electronic Computational Homotopy TheoryThe 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 2024-2025 academic year, the seminar will meet about once per month. During Fall 2024, the seminar meets on Thursdays from 10:00 am-11:45 am Eastern Time. The format is two 45-minute talks separated by a 15-minute coffee break.
For the 2024-2025 academic year, the Zoom meeting ID is 97882184731. Please contact an organizer for the password.
For some talks, a reading group will meet in advance to prepare.
The current organizers are Dan Isaksen (Wayne State University), J.D. Quigley (University of Virginia), and Jack Carlisle (Notre Dame). Contact any of us for more information. The graduate student assistants are Scotty Tilton (UCSD) and Hassan Abdallah (Wayne State University).
From previous years, talk recordings are available on 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.
See you in 2025 for the next iteration of the eCHT Research Seminar!
5 December 2024 at 11:00 AM (Eastern US time)
Yutao Liu (University of Washington)
Title: The generalized Tate diagram and slice spectral sequences
Abstract: The generalized Tate diagram developed by Greenlees and May is a fundamental tool in equivariant homotopy theory. In this talk, I will discuss an integration of the generalized Tate diagram with the equivariant slice filtration of Hill-Hopkins-Ravenel, resulting in a generalized Tate diagram for equivariant spectral sequences. This new diagram provides valuable insights into various equivariant spectral sequences and sets up relations among them, which motivate more computations and structural results in the slice spectral sequence.
As an application, we will study the slice spectral sequences of periodic Hill-Hopkins-Ravenel and Lubin-Tate theories. Building upon the work of Meier-Shi-Zeng, we will establish shearing isomorphisms, explore transchromatic phenomena, and analyze vanishing lines in the negative cone of the slice spectral sequences. Finally, we utilize our analysis to identify different duality phenomena between the two cones and explain how these phenomena can be applied in computations. This is joint work with XiaoLin Danny Shi and Guoqi Yan.
5 December 2024 at 10:00 AM (Eastern US time)
Hana Jia Kong (Zhejiang University)
Title: Motivic Milnor formulas
Abstract: In the latest research, Lin, Wang, and Xu have advanced the Adams computation of stable homotopy groups using computer programs. Developing these algorithms requires a deep understanding of the Steenrod algebra. While Milnor’s work provides a foundation in the classical case, the motivic case is much more complex. In this talk, I’ll share my work with Weinan Lin on the motivic formulas. This is part of a larger project with Lin, Wang, and Xu, aiming to compute real motivic stable stems over a large range with the assistance of computers.
7 November 2024 at 11:00 AM (Eastern US time)
Ishan Levy (University of Copenhagen)
Title: The stable homology of Hurwitz spaces and Cohen–Lenstra moments for functions fields
Abstract: The Cohen–Lenstra heuristics predict the distribution of the odd part of class groups of quadratic fields, and are one of the driving conjectures in arithmetic statistics. I will explain work with Aaron Landesman, where we compute the moments of the Cohen–Lenstra distribution for function fields, when the size of the finite field is sufficiently large (depending on the moment). We follow an approach to this problem due to Ellenberg–Venkatesh–Westerland, and the key new input is the computation of the stable homology of Hurwitz spaces associated to certain conjugacy classes in generalized dihedral groups. I will explain the ideas in our computation of the stable homology in the case of the group S_3 with conjugacy class transpositions.
7 November 2024 at 10:00 AM (Eastern US time)
Ben Spitz (Virginia)
Title: Algebraically Closed Tambara Functors
In equivariant stable homotopy theory, objects called “Tambara functors” play the role of commutative rings. Tambara functors are abstract algebraic objects: they consist of sets with certain operations satisfying certain axioms; however, the theory of Tambara functors is much less developed than the theory of commutative rings, in part because it is not clear exactly how to define the “Tambara analogues” of many classical notions.
In their proof of the Redshift Conjecture, Burklund-Schlank-Yuan introduced the notion of a “Nullstellensatzian object” in a category C. When C is the category of commutative rings, this recovers precisely the notion of an algebraically closed field — we therefore define an “algebraically closed Tambara functor” to be a Nullstellensatzian object in the category of Tambara functors.
In this talk, I will present a classification of algebraically closed G-Tambara functors (for any finite group G). As a corollary, we reduce the K-theory of algebraically closed Tambara functors to the ordinary K-theory of algebraically closed fields. This is joint work with Jason Schuchardt and Noah Wisdom.
10 October 2024 at 11:00 AM (Eastern US time)
Liam Keenan (Brown University)
Title: Revisiting some THH computations
Abstract: By work of Dundas, Goodwillie, and McCarthy, we know that topological cyclic homology and algebraic K-theory agree on an “infinitesimal neighborhood” of a ring spectrum. Key to this result is an understanding of the THH of a trivial square zero extension, which decomposes into a sum whose terms are given by cyclic analogues of THH with coefficients. These cyclic variants were originally constructed by Lindenstrauss–McCarthy and were recently revisited in work by Krause–McCandless–Nikolaus. I will explain a general construction associating to any ring spectrum A and non-unital A-algebra I, a cyclotomic spectrum T(A,I) which naturally recovers the aforementioned decomposition, and some other known THH computations.
10 October 2024 at 10:00 AM (Eastern US time)
Liz Tatum (University of Bonn)
Reading group meets on 8 October
Title: Applications of Equivariant Brown-Gitler Spectra
Abstract: In the 1980s, Mahowald and Kane used integral Brown-Gitler
spectra to construct splittings of the cooperations algebras for ko,
connective real k-theory, and ku, connective complex k-theory. These
splittings helped make it feasible to do computations using the ko- and
ku-based Adams spectral sequences.
In recent work, Guchuan Li, Sarah Petersen, and I have constructed models
for C_2-equivariant analogues of the integral Brown-Gitler spectra. In
this talk, I will report on our progress towards using these spectra to
construct C_2-equivariant analogues of these splittings, and related
results.
12 September 2024 at 11:00 AM (Eastern US time)
Nikolay Konovalov (University of Chicago)
Title: Algebraic Goodwillie spectral sequence
Abstract: I will discuss two tools for computing homotopy groups of a space:
Goodwillie and Adams spectral sequences. The first one is more
fundamental, however less known and less understood. Vice versa, the
second one is less fundamental, but far more investigated. I will also
talk about how these two spectral sequences interact with each other.
12 September 2024 at 10:00 AM (Eastern US time)
Lennart Meier (University of Utrecht)
Reading group meets on 10 September
Title: Duality in Equivariant Elliptic Cohomology
Abstract: Equivariant elliptic cohomology is in many ways a higher analogue of equivariant K-theory. One difference though is that G-equivariant elliptic cohomology of a point is finite for all compact Lie groups G and not just for finite groups. We will give an introductiont to equivariant elliptic cohomology and equivariant topological modular forms and then explain how to obtain this property and how to compute the dual of these modules.
25 April 2024 at 11:00 AM (Eastern US time)
Benjamin Antieau (Northwestern)
Title: Binomial rings and stacks in topology.
Abstract: I will explain my approach to the recovery of the homotopy type of a space X from its derived binomial ring of integer-valued chains C*(X,Z). My approach is very similar to independent work of Horel and Kubrak—Shuklin—Zakharov, and I will explain the relationship. This talk will start in characteristic p, use \delta-rings over Z_p, and glue them together to obtain integral information. It will be aimed at a general homotopy theory audience.
25 April 2024 at 10:00 AM (Eastern US time)
David Chan (Michigan State)
Title: A spherical group ring model for equivariant A-theory
Abstract: The equivariant A-theory spectrum, constructed by Malkiewich—Merling, can be used to encode geometric data about G-spaces such as Wall’s finiteness obstruction and Whitehead torsion. In this talk we discuss a new construction of equivariant A-theory which provides an interpretation of the algebraic K-theory of G-spaces as the K-theory of a genuine G-ring spectrum. As an application, we explain how this construction can be used in conjunction with a version of the Dennis trace to recover a classical splitting theorem of Waldhausen in this context. This is joint work with Maxine Calle, Anish Chedalavada, and Andres Mejia.
28 March 2024 at 11:00 AM (Eastern US time)
Andrew Senger (Harvard)
Reading group meets on 27 March
Title: On the K-theory of Z/p^n
Abstract: I will explain how to compute the mod (p,v_1 ^{p^n}) algebraic K-theory of Z/p^{n+2}, using a novel crystallinity property of mod (p,v_1 ^{p^n}) syntomic cohomology. This is joint work with Jeremy Hahn and Ishan Levy.
28 March 2024 at 10:00 AM (Eastern US time)
Ayelet Lindenstrauss (Indiana)
Title: Loday Constructions on Spheres and Tori
Abstract: (All joint work with Birgit Richter, and parts also with others as detailed during the talk)
Given a commutative ring or ring spectrum, there is a functor from finite sets to commutative rings sending a set to the tensor product of copies of the ring (or ring spectrum) indexed by that set. The Loday construction extends this to simplicial sets. The simplest nontrivial example is the Loday construction over a circle, which gives Hochschild homology (or, respectively, topological Hochschild homology).
I will discuss Loday constructions on higher spheres and on higher tori. The calculations for spheres are inductive, using the splitting of an n-sphere into two hemispheres, joined along the equator (n-1)-sphere. The calculations for tori which we know are for cases where the Loday construction on the torus is equivalent to the Loday construction on the wedge of spheres whose suspension is homotopy equivalent to the suspension of the torus. This kind of stability is unexpected, definitely not generally true, and yet there are surprisingly many cases where it holds.
29 February 2024 at 11:00 AM (Eastern US time)
Mark Behrens (Notre Dame)
Reading group meets on 28 February
Title: F_p-synthetic bo resolutions
Abstract: I will run the classical Mahowald bo-resolution story in the F_p-synthetic category. Why would you possibly want to do this? The F_p-synthetic homotopy groups encode the mod p Adams spectral sequence. Mahowald used bo-resolutions to compute the v1-periodic 2-primary stable homotopy groups of spheres. I will use the F_p-synthetic bo-resolution to compute the v1-local Adams spectral sequence converging to these p-primary v1-periodic stable homotopy groups of spheres. This amounts to the determination of a huge amount of differentials which kill most of v_1-periodic Ext, leaving just the image of J. Specifically I will discuss:
1) odd primary sphere – this recovers the results of Michael Andrews’ MIT thesis
2) 2-primary tmf – amusingly, Ext_A(2) is entirely v1-periodic – so the determination of the differentials in the v1-periodic Adams SS for tmf basically determine the differentials in the ASS for tmf
3) 2-primary sphere – Surprisingly, as far as I know, v_1 periodic Ext_A is not completely known at p = 2 – but modulo this difficulty I will discuss the v1-periodic ASS for the 2-primary sphere, which basically amounts to understanding the mod 2 ASS above the slope 1/5 line.
The subject matter of this talk is essentially an elaboration of the techniques of Burkland-Hahn-Senger and dovetails nicely with recent work of Carrick-Davies
29 February 2024 at 10:00 AM (Eastern US time)
Eva Belmont (Case Western Reserve University)
Title: A deformation of Borel-complete equivariant homotopy theory
Abstract: Synthetic homotopy theory is a general framework for constructing interesting contexts for doing homotopy theory: using the data of a spectral sequence in some category C, one can construct another category which can be viewed as a deformation of C. The motivating example is the fact, due to Gheorghe-Wang-Xu, that (p-complete, cellular) $\mathbb{C}$-motivic homotopy theory can be described as a deformation of the ordinary stable homotopy category, simply using the data of the Adams-Novikov spectral sequence. Burklund, Hahn, and Senger used this framework to study $\mathbb{R}$-motivic homotopy theory as a deformation of $C_2$-equivariant homotopy theory. In joint work with Gabriel Angelini-Knoll, Mark Behrens, and Hana Jia Kong, we give (up to completion) a different synthetic description of this deformation, which generalizes to give a deformation of (Borel-complete) G-equivariant homotopy theory for other groups G.
1 February 2024 at 11:00 AM (Eastern US time)
Morgan Opie (UCLA)
Reading group meets on 31 January
Title: Applications of higher real K-theory to vector bundle enumeration (Video)
Abstract:
The zeroeth complex topological K-theory of a space encodes complex vector bundles up to stabilization. Since complex topological K-theory is highly computable, this is a great place to start when asking bundle-theoretic questions. However, in general, there are many non-equivalent bundles that represent the same class in $K_0$. Bridging the gap between K-theory and actual bundles is challenging even for the simplest CW complexes.
For example, given a fixed rank r, the number of rank r bundles on $\mathbb CP^n$ that are stably trivial is, in general, unknown. In this talk, we give lower bounds for the number of rank r bundles on $\mathbb CP^n$ in infinitely many cases. For example, we can show that there are at least $p$ rank $p-1$, stably-trivial bundles on $\mathbb CP^{2p-1}$ for all primes $p$.
Our story will start with classical results of Atiyah and Rees for rank 2 bundles on $\mathbb CP^3$, before taking a detour through Weiss calculus. Building on work of Hu, we will use Weiss-theoretic techniques in tandem with a little chromatic homotopy theory to translate bundle enumeration to a computation of the higher real K-theory of particular simple spectra. The result will involve actual numbers! This is joint work with Hood Chatham and Yang Hu.
1 February 2024 at 10:00 AM (Eastern US time)
Tyler Lawson (Minnesota)
Title: Filtrations and spectral sequences for structured algebras (Video)
Abstract: The Adams spectral sequence comes from an inverse filtration of the sphere spectrum, and its method of construction gives it a particular interaction with structured multiplicative operations. In this talk I’ll discuss this structure, as well as techniques for producing variant filtrations that are geared towards giving higher or lower visibility to certain operations.
30 Nov 2023 at 11:00 (Eastern US Time)
Thomas Brazelton, Harvard
Title: Equivariant Enumerative Geometry
Abstract: Classical enumerative geometry asks geometric questions of the form “how many?” and expects an integral answer. For example, how many circles can we draw tangent to a given three? How many lines lie on a cubic surface? The fact that these answers are well-defined integers, independent upon the initial parameters of the problem, is Schubert’s principle of conservation of number. In this talk we will outline a program of “equivariant enumerative geometry”, which wields equivariant homotopy theory to explore enumerative questions in the presence of symmetry. Our main result is equivariant conservation of number, which states roughly that the orbits of solutions to an equivariant enumerative problem are conserved. We leverage this to compute the S4 orbits of the 27 lines on any smooth symmetric cubic surface. We will pay particular focus to the role that Pontyagin-Thom transfers play in this story, and discuss dualizability in the setting of parametrized spectra.
30 Nov 2023 at 10:00 (Eastern US Time)
Foling Zou, Chinese Academy of Science
Title: Unital operads, monoids and monads
Abstract: Operads have played an important role in both topology and algebra. It is well known that operads may be viewed as monoids in symmetric sequences. In topology, it is often sensible to work with unital operads and their (reduced) monads. I will discuss a variant of symmetric sequences monoids in which give unital operads. This is joint work with Peter May and Ruoqi Zhang.
2 Nov 2023 at 11:00 (Eastern US Time)
Agnès Beaudry, Colorado
Title: The  "RO(\Pi)")
-bundle over the fixed point set are all isomorphic representations, then there is a Thom isomorphism in  "RO(C_2)")
-graded Bredon cohomology with coefficients in the constant 
Mackey functor. However, this is generally not true. One of the easiest examples, the tautological line bundle over the projective space of the regular representation, doesn’t have a Thom isomorphism. Costenoble-Waner have established a theory of Thom isomorphism in an extended grading to the usual -grading using representations of the fundamental groupoid of the base space, which I’ll call -graded cohomology. Recently, Costenoble computed the -graded cohomology for  "B_{C_2}U(1)")
, the classifying space for complex -line bundles. In this talk, I will discuss our computation of the -graded cohomology of  "B_{C_2}O(1)")
, the classifying space for real -line bundles.2 Nov 2023 at 10:00 (Eastern US Time)
Christian Carrick, Utrecht
Title: Chromatic defect, Wood’s theorem, and higher real K-theories
Abstract: Let X(n) be Ravenel’s Thom spectrum over ΩSU(n). We say a spectrum E has chromatic defect n if n is the smallest positive integer such that E⊗X(n) is complex orientable. We will look closely at this quantity in three cases: when E is a finite spectrum, when E is the fixed points of Morava E-theory with respect to a finite subgroup of the Morava stabilizer group, and when E is an fp spectrum in the sense of Mahowald—Rezk. In this last case, we show that the property of having finite chromatic defect is closely related to the existence of splittings similar to Wood’s theorem on ko. When E admits such a splitting, we show that there is a Z-indexed version of ANSS(E) which behaves much like a Tate spectral sequence, and we explore this in detail for E=ko.
Chapters 2 and 6 of Carrick’s thesis: Stacks and Real-Oriented Homotopy Theory
(If time permits) Mahowald and Rezk, Brown-Comenetz duality and the Adams spectral sequence
5 Oct 2023 at 11:00 (Eastern US Time)
Tomer Schlank, Hebrew University
Title: K-theory and the telescope conjecture
Abstract: The telescope conjecture deals with the comparison of two competing approaches to monochromatic spectra. We shall discuss how to use hyperdescent properties of algebraic K-theory to distinguish these two approaches. I’ll sketch how recent developments from ambidexterity theory and in trace methods allow the required analysis of hyperdescent for algebraic K-theory.
5 Oct 2023 at 10:00 (Eastern US Time)
Adela YiYu Zhang, Copenhagen
Title: An equivariant Adams spectral sequence for tmf(2)
Abstract: In this talk, I will explain how to compute the 
%7D "\otimes_{\mathrm{tmf}(2)}")
7 Sep 2023 at 11:00 (Eastern US Time)
Yang Hu, New Mexico State
Title: A calculus approach to the enumeration of topological vector bundles
Abstract: In the unstable range, topological vector bundles over finite CW complexes are difficult to classify in general. Over complex projective spaces, such bundles are far from being fully classified, or even enumerated, beyond a few small dimensional cases. It is a classical problem in topology to enumerate complex vector bundles of rank r over CP^n (where 1 < r < n) with fixed Chern class data. A particular case is when the Chern data is trivial, which we call the vanishing Chern enumeration. In this talk, we apply the unitary calculus of Weiss to produce the vanishing Chern enumeration in the first two unstable cases (which belong to what we call the “metastable” range, following Mark Mahowald), namely rank (n-1) bundles over CP^n for n > 2, and rank (n-2) bundles over CP^n for n > 3. Time permitting, we will discuss work in progress (joint with Hood Chatham and Morgan Opie) where more metastable vector bundles are detected using EO-theory.
7 Sep 2023 at 10:00 (Eastern US Time)
Jack Carlisle, Notre Dame
Title: Isomorphisms of equivariant formal group laws
Abstract: In the non-equivariant setting, Quillen proved that the complex cobordism ring 
%5F* "(MU \wedge MU)_*")
%5F* "\left(MU_{C_2} \wedge MU_{C_2}\right)_*")
20 Apr 2023 at 11:00
Speaker: Cary Malkiewich, Binghamton University
Title: Higher scissors congruence (notes)
Abstract: Hilbert’s Third Problem asks for sufficient conditions that determine when two polyhedra in three-dimensional Euclidean space are scissors congruent. Classically, the attempts to solve this problem (in this and other geometries) lead into group homology and algebraic K-theory, in a somewhat ad-hoc way. In the last decade, Zakharevich has shown that the presence of K-theory here is not ad-hoc, but is integral to the definition of scissors congruence itself. This leads to a natural notion of “higher” scissors congruence groups, namely, the homotopy groups of an algebraic K-theory spectrum K(P).
In this talk, I’ll describe a surprising recent result that K(P) is actually a Thom spectrum. Its base space is the homotopy orbit space of a Tits complex, and the vector bundle is the negative tangent bundle of the underlying geometry. Using this result, we can explicitly compute the higher scissors congruence groups in new cases. Much of this is joint work with Anna-Marie Bohmann, Teena Gerhardt, Mona Merling, and Inna Zakharevich.
20 Apr 2023 at 9:30 Pre-Talk, 10:00 Talk
Speaker: Vesna Stojanoska, University of Illinois at Urbana Champaign
Pre-Talk Title: Representation Spheres and the J-homomorphism (pretalk notes)
Pre-Talk Abstract: I’ll show some toy examples to motivate the construction of the “star” of the main talk.
Title: A particularly neat invertible K(2)-local spectrum at the prime 2 (notes)
Abstract: At each prime p and height n, the K(n)-local sphere admits a Galois extension to Lubin-Tate E-theory, with the Morava stabilizer group G_n as the automorphism group. Thus, the group G_n governs the behavior of K(n)-local spectra and their invariants, such as the Picard group of the K(n)-local stable homotopy category. For one, it helps separate this Picard group into an algebraic part, which is seen by the Picard group of Morava modules, and an exotic part, which is invisible to the algebra.
The exotic Picard group also admits a useful decomposition obtained by asking the following question: can an invertible spectrum be seen as non-trivial through the eyes of well-chosen closed subgroups of G_n? This question and its answers were key in determining the exotic Picard group at height 2 and the prime 2, in recent work with Beaudry, Bobkova, Goerss, Henn, and Pham (arXiv:2212.07858). In this talk I will focus on a small but interesting piece of this exotic Picard group, namely the subgroup of order 2 seen by the kernel of the reduced determinant but not by the maximal finite subgroup.
30 Mar 2023 at 11:00
Speaker: David Mehrle, University of Kentucky
Title: The universal exponential formula relating multiplicative and additive power operations (notes)
Abstract: In finite group representation theory, the symmetric powers and Adams operations are related by an exponential formula — the same formula that relates homogeneous symmetric polynomials and power sums. This can be viewed as an integral version of the height 1 case of a theorem of Ganter, which asserts an exponential relationship between symmetric powers and Hecke operations in Morava E-theories. We ask: for which cohomology theories do such exponential formulas exist? For any global power functor R (with a few additional hypotheses), we construct an object whose universal property yields an exponential relationship between multiplicative power operations and additive power operations in R. We apply our construction to recover Ganter’s theorem and produce a new exponential formula for E-theory at height 2 and prime 2. This is joint work with Nat Stapleton and Davis Deaton.
Note: Due to technical difficulties, there is no recording of this talk.
30 Mar 2023 at 10:00
Speaker: Yunze Lu, UC San Diego
Title: Computations of height 2 higher real K-theory spectra at prime 2 (notes)
Abstract: The homotopy fixed points of Lubin-Tate theories are central objects in chromatic homotopy theory, as they are building blocks of K(n)-local spheres. This talk will describe the use of equivariant structure and vanishing line result to compute the RO(Q_8)-graded homotopy fixed points spectral sequence at height 2 and prime 2. This is joint work with Zhipeng Duan, Hana Jia Kong, Guchuan Li, and Guozhen Wang.
Note: Due to technical difficulties, there is no recording of this talk.
23 Feb 2023 at 11:00
Panel discussion on Academic careers at primarily undergraduate institutions
Panelists: Eric Hogle (Gonzaga), Angélica Osorno (Reed), Carolyn Yarnall (CSU Dominguez Hills)
The panel discussion will be followed by a speed-networking event.
23 Feb 2023 at 9:25 Pre-Talk, 10:00 Talk
Speaker: Nathalie Wahl, University of Copenhagen
Title: Computing string topology operations
Abstract: String topology, as introduced by Chas and Sullivan 20 years ago, is a family of operations one can define on the homology of the free loop space of a manifold, lifting the classical intersection product. Naef showed recently that homotopic lens spaces can have distinct string coproduct, a computation we extended with Rivera and Naef to show that a homotopy equivalence between 3-dimensional lens spaces is homotopic to a homeomorphism if and only if it respects the string coproduct. In this talk, after a short introduction to string topology, I’ll explain how such computations are done.
26 Jan 2023 at 11:00
Speaker: Jack Davies, Universität Bonn
Title: Some arithmetic in the torsion of TMF
Abstract: The rational homotopy groups of the universal elliptic cohomology theory TMF (topological modular forms) come with a natural isomorphism to the ring of rational meromorphic modular forms (sometimes called weakly holomorphic modular forms). In this talk, we will discuss the use of cohomology operations to obtain simple congruences in arithmetic from the torsion in the homotopy groups of TMF. Some simple well-known examples will be shown in detail, as well as some generalisations and applications to a conjecture in number theory. Along the way, we will see how the homotopy groups of TMF have a kind of rigidity with respect to cohomology operations.
26 Jan 2023 at 10:00
Speaker: Lior Yanovski, Hebrew University of Jerusalem
Title: Higher descent in chromatically localized algebraic K-theory
Abstract: Algebraic K-theory is a fundamental invariant of a ring R, constructed from the category of perfect R-modules, which in modern language is understood to be a spectrum denoted K(R). More generally, one can associate a K-theory spectrum K(C) to every small stable oo-category C. The far reaching conjectural redshift philosophy of Ausonni-Rognes states that this construction shifts up the “chromatic height” by 1. A large part of this conjectural assertion was established in two recent works of Land, Mathew, Meier, and Tamme and of Clausen, Mathew, Naumann and Noel. A key step in their work, which is of independent interest, is to show the following descent result: If a finite p-group G acts on a stable oo-category C of chromatic height n, then the canonical map K(C^hG) -> K(C)^hG becomes an equivalence after height n+1 localization. In this talk, I will discuss a joint work in progress with Ben-Moshe, Carmeli and Schlank concerning the “higher version” of this result when the finite p-group G is replaced by a ”pi-finite p-group”. Namely, when we consider instead of homotopy fixed point of a finite p-group, the global sections of a local system on a space with finitely many non-vanishing homotopy groups all of which are finite p-groups. I will discuss the relevant background, the relation to higher semiadditivity, and if time permits, the applications to higher cyclotomic extensions.
17 Nov 2022 at 11:00
Speaker: Kathryn Hess, École Polytechnique Fédérale de Lausanne
Title: A shadow perspective on equivariant Hochschild homologies (notes)
Abstract: Shadows for bicategories, first defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this talk, I will explain how to define Hochschild-type invariants for monoids in a symmetric monoidal, simplicial model category V, as well as for small V-categories. Each of these constructions extends to a shadow on an appropriate bicategory, which implies in particular that they are Morita invariant. I will also define a generalized theory of Hochschild homology twisted by an automorphism and show that it is Morita invariant. I will show in particular that Hochschild homology of Green functors and C_n-twisted topological Hochschild homology fit into this framework, implying that these theories are Morita invariant.
17 Nov 2022 at 10:00
Speaker: Tom Bachmann, University of Oslo
Title: Cohomology of motivic Eilenberg-MacLane spaces
Abstract: I will report on joint work (somewhat in progress) with Mike Hopkins.
Working over a field k of characteristic different from p, we determine
H^**(K(Z/p(i), j), Z/p) for j >= 2i. We also determine the homology.
Already, Voevodsky remarked that his motivic cohomology operations are
insufficient to generate these unstable cohomology groups (this is in
contrast with the stable situation, where the motivic Steenrod algebra
is famously generated by Voevodsky operations). Recall that there is a
second source of cohomology operations, namely the E-infinity ring
structure on motivic cohomology. Our main observation is that these
almost tautological operations can be used together with Voevodsky’s to
generate the entire cohomology.
20 Oct 2022 at 11:00
Speaker: Ang Li, University of California Santa Cruz
Title: The real motivic Steenrod subalgebra A(1) and its 128 module structures (notes)
Abstract: Self maps of small CW-complexes contribute constructing elements in the stable homotopy group of spheres. Davis and Mahowald constructed v_1 self maps of the spectrum Y, and analyzed the structure of their cofibers. Let A(1) be the Steenrod subalgebra generated by Sq^1 and Sq^2. It turns out that A(1) has 4 different A-module structures distinguished by the action of Sq^4, which match up with different homotopy types of the cofibers of the v_1 self maps of Y. I will talk about the analogy in the real motivic setting and establish the 128 realizations of the real motivic Steenrod subalgebra A(1). This is joint work with Prasit Bhattacharya and Bertrand Guillou.
20 Oct 2022 at 10:00
Speaker: Noah Riggenbach, Northwestern University
Title: A prismatic description of the K-theory of truncated polynomial algebras
Abstract: The algebraic K-theory of truncated polynomial algebras in a single variable has been studied by many groups, including Soule; Staffeldt; Hesselholt and Madsen; Angeltveit, Gerhardt, and Hesselholt; Speirs; Mathew; and Sulyma among many others. This focus has lead to a complete calculation of these groups when the base ring is a perfect ring of positive characteristic, and a computation of the rank and cardinality of these groups when the base ring is the integers. In this talk I will discuss how to use the prismatic cohomology and quasisyntomic topology of Bhatt, Morrow, and Scholze to describe the K-theory of truncated polynomial algebras for a class of rings which include perfect algebras of characteristic p, perfectoid rings, smooth curves over perfectoid rings, and complete mixed characteristic discrete valuation rings.
22 Sep 2022 at 11:00
Speaker: Mona Merling, University of Pennsylvania
Title: Derived scissors congruence for manifolds (notes)
Abstract: The classical scissors congruence problem asks whether given two polyhedra with the same volume, one can cut one into a finite number of smaller polyhedra and reassemble these to form the other. There is an analogous definition of an SK (German “schneiden und kleben,” cut and paste) relation for manifolds and classically defined scissors congruence (SK) groups for manifolds. Recent work of Jonathan Campbell and Inna Zakharevich has focused on building machinery for studying scissors congruence problems via algebraic K-theory, and applying these tools to studying the Grothendieck ring of varieties. I will talk about a new application of this framework: we will construct a K-theory spectrum of manifolds, which lifts the classical SK group, and a derived version of the Euler characteristic. I will discuss what this invariant sees on the level of K1.
22 Sep 2022 at 10:00
Speaker: Ben Knudsen, Northeastern University
Title: Stable and unstable homology of graph braid groups (notes)
Abstract: The homology of the configuration spaces of a graph forms a finitely generated module over the polynomial ring generated by its edges; in particular, each Betti number is eventually equal to a polynomial in the number of particles, an analogue of classical homological stability. The degree of this polynomial is captured by a connectivity invariant of the graph, and its leading coefficient may be computed explicitly in terms of cut counts and vertex valences. This “stable” (asymptotic) homology is generated entirely by the fundamental classes of certain tori of geometric origin, but exotic non-toric classes abound unstably. These mysterious classes are intimately tied to questions about generation and torsion whose answers remain elusive except in a few special cases. This talk represents joint work with Byung Hee An and Gabriel Drummond-Cole.
Pretalk 9:25 (notes)
Title: How to build a surface of genus six
Abstract: The study of configuration spaces of graphs began in the context of motion planning problems in robotics. A surge of recent research has been re-examining them through the lenses of algebraic topology, geometric group theory, representation stability, physics, tropical geometry, and equivariant stable homotopy theory. This talk will be a low-tech and example-driven introduction to these spaces, which, along the way, will provide an explanation for its title.
25 Aug 2022 at 11:00
Speaker: Sarah Petersen, Max Planck Institute for Mathematics Bonn
Title: Ravenel-Wilson Hopf ring methods in C2-equivariant homotopy theory and the HF2-homology of C2-equivariant Eilenberg-MacLane spaces (notes)
Abstract: This talk describes an extension of Ravenel-Wilson Hopf ring techniques to C2-equivariant homotopy theory. Our main application and motivation for introducing these methods is a computation of the RO(C2)-graded homology of C2-equivariant Eilenberg-MacLane spaces. The result we obtain for C2-equivariant Eilenberg-MacLane spaces associated to the constant Mackey functor F2 gives a C2-equivariant analogue of the classical computation due to Serre at the prime 2. We also investigate a twisted bar spectral sequence computing the homology of these equivariant Eilenberg-MacLane spaces and suggest the existence of another twisted bar spectral sequence with E2-page given in terms of a twisted Tor functor.
25 Aug 2022 at 10:00
Speaker: Robert Burklund, University of Copenhagen
Title: Multiplicative structures on Moore spectra
Abstract: One of the distinguishing features of higher algebra is the difficulty of constructing quotients. In this talk I will explain a new technique for constructing algebra structures on quotients.
This technique allows us to prove that 𝕊/8 is an E1-algebra, 𝕊/32 is an E2-algebra, 𝕊/pn+1 is an En-algebra at odd primes and, more generally, for every h and n there exist generalized Moore spectra of type h which admit an En-algebra structure.
21 Apr 2022 at 11:00
Speaker: Anna Marie Bohmann, Vanderbilt University
Title: Multiplicative uniqueness of rational equivariant K-theory
Abstract: Topological K-theory is one of the classical motivating examples of a commutative ring spectrum, and it has a natural equivariant generalization. The equivariant structure here has the strongest possible type compatibility with the multiplication, making K-theory an example of a “genuine-commutative” ring spectrum. There’s quite a lot of structure involved here, so in order to understand it, we employ a classic strategy and rationalize. After rationalizing, we can use algebraic models due to Barnes–Greenlees–Kedziorek and to Wimmer to show that all of the additional “norm” structure is determined by the equivariant homotopy groups and the underlying multiplication. This is joint work with Christy Hazel, Jocelyne Ishak, Magdalena Kedziorek, and Clover May.
21 Apr 2022 at 10:00
Speaker: Allen Yuan, Columbia University
Title: The chromatic Nullstellensatz (notes)
Abstract: Hilbert’s Nullstellensatz is a fundamental result in commutative algebra which is the starting point for classical algebraic geometry.
In this talk, I will discuss joint work with Robert Burklund and Tomer Schlank on a chromatic version of Hilbert’s Nullstellensatz in which Lubin-Tate theories play the role of algebraically closed fields. I will then sample some applications of our results to chromatic support, redshift, and orientation theory for E-infty rings.
24 Mar 2022 at 11:00
Speaker: Gabe Angelini-Knoll, Freie Universität Berlin
Title: Topological cyclic homology of elliptic cohomology (notes)
Abstract: I will discuss joint work C. Ausoni, D. Culver, E. Höning, and J. Rognes on the topological cyclic homology (TC) of the truncated Brown-Peterson spectrum. In particular, we show that the mod (p,v1,v2)-homotopy of TC of BP<2> is a finitely generated free P(v3)-module in 12p+4 generators in explicit degrees -2<s<2p³+2p²+2p-2 for p>5. This verifies the expectation of C. Ausoni and J. Rognes based on their seminal calculation of mod (p,v1)-homotopy of TC of the Adams summand in 2002 and the foundational computation of mod (p)-homotopy of TC of the p-local integers by M. Bökstedt and I. Madsen in 1995. This also verifies a strong form of the red-shift conjecture as proposed by J. Rognes in 2000 and therefore provides a strengthening of the groundbreaking result of J. Hahn and D. Wilson in 2020 in the special case n=2 and p>5.
24 Mar 2022 at 10:00
Speaker: Teena Gerhardt, Michigan State University
Title: Free loop spaces and topological coHochschild homology (notes)
Abstract: The study of free loop spaces, and in particular their homology, has broad applications in topology and geometry. This talk will describe a new approach to the homology of free loop spaces via topological coHochschild homology (coTHH), a recently developed invariant of coalgebras in spectra. In this talk I will introduce coTHH and discuss how computational tools for coTHH yield new computations of the homology of free loop spaces. In particular, I will introduce a spectral sequence computing the homology of free loop spaces which has interesting algebraic structure. This work is joint with Anna Marie Bohmann and Brooke Shipley.
24 Feb 2022 at 11:00
Speaker: William Balderrama, University of Virginia
Title: Some chromatic equivariant homotopy theory (notes)
Abstract: Chromatic homotopy theory tells us that one can build the sphere out of smaller pieces, these pieces being the K(n)-local spheres. This talk will discuss what happens when you try to do this in the equivariant setting, focusing on the question of what plays the role of a “G-equivariant K(n)-local sphere”. Once this question is answered, I will describe some computations, focusing on n = 1 and G = C_2. If time allows, this will include a discussion of how equivariant norms manifest in spectral sequence computations.
24 Feb 2022 at 10:00
Speaker: Prasit Bhattacharya, University of Notre Dame
Title: Equivariant Steenrod Operations (notes)
Abstract: Classical Steenrod algebra is one of the most fundamental algebraic gadgets in stable homotopy theory. It led to the theory of characteristic classes, which is key to some of the most celebrated applications of homotopy theory to geometry. The G-equivariant Steenrod algebra is not known beyond the group of order 2. In this talk, I will recall a geometric construction of the classical Steenrod algebra and generalize it to construct G-equivariant Steenrod operations, for any finite group G. Time permitting, I will discuss potential applications to equivariant geometry.
27 Jan 2022 at 11:00
Speaker: Aaron Mazel-Gee, California Institute of Technology
Title: Derived Mackey functors and C_{p^n}-equivariant cohomology via stratifications
Abstract: In this talk, I’ll explain a new approach to computing equivariant cohomology, developed jointly with David Ayala and Nick Rozenblyum. This is an application of our theory of “stratifications” to the category of derived Mackey functors. This is analogous to a stratification of a space into simpler pieces, which allows for computations via local-to-global principles (e.g. Mayer–Vietoris). I’ll illustrate this approach with some examples: taking our group to be G=C_{p^n}, we compute the Picard group of genuine G-spectra as well the RO(G)-graded G-equivariant cohomology of a point.
27 Jan 2022 at 10:00
Speaker: Inna Zakharevich, Cornell University
Title: Detecting non-permutative elements of K_1(Var) using point counting
Abstract: The Grothendieck ring of varieties is defined to be the free abelian group generated by varieties (over some base field k) modulo the relation that for a closed immersion Y –> X there is a relation that [X] = [Y] + [X \ Y]. This structure can be extended to produce a space whose connected components give the Grothendieck ring of varieties and whose higher homotopy groups represent other geometric invariants of varieties. This structure is compatible with many of the structures on varieties. In particular, if the base field k is finite for a variety X we can consider the “almost-finite” set X(\bar k), which represents the local zeta function of X. In this talk we will discuss how to detect interesting elements in K_1(\Var) (which is represented by piecewise automorphisms of varieties) using this zeta function and precise point counts on X.
11 Nov 2021 at 11:30
Speaker: Emanuele Dotto, University of Warwick
Title: The geometric fixed points of real topological cyclic homology
Abstract: Real topological Hochschild and cyclic homology are refinements of the classical topological Hochschild and cyclic homology which informally take into account the reflection of the circle.
The talk will introduce these theories, with a focus on their geometric fixed points under the action of a reflection. I will give a formula for these geometric fixed-points in terms of tensor products of Hill-Hopkins-Ravenel norms, in line with a similar formula for normal L-theory of Harpaz-Nikolaus-Shah. We will then use this formula to carry out calculations for perfect fields and for the ring of integers.
This is joint work with Kristian Moi and Irakli Patchkoria.
11 Nov 2021 at 10:15
Speaker: Andrew Blumberg, Columbia University
Title: Chromatic convergence for the algebraic K-theory of the sphere spectrum
Abstract: I will discuss joint work with Mandell and Yuan in which we prove a version of Waldhausen’s chromatic convergence conjecture for K(S).
14 Oct 2021 at 11:30
Speaker: Ningchuan Zhang, University of Pennsylvania
Title: Exotic K(h)-Picard groups when 2p-1=h^2 and the Vanishing Conjecture (slides)
Abstract: In this talk, I will explain how to use the Gross-Hopkins duality and Miller-Ravenel-Wilson’s computation of Greek letter elements to study the exotic K(h)-local Picard group, when the prime p satisfies 2p-1=h^2. I will also describe its relation with a special case of Hopkins’ Chromatic Vanishing Conjecture. This is joint work in progress with Dominic Culver.
14 Oct 2021 at 10:15
Speaker: Dominic Culver, Max Planck Institute for Mathematics in Bonn
Title: The v_2-local structure of the algebraic tmf resolution
Abstract: In the 1980s Mahowald was able to prove the height 1 telescope conjecture at the prime p=2 by a careful analysis of the Adams spectral sequence based on connective real K-theory. Inspired by this, many people have studied the Adams spectral sequence based on connective topological modular forms. Essential to analyzing this spectral sequence is computing the A(2)-Ext groups of the ko-Brown-Gitler spectra. In this talk, I will discuss joint work with Behrens and Bhattacharya where we determine these Ext groups v_2-locally.
16 Sept 2021 at 11:30
Speaker: Elden Elmanto, Harvard University
Title: Birational geometry and the slice spectral sequence (notes)
Abstract: I will explain a perspective, joint with Tom Bachmann, on how birational geometry controls one of the main computational tools in motivic homotopy theory: the slice spectral sequence. This reproves parts of Levine’s resolution of Voevodsky’s conjectures. I will then explain joint work with Nardin and Yakerson on how to twist the motivic spectral sequence by a Brauer class, improving earlier results of Levine and Kahn. In this talk I will make a special effort to make the necessary algebraic geometry accessible and picturesque to this audience.
16 Sept 2021 at 10:15
Speaker: Andy Baker, University of Glasgow
Title: MSp: something old and something new (slides)
Abstract: This is a semi-historical survey of results on the E-infinity Thom spectrum MSp which represents symplectic(=quaternionic) cobordism.
I will describe some results from the past half century due to Nigel Ray and others. Then I will discuss more recent results that are joint work with Jan Holz and Gerd Laures on K-local and K(1)-local aspects.
22 April 2021 at 11:30
Speaker: J.D. Quigley, Cornell University
Title: Free incomplete Tambara functors are almost never flat
Abstract: In equivariant algebra, Mackey functors play the role of abelian groups, and incomplete Tambara functors play the role of commutative rings. In this talk, I will discuss a surprising result in equivariant algebra: free incomplete Tambara functors are almost never flat as Mackey functors. This result is in stark contrast with the situation in ordinary algebra, where free commutative rings are always flat (in fact, free!) as abelian groups. I plan to discuss connections to equivariant stable homotopy theory, describe specific examples of free incomplete Tambara functors, and sketch the proof of the titular result. If time permits, I will also indicate how these phenomena can be resolved using localization. This is joint work with Mike Hill and David Mehrle.
22 April 2021 at 10:15
Speaker: Birgit Richter, Universität Hamburg
Title: Detecting and describing ramification for structured ring spectra
Abstract: This is a report on joint work with Eva Höning. For rings of integers in an extension of number fields, there are classical methods for detecting ramification and for identifying ramification as
being tame or wild. Noether’s theorem characterizes tame ramification in terms of a normal basis, and tame ramification can also be detected via the surjectivity of the norm map. We take the latter fact and use the Tate cohomology spectrum to detect wild ramification in the context of commutative ring spectra. I will discuss several examples in the context of topological K-theory and modular forms.
25 March 2021 at 11:30
Speaker: Piotr Pstrągowski, Harvard University
Title: Morava K-theory and filtrations by powers
Abstract: The recent advances in knowledge of stable homotopy due to Isaksen, Wang and Xu rest on the relationship between the Adams-Novikov and Adams spectral sequences, as exhibited by the Miller square. In this joint project with Tobias Barthel, we focus on the finite height analogue of this picture. We show that at large primes, the Adams spectral sequence based on Morava K-theory can be identified with a spectral sequence obtained by filtration by powers. The same technique applied to homology of K-local spectra relates the latter to derived completion of comodules, which we express in terms of the cohomology of the Morava stabilizer group.
25 March 2021 at 10:15
Speaker: Yuri Sulyma, Brown University
Title: Slice Bökstedt periodicity and RO(G)-graded homotopy
Abstract: I will propose a definition of a C_n-spectrum being “slice 2-periodic”, motivated by recent work in which I showed that the topological Hochschild homology of perfectoid rings has this property. I will then explain in detail how to carry out the calculations of RO(G)-graded homotopy that go into this result. This has interesting connections with arithmetic geometry, which I will briefly summarize, but the emphasis will be on computational techniques.
25 February 2021 at 11:30
Speaker: Christy Hazel, UCLA
Title: Rational equivariant K-theory for finite abelian groups
Abstract: When working rationally, we can often translate topological questions to algebraic questions through the use of algebraic models. Recent work of Wimmer gives an algebraic model for rational genuine-commutative equivariant ring G-spectra for G a finite group. In this talk, we describe how we use this model to see there is a unique genuine commutative structure on rational equivariant K-theory for G an abelian group. This is joint work with Anna Marie Bohmann, Jocelyne Ishak, Magdalena Kędziorek, and Clover May.
25 February 2021 at 10:15
Speaker: Lennart Meier, Utrecht
Title: Equivariant topological modular forms
Abstract: Work of Lurie (with elaborations by Gepner and myself) constructs a theory of equivariant topological modular forms. In particular, this defines for every compact Lie group G a G-equivariant spectrum TMF. In this talk I will speak about the necessary framework and identify the computational challenges. For example, the fixed points TMF^G are by now understood for G a torus or G = C_2, but remain uncomputed for most other groups.
28 January 2021 at 11:30
Speaker: Dylan Wilson, Harvard University
Title: Redshift, Lichtenbaum-Quillen, and multiplication on BP<n>
Abstract: We describe recent work, joint with Jeremy Hahn, showing that the algebraic K-theory of BP<n> has ‘chromatic complexity n+1’. This gives the first arbitrary height examples of the redshift philosophy of Ausoni-Rognes. There are some ingredients to the argument that may be of independent interest, including the construction of an E_3-BP-algebra form of BP<n> and the construction of several spectral sequences for studying Hochschild homology and its variants.
28 January 2021 at 10:15
Speaker: Bert Guillou, University of Kentucky
Title: R-motivic and C_2-equivariant v_1-self-maps
Abstract: I will describe joint work with Prasit Bhattacharya and Ang Li that considers v1-periodicity in the R-motivic and C_2-equivariant contexts. Along the way, we will also produce a finite R-motivic spectrum whose cohomology realizes the subalgebra A^R(1) of the R-motivic Steenrod algebra.
19 November 2020 at 11:30
Speaker: Foling Zou, University of Michigan
Title: Loday constructions on twisted products and on tori
Abstract: For a commutative algebra A and a simplicial set X, the Loday construction L_X(A) is a simplicial commutative ring whose homology generalizes the Hochschild homology of A. In this talk, I will discuss some computational aspects of the Loday construction and answer a stability question of A = Q[t] / t^n, generalizing a result by Dundas-Tenti. This is joint work with Alice Hedenlund, Sarah Klanderman, Ayelet Lindenstrauss and Birgit Richter.
19 November 2020 at 10:15
Speaker: Paul Arne Østvær, University of Oslo
Title: Motivic homotopy theory at infinity
Abstract: We will discuss motivic homotopy theory at infinity for algebraic varieties. This is joint work in progress with Frederic Deglise and Adrien Dubouloz.
22 October 2020 at 11:30
Speaker: Irina Bobkova, Texas A&M University
Title: Invertible E_2^{hC_4}-modules
Abstract: For Morava E-theory E_n and a finite subgroup F of the Morava stabilizer group, the homotopy fixed points spectrum E_n^hF is periodic and the Picard group of the category of modules over the ring spectrum E_n^hF contains the cyclic subgroup generated by Sigma E_n^hF. In most known examples, the Picard group is found to be precisely this cyclic group. However, at n = p = 2, the Picard group of the category of E_2^{hC_4}-modules is not cyclic and contains an extra element of order 2. I will describe the tools we use to compute this Picard group: a group homomorphism from RO(C_4) to it and the Picard spectral sequence. This talk is based on joint work with Agnes Beaudry, Mike Hill and Vesna Stojanoska.
22 October 2020 at 10:15
Speaker: Tom Bachmann, LMU Munich
Title: The eta-periodic motivic image of j spectrum over fields
Abstract: (joint work with Mike Hopkins) Let ko and ksp denote the very effective motivic spectra corresponding to orthogonal and symplectic K-theory, respectively. One may show that the Fasel-Haution Adams operation Psi^3: ko_(2) -> ko_(2) lifts to Sigma^4,2 ksp_(2); we denote by jo its fiber. In this talk I will explain how to show that after inverting the motivic Hopf element eta, jo just becomes the (periodized, 2-local) sphere spectrum. This determines in particular all the eta-periodic motivic stable stems.
24 September 2020 at 11:30
Speaker: Hood Chatham, UCLA
Title: Snaith Constructions on Wilson Spaces
Abstract: Snaith showed that complex K theory can be constructed from CP^\infty by taking the suspension spectrum and inverting a “Bott class”. We generalize this construction and build a new Landweber exact complex oriented E_\infty ring spectrum R_h for each prime p and height h. Unlike prior arbitrary height ring spectra, R_h is both simple to construct and has simple homotopy groups. We relate R_h to work of Peterson and Westerland constructing a fixed point spectrum of Morava E theory from K(Z, h+1). We use this connection to investigate when there exist structured ring maps from R_h to Morava E theories. This depends on the arithmetic of the formal group. For instance, there are structured ring maps from R_2 to a Morava E theory associated to a supersingular elliptic curve but not to general Morava E theories. It is our hope to identify some collection of Morava E theories oriented by R_3 as “geometric” E theories, analogous to complex K theory at height 1 and Elliptic E theories at height 2. This work is joint with Jeremy Hahn and Allen Yuan.
24 September 2020 at 10:15
Speaker: Stefan Schwede, Universität Bonn
Title: Splittings of global Mackey functors and regularity of equivariant Euler classes
Reference: arxiv.org/abs/2006.0943
Abstract: In 1962, Dold published an elegant proof of Nakaoka’s splitting of the cohomology of symmetric groups. Dold’s proof only uses formal aspects of group cohomology, and I’ll explain that his argument is way more general and provides a splitting of the values of global Mackey functors at symmetric groups. The relevant kind of global Mackey functor is also known as `inflation functors’, or `biset functors with restriction, transfers and inflations’ (but possibly no deflations).
Equivariant homotopy theory provides an even more general kind of global Mackey functor with values at all compact Lie groups. The main result of the talk is an analog of Dold’s splitting for the values of these `global functors’ at orthogonal, unitary and symplectic groups. As a consequence of these splittings, certain long exact sequences of equivariant homotopy groups decompose into short exact seqeunces. This in turn implies that the Euler class of the tautological U(n)-representation in homotopical equivariant bordism is a non-zero divisor.
16 April 2020
Speaker: Mark Behrens, University of Notre Dame
Title: 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 
be any prime. When k "n = (p-1)k")
, the 
-th Morava stabilizer group contains 
as a subgroup. 
, the homotopy fixed point of height prime Morava E-theory 
, can be thought of as the higher height analogues of real 
-theory 
. Indeed, 
is 
-completed 
and 
is -completed . It is a classical result that the tautological line bundle 
over 
is not -orientable, but 
is, meaning the -orientation order of is . In this talk, we generalize this result to show that the -orientation order of divides %09=%09p%5E%7Bp%5E%7Bk%7D%09-1%7D "o(n, p) = p^{p^{k} -1}")
. Consequently, we conclude that the  "o(n,p)")
-fold direct sum of any complex vector bundle is -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
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
1 November 2018
Speaker: Zhouli Xu, MIT
Title: The intersection form of spin 4-manifolds and Pin(2)-equivariant Mahowald invariants
Abstract: A fundamental problem in 4-dimensional topology is the following geography question: “which simply connected topological 4-manifolds admit a smooth structure?” After the celebrated work of Kirby-Siebenmann, Freedman, and Donaldson, the last uncharted territory of this geography question is the “11/8-Conjecture”. This conjecture, proposed by Matsumoto, states that for any smooth spin 4-manifold, the ratio of its second-Betti number and signature is least 11/8.
Furuta proved the ”10/8+2”-Theorem by studying the existence of certain Pin(2)-equivariant stable maps between representation spheres. In this talk, we will present a complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald invariants of powers of certain Euler classes in the RO(Pin(2))-graded equivariant stable homotopy groups of spheres. In particular, we improve Furuta’s result into a ”10/8+4”-Theorem. Furthermore, we show that within the current existing framework, this is the limit. For the proof, we use the technique of cell-diagrams, known results on the stable homotopy groups of spheres, and the j-based Atiyah-Hirzebruch spectral sequence.
4 October 2018
Speaker: Craig Westerland, University of Minnesota
Title: Structure theory for braided Hopf algebras and their cohomology
Abstract: Braided Hopf algebras are Hopf algebra objects in a braided monodical category (e.g., the category of Yetter-Drinfeld modules). Computation of their cohomology can be closely related to computations of the cohomology of the braid groups with certain families of coefficients. When working in the category of graded vector spaces, particularly over a field of characteristic zero, the Milnor-Moore and Poincare-Birkhoff-Witt theorems yield a characterization of primitively generated Hopf algebras which are particularly amenable to cohomology computations (e.g. via Lie algebra cohomology and various May-type spectral sequences). In the genuinely braided (and not symmetric) setting, very little of this structure theory carries over. The purpose of this work is to develop some of that machinery, which will be phrased in the language of braided operads. While still very much in progress, this is already elucidating some cohomology computations.
20 September 2018
Speaker: Vigleik Angeltveit, Australian National University
Title: Picard groups and the algebraic K-theory of cuspidal singularities.
6 September 2018
Speaker: Kristen Wickelgren, Georgia Institute of Technology
Title: An arithmetic count of the lines through 4 lines in 3-space
Abstract: Given four general lines in 3-dimensional space, it is a classical result that the number of lines intersecting all four is two, provided you allow the coefficients of the lines to be complex numbers. Over a general field k, say with characteristic not 2, and for example the real numbers, the two lines may be a conjugate pair over a quadratic extension of the field. We give a count of the lines weighted by their field of definition and arithmetic-geometric information about the configuration, by using an Euler class in A1-homotopy theory. Because the target of Morel’s degree homomorphism is the Grothendieck-Witt group GW(k) of quadratic forms over the field, this count takes the form of an equality in GW(k). More generally, we give such a count for the lines intersecting 2n-2 codimension 2 hyperplanes in P^n for n odd. This is joint work with Padmavathi Srinivasan, building on joint work with Jesse Kass.
3 May 2018
Speaker: Justin Noel, Universitaet Regensburg
Title: Nilpotence and periodicity in equivariant stable homotopy theory
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.