The Electronic Computational Homotopy Theory Seminar is an international research seminar on the topic of computational homotopy theory.  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.The seminar meets on Thursdays at 11:30am in Detroit (Eastern Time).Contact Dan Isaksen (isaksen@wayne.edu) for more information, or to be added to the seminar mailing list.The ECHT calendar lists all scheduled talks.See below for the schedule of talks, in reverse chronological order.You might also be interested in other electronic seminars in mathematics:EMES: Electronic Mathematics Education SeminarHoTTEST: Homotopy Type Theory Electronic Seminar Talks3 May 2018Speaker: Justin Noel, Universitaet Regensburg19 April 2018Speaker: Dominic Culver, University of Illinois Urbana-ChampaignTitle: 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 2018Speaker: Yifei Zhu, Southern University of Science and Technology, ChinaTitle: Toward calculating unstable higher-periodic homotopy typesAbstract: 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.22 March 2018No meeting8 March 2018Speaker: Niko Naumann, Universitaet RegensburgTitle: 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 2018Speaker: Drew Heard, University of HaifaTitle: Picard groups of higher real K-theory spectraAbstract: 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 2018Speaker: Sean Tilson, Universitaet WuppertalTitle: Squaring operations in C_2 and motivic Adams spectral sequencesAbstract: 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 2018Speaker: Tyler Lawson, University of MinnesotaTitle: 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.11 January 2018Canceled14 December 2017Speaker: Teena Gerhardt, Michigan State UniversityTitle: Computational tools for algebraic K-theoryAbstract: 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.30 November 2017Cancelled16 November 2017Speaker: Dan Dugger, University of OregonTitle: 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 2017Speaker: Vitaly Lorman, University of RochesterTitle: 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 2017Speaker: Glen Wilson, University of OsloTitle: 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 2017Speaker: Prasit Bhattacharya, University of VirginiaTitle: 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 2017Speaker: Bogdan Gheorghe, Max Planck InstituteTitle: Tau-obstruction theory and the cooperations of kq/tauAbstract: 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 2017Speaker: Dan Isaksen, Wayne State University; Guozhen Wang, Fudan UniversityTitle: Stable stems - a progress reportAbstract: 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 2017Speaker: Mark Behrens, University of Notre DameTitle: Generalized Adams spectral sequencesAbstract: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 2017Speaker: Nat Stapleton, Universitaet RegensburgTitle: The character of the total power operationAbstract: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 2017Speaker: Oliver Roendigs, Universitaet OsnabrueckTitle: The first and second stable homotopy groups of motivic spheres over a fieldAbstract: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 2017Speaker: Kyle Ormsby, Reed CollegeTitle: Vanishing in motivic stable stemsAbstract: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 2017Speaker: Andrew Salch, Wayne State UniversityTitle: 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 2017Speaker: Bert Guillou, University of KentuckyTitle: 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 2017Speaker: Doug Ravenel, University of RochesterTitle: The Lost Telescope of ZAbstract: 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 2017Speaker: Vesna Stojanoska, UIUCTitle: The Gross-Hopkins duals of higher real K-theory spectraAbstract: 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 2017Speaker: Michael Hill, UCLATitle: Equivariant derivations with applications to slice spectral sequence computationsAbstract: 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 2017Speaker: Lennart Meier, Universitaet BonnTitle: Real spectra and their Anderson dualsAbstract: 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 theReal truncated Brown-Peterson spectra BPR 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 2016Speaker: Agnes Beaudry, University of ColoradoTitle: Duality and K(n)-local Picard groupsAbstract: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. ↑ back to top