eCHT Research Seminar

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

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

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: C2-equivariant stable stems

Abstract: I will explain how to compute C2-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 G3, which is not quite a compact Lie group but looks like a 2-completion of one. The smallest currently known nontrivial homotopy representation of G3 has dimension 246.
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 Bökstedt’s calculation of the topological Hochschild homology of finite prime fields.

27 February 2020

Speaker: Inbar Klang, Columbia University
Title: Computing Cn-relative topological Hochschild homology

Abstract: Let Cn denote the cyclic group of order n. Given a Cn-ring spectrum, Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell defined its Cn-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 Cn-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 EO-orientation order of complex vector bundles

Abstract: Let p be any prime. When n = (p-1)k, the n-th Morava stabilizer group contains C_p as a subgroup. EO_{n,p}, the homotopy fixed point of height n prime p Morava E-theory E_{n,p}, can be thought of as the higher height analogues of real K-theory KO. Indeed, E_{1,2} is 2-completed KU and EO_{1,2} is 2-completed KO. It is a classical result that the tautological line bundle \gamma_1 over CP^\infty is not KO-orientable, but 2 \gamma_1 is, meaning the KO-orientation order of \gamma_1 is 2. In this talk, we generalize this result to show that the EO_{n,p}-orientation order of \gamma_1 divides o(n, p) = p^{p^{k} -1}. Consequently, we conclude that the o(n,p)-fold direct sum of any complex vector bundle is EO_{n,p}-orientable. This is joint work with Hood Chatham.

16 January 2020

Speaker: Oscar Randal-Williams, University of Cambridge
Title: E_k-algebras and homological stability

Presentation slides

Abstract: Many sequences of moduli spaces may be combined to form E_k-algebras, and in recent and ongoing work with S. Galatius and A. Kupers we have shown that analysing cellular models for these in the category of E_k-algebras can lead to new insights about the homological stability of moduli spaces, and about new phenomena “beyond homological stability”. The analysis typically has two steps. In the first step one proves that the derived E_k-algebra indecomposables (= topological Quillen homology) vanishes in a certain range, meaning that the E_k-algebra may be constructed only using cells of controlled dimensions. In the second step one explicitly analyses the situation in low degrees, to extract finer information about low-dimensional cells and their attaching maps. I will explain this general picture, but focus on the computational problems which arise in the last step, when studying cellular E_k-algebras with few cells.

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 En with its C2-action by the formal inverse. Then, I will talk about the slice spectral sequence of a C4-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 E4 with C4-action induced from the Goerss-Hopkins-Miller theorem. In particular, our computation shows that E4hC4 is 128-periodic, and E4hC12 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

Abstract: Computations in RO(G)-graded Bredon cohomology can be challenging and are not well understood even for G = C_2, the cyclic group of order two. I will present a structure theorem for RO(C_2)-graded cohomology with constant Z/2 coefficients that substantially simplifies computations. The structure theorem says the cohomology of any finite C_2-CW complex decomposes as a direct sum of two basic pieces: cohomologies of representation spheres and cohomologies of spheres with the antipodal action. I will sketch the proof, which depends on a Toda bracket calculation, and give some examples. Further work toward a structure theorem for RO(C_p)-graded cohomology with constant Z/p coefficients again requires two types of spheres, as well as a new space that is not a sphere at all.


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.

This is joint work with Mike Hopkins, Jianfeng Lin and XiaoLin Danny Shi.

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.

Abstract: Hesselholt has a conjectural calculation of the algebraic K-theory of k[x,y]/(x^b-y^a). It has remained a conjecture until now because nobody has been able to prove that a certain S^1-equivariant space that comes up in the calculation is built from representation spheres in a specified way. I will explain how to sidestep this issue by computing the Picard group of the category of p-complete C_{p^n}-spectra. This lets us use homological data to recognize, up to p-completion, when a C_{p^n}-spectrum looks like a virtual representation sphere.

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

Abstract: I will survey some joint work on nilpotence and periodicity in equivariant stable homotopy theory. I will discuss applications to conceptual and computational problems. Time permitting, I will then try to discuss a few related open questions.

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

Abstract: The Picard group of the category of spectra is known to contain only suspensions of the sphere spectrum. Working K(n)-locally, however, the story is much richer. For a finite subgroup K of the Morava stabilizer group, there is a homotopy fixed point spectrum E_n^{hK} which is an approximation to the K(n)-local sphere. We compute the Picard groups of these spectra when n = p – 1, showing that they are always cyclic. Joint work with Akhil Mathew and Vesna Stojanoska.

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

Abstract: The setting is motivic homotopy theory over Spec C. After p-completing, the Tate twist originating in the motivic mod p cohomology of a point lifts to an element \tau in the stable homotopy groups of the (p-completed) motivic sphere. Inverting this element recovers classical homotopy theory, while killing it produces a homotopy theory that is equivalent to the (algebraic) derived category of the Hopf algebroid BP_* BP. One can use this element tau to formulate an obstruction theory to construct motivic spectra which starts in the algebraic category, and with obstructions in algebraic Ext-groups (similar to Toda’s obstruction theory). We will illustrate this obstruction theory by reconstructing the motivic spectrum kq representing hermitian K-theory, and by also computing the cooperations of kq/tau along the way, which proves to be similar but easier to the classical computation for kO.

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

Abstract: The E-based Adams Spectral Sequence (E-ASS) famously has E_2-term given by Ext over E_*E if E_*E is flat over E_*. What do you do if this is not the case?? Lellmann-Mahowald, in their analysis of the bo-ASS, had to figure this out. In their case, the E_1 term decomposed into a v_1-periodic summand and an Eilenberg-MacLane summand. They completely computed the cohomology of the v_1-periodic summand, and left Don Davis to use a computer to attack the Eilenberg-MacLane summand (which petered out around the 20 stem). I will discuss a new technique, joint with Agnes Beaudry, Prasit Bhattacharya, Dominic Culver, and Zhouli Xu, which instead computes the Eilenberg-MacLane summand in terms of Ext over the Steenrod algebra (and thus is much more robust). This technique applies whenever such a decomposition occurs, and I will discuss applications to the BP<2>-ASS and the tmf-ASS.

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

Abstract: Recent work of Röndigs-Spitzweck-Østvær sharpens the connection between the slice and Novikov spectral sequences. Using classical vanishing lines for the E_2-page of the Adams-Novikov spectral sequence and the work of Andrews-Miller on the alpha_1-periodic ANSS, I will deduce some new vanishing theorems in the bigraded homotopy groups of the eta-complete motivic sphere spectrum. In particular, I will show that the m-th eta-complete Milnor-Witt stem is bounded above (by an explicit piecewise linear function) when m = 1 or 2 mod 4, and then lift this result to integral Milnor-Witt stems (where an additional constraint on m appears). This is joint work with Oliver Röndigs and Paul Arne Østvær.

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

Abstract: I’ll talk about a genuine equivariant notion of a derivation which not only takes products to sums but also takes norms to transfers. This arises naturally from genuine equivariant multiplicative filtrations, like the slice filtration, and gives some techniques for producing differentials. As an application, I’ll discuss in some detail the slice spectral sequence for a $C_4$-analogue of $BP\langle 1\rangle 1$, the simplified $C_4$ version of the spectrum used in the solution of the Kervaire invariant one problem.

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.