Schedule for: 16w5108 - Operations in Highly Structured Homology Theories

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Let \(E\) be a \(G\)-spectrum for a finite group \(G\). It's reasonably well understood that the homotopy groups of \(E\) have the structure of Mackey functors. If \(E\) is \(G\) commutative ring spectrum, then work of Strickland and of Brun shows that the zeroth homotopy groups of \(E\) form a Tambara functor, which is more structure than just a Mackey functor with commutative multiplication. I will discuss work with Angeltveit that extends this result to include the higher homotopy groups of \(E\). Specifically, if \(E\) has a commutative multiplication that enjoys lots of structure with respect to the \(G\) action, the homotopy groups of \(E\) form a graded Tambara functor. In particular, genuine commutative \(G\) ring spectra enjoy this property.

In this talk, I'll describe the basics of a family of equivariant operads which generalize the classical notion of an \(E_\infty\) operad. These \(N_\infty\) operads parameterize multiplications in which the group action is intermingled with the ordering of the coordinates, explaining some of the exciting peculiarities that arise in equivariant homotopy theory. This talk will have an introductory feel to it, and I'll discuss more generally how maps into an operad give rise to natural operations on the algebras over it, and then specialize to the case of \(N_\infty\) operads and equivariant infinite loop spaces. Time permitting, I will also sketch out how a similar story holds motivically.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

We describe algebraic data in a symmetric monoidal 2-category which correspond to the bottom two Postnikov invariants of its \(K\)-theory spectrum. Our applications include: an obstruction to strictification of the symmetric monoidal structure, a model for the \(2\)-type of the sphere, and the units-Picard-Brauer sequence of commutative ring spectra. This is work in progress, joint with Nick Gurski and Angélica Osorno.

By work of Hu-Kriz-Ormsby, Isaksen exhibits a motivic \(2\)-cell complex that has bigraded homotopy groups isomorphic to the classical Adams-Novikov \( E_2 \)-page for the sphere \( S^0 \). We show how to endow this \(2\)-cell complex with an \( E_{\infty}\)-ring structure, upgrading this isomorphism to an isomorphism of highly structured rings. We then show how to exploit this \(2\)-cell to construct a spectrum which we call \(wBP\), whose homotopy is polynomial in the new periodic motivic operators \(w_i\) introduced by Michael Andrews et al.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk I will explain how the use of functors defined on the category \(I\) of finite sets and injections makes it possible to replace \(E\)-infinity objects by strictly commutative ones. For example, an \(E\)-infinity space can be replaced by a strictly commutative monoid in \(I\)-diagrams of spaces. The quasi-categorical version of this result is one building block for an interesting rigidification result about multiplicative homotopy theories: we show that every presentably symmetric monoidal infinity-category is represented by a symmetric monoidal model category. This is a report on joint work with Christian Schlichtkrull, with Dimitar Kodjabachev, and with Thomas Nikolaus.

This is joint work with S. Basu and S. Sagave. We consider \(R\)-algebra Thom spectra for a commutative symmetric (or orthogonal) ring spectrum \(R\) and analyze their topological Hochschild homology. As an example, we consider generalized Thom ring spectra based on \(\mathrm{SU}(n)\) and show how their topological Hochschild homology may depend on the multiplicative structure. Time permitting, we also discuss graded Thom spectra and the relation to logarithmic topological Hochschild homology.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

The large number of distinct \(N_\infty\) operads produces additional confusion when considering the Dyer-Lashof algebra for an algebra over such an operad. I'll discuss ongoing work in this area, grounding the problem in explicit computations and then moving into more uncharted waters to discuss obstruction theory concerns.

Classical obstructions to the existence and uniqueness of \(A\)-infinity algebra structures on spectra live in the Hochschild cohomology of the stable homotopy ring. Hence, classical existence and uniqueness results rely on the vanishing of this cohomology. Angeltveit considered finer obstructions in the pages of a spectral sequence computing the homotopy groups of the moduli space of \(A\)-infinity algebra structures. We calculate the second differential of this spectral sequence in terms of Massey products. As an application, we obtain existence and uniqueness results even when the Hochschild cohomology algebra is highly non-trivial.

Cannibalistic classes were first studied by Bott for real \(K\)-theory in the context of spherical fibrations. I will introduce cannibalistic classes for string bundles with values in various forms of \(TMF\). I will provide some tools to compute them and talk about possible applications.

This will be an informal event, during which each speaker will tell us a bit about what they're working on, or even just something they think is cool. Every speaker will get some fixed number of minutes (probably 10 or 15, but this will depend on the number of speakers that sign up). After the set amount of time, a bell will go off, and everyone will clap heartily. Imbibing refreshments by speakers and audience members will be socially acceptable. This is a particularly good opportunity for grad students and recent graduates to introduce themselves and give a short talk in an informal setting. However, everyone is welcome to speak. Please send offers of talks to Aaron Mazel-Gee (aaron@math.berkeley.edu).

If \(\mathcal O\) is an operad (in \(S\)-modules for example), \(M\) is an \(\mathcal O\)-bimodule, and \(A\) is an \(\mathcal O\)-algebra, then \(M \circ_{\mathcal O}A\) is again an \(\mathcal O\)-algebra. A bar construction model for a derived version of this has been studied in joint work with Luis Pereira. Then one can take advantage of the fact that this construction is very friendly in the first variable to both do homotopical analysis and to find extra structure. In particular, with \(A\) an augmented commutative \(S\)-algebra, our approach allows one to define the augmentation ideal filtration of \(A\) together with composition structure (as needed in my recent work on Hurewicz maps for infinite loopspaces), and also recover the filtration I found a decade ago on the tensor product \(K \otimes A\), with \(K\) is a based space (implicitly appearing in ongoing work by Behrens and Rezk).

The notion of a braided vector space \(V\) comes from the Hopf algebra community, and examples abound, from conjugacy classes in groups to braidings coming from Cartan matrices. From this definition, the tensor powers of \(V\) form a family of braid group representations. They also may be assembled into a non-commutative, non-cocommutative braided Hopf algebra called a quantum shuffle algebra. Our main result identifies the homology of the braid groups with these coefficients as the cohomology of this algebra. Using change of rings spectral sequences, we begin to get a handle on these homology groups. If time permits, we will discuss applications to conjectures in arithmetic statistics. This is joint work with TriThang Tran and Jordan Ellenberg.

We discuss how power operations can be used to prove a nilpotence conjecture of J.P. May on elements in the homotopy of \(E_\infty\)-ring spectra. This is joint work with Niko Naumann and Justin Noel.

Ausoni and Rognes have conjectured that the algebraic \(K\)-theory of commutative \(S\)-algebras should satisfy a form of Galois descent after telescopic localization (at least in special cases). In the case of discrete commutative rings this is a celebrated result of Thomason. Now by a result of Mitchell, the algebraic \(K\)-theory of discrete rings only has chromatic information at heights \(0\) and \(1\) and the proof of Thomason's result at height \(0\) (i.e., rationally) is actually quite easy. In joint work with Dustin Clausen, Akhil Mathew, and Niko Naumann we show that the easy case of Thomason's result combined with May's nilpotence conjecture implies the hard case. Moreover, this approach can be combined with a joint result with Lennart Meier and Niko Naumann to establish important cases of the Ausoni-Rognes conjecture. We also give a new proof of Mitchell's theorem. This proof uses a general technique for bounding the chromatic complexity of commutative \(S\)-algebras using techniques from equivariant homotopy theory. In the case of \(KU\) we are able to show \(K(KU)\) has chromatic information only up to height \(2\) at the primes \(2\), \(3\), and \(5\). Ausoni and Rognes established this result for primes \(5\) and higher by explicit calculation.

The mod \(\ell\) dual motivic Steenrod algebra was computed by Voevodsky over a field of characteristic zero, and by Hoyois, Kelly, and Østvær over a field of positive characteristic \(p \neq l\). I will describe work in progress towards extending their result to the case \(p=\ell\). This is joint with Markus Spitzweck.

We use discrete Morse theory to prove a complementation formula (originally discovered by Björner-Walker) and demonstrate its applicability by computing various equivariant posets of interest in a uniform manner: fixed point spaces of the partition complex, parabolic restrictions of BT buildings in characteristic \(p\), and Young restrictions of the partition complex (thereby giving a short and purely combinatorial proof of a recent theorem of Arone). In joint work with Arone, we link these computations to `AQ for commutative monoid spaces' and provide space-level models for results of Goerss for AQ.

Organiser: Yifei Zhu

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.