinois Urbana-Champaign

Title: BP<2>-cooperations

Abstract: In this talk, I will describe two aspects of the BP<2>-cooperations algebra. I will begin with general structural results about BP<2>-cooperations. The second part of the talk will be concerned with an inductive method for computing a large portion of the cooperations algebra.

**5 April 2018
**

Speaker: Yifei Zhu, Southern University of Science and Technology, China

Title: Toward calculating unstable higher-periodic homotopy types

Abstract: The rational homotopy theory of Quillen and Sullivan identifies homotopy types of topological spaces with differential graded commutative (co)algebras, and with differential graded Lie algebras, after inverting primes. Given any non-negative integer n, we can instead invert certain “v_n self-maps” and seek algebraic models for the resulting unstable “v_n-periodic” homotopy types. I’ll explain why this is a natural and useful generalization of the classical story, and how a version of it has been achieved through Goodwillie calculus in recent work of Behrens and Rezk. I’ll then explain my work on its applications to calculating unstable homotopy types in the case of n = 2. A key tool is power operations in Morava E-theory. Time permitting, I’ll report further joint work in progress with Guozhen Wang.

**8 March 2018
**

Speaker: Niko Naumann, Universitaet Regensburg

Title: The Balmer spectrum of the equivariant homotopy category of a finite abelian group

Abstract: For a finite abelian group A, we determine the Balmer spectrum of the compact objects in genuine A-spectra. This generalizes the case A = Z / pZ due to Balmer and Sanders by establishing (a corrected version of) their log_p-conjecture for abelian groups. We work out the consequences for the chromatic type of fixed-points.

**22 February 2018**

Speaker: Drew Heard, University of Haifa

Title: Picard groups of higher real K-theory spectra

**8 February 2018**

Speaker: Sean Tilson, Universitaet Wuppertal

Title: Squaring operations in C_2 and motivic Adams spectral sequences

Abstract: Great strides were made in the computability of differentials in the classical Adams spectral sequence by Bruner. He developed a technique for computing differentials in terms of algebraic power operations on the E_2 page. These differentials can be viewed as a failure of the operations to commute with the differentials. We will present similar results for permanent cycles in the RO(C_2)-graded equivariant and Spec(\R) motivic Adams spectral sequences. We will focus on the moving parts of such machinery in the hopes that it can be adapted to other situations.

**25 January 2018**

Speaker: Tyler Lawson, University of Minnesota

Title: The MU-dual Steenrod algebra and unstable operations

Abstract: The MU-dual Steenrod algebra governs homology and cohomology operations for MU-modules, and it has a power operation structure with a number of useful applications. In this talk I’ll discuss the use of unstable homotopy theory to determine power operations that are difficult to access stably.

**14 December 2017**

Speaker: Teena Gerhardt, Michigan State University

Title: Computational tools for algebraic K-theory

Abstract: Computational techniques from equivariant stable homotopy theory have been essential to many algebraic K-theory computations. When studying algebraic K-theory of pointed monoid algebras, such as group rings or truncated polynomials, RO(S^1)-graded equivariant homotopy groups can arise. In this talk I will give an overview of the computational tools used to study the algebraic K-theory of pointed monoid algebras, and discuss some of the recent successes of these methods.

**16 November 2017**

Speaker: Dan Dugger, University of Oregon

Title: Some Bredon cohomology calculations for Z/2-spaces

Abstract: I will talk about some issues that arise in the computation of RO(Z/2)-graded Bredon cohomology for Z/2-spaces, and some recent progress for the cases of surfaces and Grassmannians.

**2 November 2017**

Speaker: Vitaly Lorman, University of Rochester

Title: Real Johnson-Wilson theories and the projective property

Abstract: The Johnson-Wilson theories E(n) carry an action of C_2 stemming from complex conjugation. Taking fixed points yields the Real Johnson-Wilson theories, ER(n). To begin, I will survey their properties and motivate why they are interesting cohomology theories to study. I will then describe a result, joint with Kitchloo and Wilson, that presents the ER(n)-cohomology of many familiar spaces (including connective covers of BO and half of the Eilenberg MacLane spaces) as a base change of their (known) E(n)-cohomology. A key ingredient in the proof is a computation of the equivariant E(n) (or MR) cohomology of spaces with the so-called projective property. This result is interesting in its own right, as, for instance, it gives us access to certain equivariant unstable cohomology operations. If time permits, I will conclude with a brief description of a potential application to the immersion problem for real projective spaces.

**19 October 2017**

Speaker: Glen Wilson, University of Oslo

Title: The eta-inverted sphere spectrum over the rationals

Abstract: Guillou and Isaksen, with input from Andrews and Miller, have calculated the motivic stable homotopy groups of the two-complete sphere spectrum after inverting multiplication by the Hopf map eta over the fields R and C. We will review these known results and show how to calculate the motivic stable homotopy groups of the two-complete eta-inverted sphere spectrum over fields of cohomological dimension at most two with characteristic different from 2 and the field of rational numbers.

**5 October 2017**

Speaker: Prasit Bhattacharya, University of Virginia

Title: Computing K(2)-local homotopy groups of a type 2 spectrum Z in $\widetilde{\mathcal{Z}}$

Abstract: $\widetilde{\mathcal{Z}}$ is a class of type 2 spectra that was introduced recently by myself and Philip Egger. Any Z in $\widetilde{\mathcal{Z}}$ admits a v_2^1-self-map. In joint work with Egger, we use the duality spectral sequence to compute the E_2 page of the descent spectral sequence for any Z in $\widetilde{\mathcal{Z}}$. In fact, the duality spectral sequence is the easy part of the computation. The hard part is to show that (E_2)_0 Z is isomorphic to F_4 [Q_8]. In this talk, I will highlight how this computation is carried out. The descent spectral sequence has potential d_3-differentials. If time permits, I will explain how the tmf-resolution can be used to eliminate the d_3-differentials.

**21 September 2017**

Speaker: Bogdan Gheorghe, Max Planck Institute

Title: Tau-obstruction theory and the cooperations of kq/tau

**7 September 2017**

Speaker: Dan Isaksen, Wayne State University; Guozhen Wang, Fudan University

Title: Stable stems – a progress report

Abstract: In the past year, Guozhen Wang, Zhouli Xu, and I have computed approximately thirty new stable homotopy groups, in dimensions 62-93. Our methodology uses motivic techniques to leverage computer calculations of both the Adams and Adams-Novikov E2-pages. I will describe our computational approach, and I will point out some interesting phenomena in the stable stems that we have uncovered. Guozhen Wang will also present some information about our computer code.

**1 June 2017**

Speaker: Mark Behrens, University of Notre Dame

Title: Generalized Adams spectral sequences

**18 May 2017**

Speaker: Nat Stapleton, Universitaet Regensburg

Title: The character of the total power operation

Abstract: In the 90’s Goerss, Hopkins, and Miller proved that the Morava E-theories are E_\infty-ring spectra in a unique way. Since then several people including Ando, Hopkins, Strickland, and Rezk have worked on explaining the affect of this structure on the homotopy groups of the spectrum. In this talk, I will present joint work with Barthel that shows how a form of character theory due to Hopkins, Kuhn, and Ravenel can be used to reduce this problem to a combination of combinatorics and the GL_n(Q_p)-action on the Drinfeld ring of full level structures which shows up in the local Langlands correspondence.

**4 May 2017**

Speaker: Oliver Roendigs, Universitaet Osnabrueck

Title: The first and second stable homotopy groups of motivic spheres over a field

Abstract: The talk will report on joint work (partly in progress) with Markus Spitzweck and Paul Arne Ostvaer. This work describes the 1-line and the 2-line of stable homotopy groups of the motivic sphere spectrum via Milnor K-theory, motivic cohomology, and hermitian K-theory. The main computational tool is Voevodsky’s slice spectral sequence.

**20 April 2017**

Speaker: Kyle Ormsby, Reed College

Title: Vanishing in motivic stable stems

**13 April 2017**

Speaker: Andrew Salch, Wayne State University

Title: Special values and the height-shifting spectral sequence

Abstract: I will explain how to use formal groups with complex multiplication to assemble the cohomology of large-height Morava stabilizer groups out of the cohomology of small-height Morava stabilizer groups, using a new “height-shifting spectral sequence.” I will describe some new computations which have been made possible by this technique, and also one of the main motivations for making computations in this way: this approach is very natural for someone who is trying to give a description of orders of stable homotopy groups of Bousfield localizations of finite spectra in terms of special values of L-functions, generalizing Adams’ 1966 description of im J in terms of denominators of special values of the Riemann zeta-function. I will explain, as much as time allows, both positive and negative results in that direction.

**23 March 2017**

Speaker: Bert Guillou, University of Kentucky

Title: From motivic to equivariant homotopy groups – a worked example

Abstract: The realization of a motivic space defined over the reals inherits an action of Z/2Z, the Galois group. This realization functor allows for information to pass back and forth between the motivic and equivariant worlds. I will discuss one example: an equivariant Adams spectral sequence computation for ko, taking the simpler motivic computation as input. This is joint work with M. Hill, D. Isaksen, and D. Ravenel.

**9 March 2017**

Speaker: Doug Ravenel, University of Rochester

Title: The Lost Telescope of Z

Abstract: I will describe a possible equivariant approach to the Telescope Conjecture at the prime 2. It uses the triple loop space approach described in a paper by Mahowald, Shick and myself of 20 years ago. The telescope we studied there is closely related to the geometric fixed point spectrum of a telescope with contractible underlying spectrum.

**2 March 2017**

Speaker: Vesna Stojanoska, UIUC

Title: The Gross-Hopkins duals of higher real K-theory spectra

Abstract: The Hopkins-Mahowald higher real K-theory spectra are generalizations of real K-theory; they are ring spectra which give some insight into higher chromatic levels while also being computable. This will be a talk based on joint work with Barthel and Beaudry, in which we compute that higher real K-theory spectra at prime p and height p-1 are Gross-Hopkins self-dual with shift (p-1)^2. We expect this will allow us to detect exotic invertible K(n)-local spectra.

**16 February 2017**

Speaker: Michael Hill, UCLA

Title: Equivariant derivations with applications to slice spectral sequence computations

**19 January 2017**

Speaker: Lennart Meier, Universitaet Bonn

Title: Real spectra and their Anderson duals

Abstract: Real spectra will be for us a loose term denoting C2-spectra built from Real bordism MR and BPR. This includes Atiyah’s kR and the Real truncated Brown-Peterson spectra BPR<n> and the Real Johnson-Wilson spectra ER(n). We will recall how to calculate the RO(C2)-graded homotopy groups of these C2-spectra. Then we will see how these exhibit a hidden duality, which can be explained by the computation of Anderson duals.

**15 December 2016**

Speaker: Agnes Beaudry, University of Colorado

Title: Duality and K(n)-local Picard groups

Abstract: I will discuss the different types of exotic elements in the K(n)-local Picard group and methods for producing non-trivial elements at height n=2. Then I will describe how the relationship between Spanier-Whitehead and Brown-Comenetz duality could be used to prove the non-triviality of exotic Picard groups at higher chromatic heights.