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.


19 April 2018

Speaker: Dominic Culver, University of Illinois Urbana-Champaign


5 April 2018

Speaker: Hans-Werner Henn, Universite de Strasbourg


22 March 2018

Speaker: TBA


8 March 2018

Speaker: TBA


22 February 2018

Speaker: Drew Heard, Universitaet Hamburg


8 February 2018

Speaker: Sean Tilson, Universitaet Wuppertal


25 January 2018

Speaker: Tyler Lawson, University of Minnesota


11 January 2018

Speaker: Kirsten Wickelgren, Georgia Tech


14 December 2017

Speaker: Teena Gerhardt, Michigan State University


30 November 2017

Cancelled


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.


 

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