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 Ktheory. The seminar meets on Thursdays at 11:30am in Detroit (Eastern Time). In the 20182019 academic year, the meeting link is zoom.us/j/612660457. 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:  ESME: Electronic Seminar on Mathematics Education
 HoTTEST: Homotopy Type Theory Electronic Seminar Talks
 AATRN: Applied Algebraic Topology Research Network
7, 9, 14, 16 May 2018 4hour online minicourse, offered on consecutive Tuesdays and Thursdays at 11:30am in Detroit (Eastern Time). Speaker: Agnes Beaudry, University of Colorado Title: TBA
25 April 2019 Speaker: Mingcong Zeng, University of Rochester
11 April 2019 Speaker: Eva Belmont, Northwestern University
28 March 2019 Speaker: Nick Kuhn, University of Virginia
14 March 2019 Speaker: Thomas Nikolaus, Universität Münster
28 February 2019 Speaker: Martin Frankland, University of Regina
31 January 2019 Speaker: Gabe AngeliniKnoll, Michigan State University Title: Iterated algebraic Ktheory of the integers and Higher LichtenbaumQuillen conjectures Abstract: The LichtenbaumQuillen conjecture (LQC) suggests a relationship between special values of zeta functions and algebraic Ktheory groups. For example, the algebraic Ktheory 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 redshift conjectures of AusoniRognes, which generalize the LQC to higher chromatic heights, I conjecture that the nth Greek letter family is detected in the Hurewicz image of the nth iteration of algebraic Ktheory 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 Ktheory detects information about modular forms.
17 January 2019 Speaker: Lukas Brantner, University of Oxford Title: On the Etheory of Configuration Spaces
Abstract: Given natural numbers n and h, one can investigate the Morava K and Etheory of nfold 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 Etheory of configuration spaces in nmanifolds and, in good cases, identify its E_2 page as the purely algebraic ChevalleyEilenberg 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
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_ooring spectrum (this is similar to the notion of Gcommutative 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 EilenbergMacLane.
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_2CW 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 4manifolds and Pin(2)equivariant Mahowald invariants Abstract: A fundamental problem in 4dimensional topology is the following geography question: "which simply connected topological 4manifolds admit a smooth structure?" After the celebrated work of KirbySiebenmann, Freedman, and Donaldson, the last uncharted territory of this geography question is the "11/8Conjecture''. This conjecture, proposed by Matsumoto, states that for any smooth spin 4manifold, the ratio of its secondBetti 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 celldiagrams, known results on the stable homotopy groups of spheres, and the jbased AtiyahHirzebruch spectral sequence. This is joint work with Mike Hopkins, Jianfeng Lin and XiaoLin Danny Shi.
16, 18, 23, 25 October 2018
This is a special event. The ECHT community is experimenting with a 4hour online minicourse, offered on consecutive Tuesdays and Thursdays at 11:30am in Detroit (Eastern Time).
Speaker: Akhil Mathew, University of Chicago Title: Topological Hochschild homology and its applications
First talk video (Beware that the video quality in the later part of this talk is poor.) Second talk video Second talk notes Third talk (Due to technical difficulties, this file is audio only.) Third talk notes Fourth talk Fourth talk notes
Abstract: Topological Hochschild homology (THH) is an invariant of rings, a variant of ordinary Hochschild homology when the base ring is replaced with the sphere spectrum and all relevant constructions are carried out in spectra rather than chain complexes. The construction THH contains more information than Hochschild homology, and it acquires a rich additional structure called a cyclotomic spectrum. The resulting construction of topological cyclic homology (TC) has been used in many fundamental calculations of algebraic Ktheory. More recently, THH has been used in the work of Bhatt, Morrow, and Scholze to build padic cohomology theories, which provide oneparameter deformations of algebraic de Rham cohomology in mixed characteristic.
In these talks, I will give an introduction to topological Hochschild homology and to the work of BhattMorrowScholze.
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 YetterDrinfeld 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 MilnorMoore and PoincareBirkhoffWitt theorems yield a characterization of primitively generated Hopf algebras which are particularly amenable to cohomology computations (e.g. via Lie algebra cohomology and various Maytype 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 Ktheory of cuspidal singularities. Abstract: Hesselholt has a conjectural calculation of the algebraic Ktheory of k[x,y]/(x^by^a). It has remained a conjecture until now because nobody has been able to prove that a certain S^1equivariant 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 pcomplete C_{p^n}spectra. This lets us use homological data to recognize, up to pcompletion, 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 3space
Abstract: Given four general lines in 3dimensional 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 arithmeticgeometric information about the configuration, by using an Euler class in A1homotopy theory. Because the target of Morel’s degree homomorphism is the GrothendieckWitt 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 2n2 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 UrbanaChampaign 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 higherperiodic 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 nonnegative integer n, we can instead invert certain "v_n selfmaps" and seek algebraic models for the resulting unstable "v_nperiodic" 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 Etheory. 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 Aspectra. This generalizes the case A = Z / pZ due to Balmer and Sanders by establishing (a corrected version of) their log_pconjecture for abelian groups. We work out the consequences for the chromatic type of fixedpoints.
22 February 2018 Speaker: Drew Heard, University of Haifa Title: Picard groups of higher real Ktheory 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 MUdual Steenrod algebra and unstable operations
Abstract: The MUdual Steenrod algebra governs homology and cohomology operations for MUmodules, 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 Ktheory
Abstract: Computational techniques from equivariant stable homotopy theory have been essential to many algebraic Ktheory computations. When studying algebraic Ktheory 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 Ktheory 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/2spaces Abstract: I will talk about some issues that arise in the computation of RO(Z/2)graded Bredon cohomology for Z/2spaces, and some recent progress for the cases of surfaces and Grassmannians.
2 November 2017 Speaker: Vitaly Lorman, University of Rochester Title: Real JohnsonWilson theories and the projective property Abstract: The JohnsonWilson theories E(n) carry an action of C_2 stemming from complex conjugation. Taking fixed points yields the Real JohnsonWilson 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 socalled 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 etainverted sphere spectrum over the rationals
Abstract: Guillou and Isaksen, with input from Andrews and Miller, have calculated the motivic stable homotopy groups of the twocomplete 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 twocomplete etainverted 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_2^1selfmap. 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_3differentials. If time permits, I will explain how the tmfresolution can be used to eliminate the d_3differentials.
21 September 2017 Speaker: Bogdan Gheorghe, Max Planck Institute Title: Tauobstruction theory and the cooperations of kq/tau Abstract: The setting is motivic homotopy theory over Spec C. After pcompleting, 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 (pcompleted) 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 Extgroups (similar to Toda's obstruction theory). We will illustrate this obstruction theory by reconstructing the motivic spectrum kq representing hermitian Ktheory, 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 6293. Our methodology uses motivic techniques to leverage computer calculations of both the Adams and AdamsNovikov E2pages. 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 Ebased Adams Spectral Sequence (EASS) famously has E_2term given by Ext over E_*E if E_*E is flat over E_*. What do you do if this is not the case?? LellmannMahowald, in their analysis of the boASS, had to figure this out. In their case, the E_1 term decomposed into a v_1periodic summand and an EilenbergMacLane summand. They completely computed the cohomology of the v_1periodic summand, and left Don Davis to use a computer to attack the EilenbergMacLane 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 EilenbergMacLane 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 tmfASS.
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 Etheories are E_\inftyring 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 1line and the 2line of stable homotopy groups of the motivic sphere spectrum via Milnor Ktheory, motivic cohomology, and hermitian Ktheory. 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 RöndigsSpitzweckØstvær sharpens the connection between the slice and Novikov spectral sequences. Using classical vanishing lines for the E_2page of the AdamsNovikov spectral sequence and the work of AndrewsMiller on the alpha_1periodic ANSS, I will deduce some new vanishing theorems in the bigraded homotopy groups of the etacomplete motivic sphere spectrum. In particular, I will show that the mth etacomplete MilnorWitt stem is bounded above (by an explicit piecewise linear function) when m = 1 or 2 mod 4, and then lift this result to integral MilnorWitt 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 heightshifting spectral sequence Abstract: I will explain how to use formal groups with complex multiplication to assemble the cohomology of largeheight Morava stabilizer groups out of the cohomology of smallheight Morava stabilizer groups, using a new "heightshifting 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 Lfunctions, generalizing Adams' 1966 description of im J in terms of denominators of special values of the Riemann zetafunction. 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 GrossHopkins duals of higher real Ktheory spectra
Abstract: The HopkinsMahowald higher real Ktheory spectra are generalizations of real Ktheory; 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 Ktheory spectra at prime p and height p1 are GrossHopkins selfdual with shift (p1)^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 C2spectra built from Real bordism MR and BPR. This includes Atiyah's kR and the Real truncated BrownPeterson spectra BPR<n> and the Real JohnsonWilson spectra ER(n). We will recall how to calculate the RO(C2)graded homotopy groups of these C2spectra. 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 nontrivial elements at height n=2. Then I will describe how the relationship between SpanierWhitehead and BrownComenetz duality could be used to prove the nontriviality of exotic Picard groups at higher chromatic heights.
