We investigate several versions of the telescope conjecture on localized categories of spectra, and implications between them. Generalizing the “finite localization” construction, we show that on such categories, localizing away from a set of strongly dualizable objects is smashing. We classify all smashing localizations on the harmonic category, \(H\mathbb{F}_p\)-local category and I-local category, where I is the Brown-Comenetz dual of the sphere spectrum; all are localizations away from strongly dualizable objects, although these categories have no nonzero compact objects. The Bousfield lattices of these categories are also computed.

Stratified spaces occur in many contexts, e.g. in (mathematical models for) phase transitions in physics, as quotient spaces of proper Lie group actions, or more generally for proper Lie groupoids.

Applications to higher category theory suggest the interest of a version of \(\pi_0\) for topological groupoids which takes values in a category more general than discrete sets, eg in profinite topological spaces or perhaps databases.

Some references:

• arxiv/1101.0180
• arxiv/1202.0684
• arxiv/1212.5303

Real Johnson-Wilson theories are fixed points of standard Johnson-Wilson theories under the action of an involution. They have nice properties that make them useful for computations and “see” more information than the E(n)'s. Their connective covers are fixed points of truncated Brown-Peterson theories. I will discuss work-in-progress to compute the cohomology of these connective covers.

We discuss work in progress to extend the classical theory of descent cohomology and twisted forms to the homotopical setting. We will briefly review the classical case and also discuss a potential application of the theory to Galois and Hopf-Galois extensions of ring spectra.

I will discuss ongoing work with Arone on computing the orthogonal derivatives of spaces of link maps, making comparisons along the way with spaces of embeddings for motivation. Work of Goodwillie-Klein and Arone give some answers to questions about embeddings whose analogs for link maps are still unknown.

I will construct the suspension functor in the category of directed graphs as well as in the category of small categories with the canonical model structure. Then I will describe the stabilizations of these categories. This work is motivated by efforts to find examples which link Goodwillie derivatives of the identity functor to operads.

We give a brief introduction to colored props and their ilk. We then propose ways of encoding the notion of up-to-homotopy prop.

Secondary and higher order homotopy operations (Toda brackets) were introduced by Toda in order to construct elements of the homotopy groups of spheres as part of his “composition method” for computing these groups. As early as the 9-stem a tertiary operation (quaternary Toda bracket) was needed to describe a generator. Unfortunately a coherence condition and considerable detail obscure the definiton of this operation. Our aim here is to define and study such operations both in the classical topological setting and, ideally, in the abstract setting of a 2-category with zeros that admits a suspension 2-functor. We focus in particular on a new tertiary operation called the box quaternary operation.

Homotopy type theory is a homotopical interpretation of a system of formal logic, providing a system of foundations with intrinsic homotopical content and a computational implementation. It forms the basis of the Univalent foundations program, which was the subject of a recent special year at IAS. In this survey talk, I will show how to compute some homotopy groups of spheres in homotopy type theory, including \(\pi_3(S^2)\). These new logical proofs of classical theorems from algebraic topology make essential use of the new ideas of higher inductive types and the Univalence axiom.

We show that an adjoint functor between quasi-categories may be extended to a simplicially enriched functor whose domain is an explicitly presented “homotopy coherent adjunction”. This adjunction data encapsulates both the homotopy coherent monad and comonad defined by the adjunction. Using this result, we construct the quasi-category of algebras associated to a homotopy coherent monad and give a formal re-proof of the classical monadicity theorem. This is joint work with Dominic Verity.

Localization is a fundamentally important tool in mathematics. Constructing the localization of a category at a given class of maps leads naturally to the notion of a model category. Bousfield localization is a method of localizing further by turning a given class of maps into weak equivalences. In this talk we will give conditions on a monoidal model category and on the class of maps being localized so that the Bousfield localization preserves strict commutative monoids.

This problem was motivated by an example due to Mike Hill which demonstrates that for the model category of equivariant spectra, even very nice localizations can fail to preserve strict commutative monoids. A recent theorem of Hill and Hopkins gives conditions on the localization to prohibit this behavior. When we specialize our general machinery to the equivariant spectra we recover this theorem. En route to solving the localization problem we introduce an axiom which guarantees us that commutative monoids inherit a model structure. If there is time we will discuss a generalization which allows preservation of structure over arbitrary operads, and relate this to the situation of algebras in equivariant spectra over equivariant operads.

The stable homotopy groups of a G-spectrum are Mackey functors, and the zeroeth stable homotopy group of a com- mutative G-ring spectrum has the extra structure of a Tambara functor. However, while Mackey functors and Tambara functors make frequent appearances in equivariant stable homotopy theory, much of their underlying algebra remains mysterious. I will discuss a new structure on the category of Mackey functors such that Tambara functors are commutative algebra-like objects. Moreover, the advantage to this new structure is that it is concrete and computable.

Topological Hochschild Homology (THH) is an important invariant that comes up in both algebraic K-theory computations and topological quantum field theories. In this talk I’ll present a duality for THH of Koszul dual E₁-algebras. This duality has possible applications for computations, and is also the shadow of a conjectured richer structure for field theories.

This talk will be about the Kervaire invariant one problem in algebraic topology.

Given two homotopy algebras and an infinity-morphism between them, it is natural to ask that, if we can modify the two homotopy algebras in some structured way, can we modify the infinity-morphism in some similar way, so as to preserve the new structures? In this talk we describe a situation in which the answer is yes, and indicate how it is possible. We will also give an application of these results, to show that Tamarkin's construction of formality morphisms is equivariant with respect to the action of the Grothendieck-Teichmuller group.

We define the adjoint action of a homotopy associative H-space on its loop space, generalizing the definition by Kono and Kozima for a Lie group (from Kono, Kozima, 1993). We proceed to use this generalized adjoint action to characterize homotopy associative H-spaces whose homology over \(\mathbb{F}_p\) is a commutative ring, generalizing a result of Iwase (from Iwase, 1997).

Chromatic homotopy theory uses the algebraic geometry of formal groups to organize calculations. In particular, at each prime p there exists a series of homology theories K(n), called Morava K-theories and we can reconstruct the homotopy type of p-local spectra from their Morava K-theories localizations. When n = 2 a lot of information can be derived from the action of a certain profinite group, called the Morava stabilizer group, on the Lubin-Tate theory. We can form homotopy fixed points spectra with respect to this action and compute their homotopy groups using continuous group cohomology. We discuss a generalization to prime 2 of work of Goerss-Henn-Mahowald-Rezk on constructing a tower of fibrations, whose inverse limit is the spectrum \(E^{h\mathbb{S}^1_2}\), a “half” of the K(2)-local sphere. The successive fibers of the tower 2 are homotopy fixed points spectra with respect to specific finite subgroups of the Morava stabilizer group. This makes the computations accessible as it is possible to make very detailed calculations with finite subgroups using the theory of elliptic curves.

In this talk we will discuss a computation of the Adams-Novikov E₂ term for the spectrum Q(2). This spectrum is built using degree 2 isogenies of elliptic curves, and is closely tied to the 3-primary K(2)-local sphere. We will also examine potential connections between our computation and algebraic Greek letter families in the Adams-Novikov spectral sequence for the 3-local sphere.

Gorenstein homological algebra is essentially the study of modules after sending certain modules to zero. In the simplest case of modular representation theory, projective and injective modules coincide and sending them to zero gives a triangulated category called the stable module category. Such a simple plan will not work for a general ring. We show, however, that by changing one's notion of a “finite” module from finitely generated or presented to modules of type FP_∞, we get good analogues of flat and injective modules that are well-behaved for any ring. This enables us to develop Gorenstein homological algebra and an associated triangulated stable module category in full generality.

An equivariant infinite loop space machine should turn categorical or algebraic data into genuine G-spectra. While infinite loop space machines have been a crucial part of homotopy theory for decades, equivariant versions are in early stages of development. I will describe joint work with A. Osorno in which we build an equivariant infinite loop space machine that starts with diagrams of categories on the Burnside category and produces a genuine G-spectrum via the work of Guillou–May. This machine readily applies to produce Eilenberg–MacLane spectra for Mackey functors and topological K-theory.

In the early 1980's, Dress and Kuku, and Fiedorowicz, Hauschild and May introduced space level equivariant versions of the plus and Q constructions in algebraic K-theory. However, back then, the methods did not allow for nontrivial group action on the input ring or category. We generalize these definitions to the case in which a finite group G acts nontrivially on a ring (or an exact or Waldhausen category) and we show how to construct a genuine equivariant K-theory spectrum with good properties from a G-ring. An example of interest is that of a Galois extension.

The equivariant constructions rely on finding categorical models for classifying spaces of equivariant bundles (a joint project with Guillou and May) and the use of equivariant infinite loop space machines such as the one developed by Guillou and May, or the equivariant version of Segal's machine. The comparison of these machines, which will allow their interchangeable use in algebraic K-theory constructions, is a joint project with May and Osorno. New ideas are needed since, among other things, the comparison theorem of May and Thomason fails equivariantly.

Let \(R_n\) denote the homotopy fixed point spectrum \(E_n^{hS\mathbb G_n}\), where \(S\mathbb G_n\) is the kernel of the determinant homomorphism \(\text{det}:\mathbb G_n\to \mathbb Z_p^\times\) with \(\mathbb G_n\) being the Morava group. Here \(E_n\) denotes the \(n\)-th Morava \(E\)-theory. We show that for a \(K(n)\)-local space \(X\) equipped with a \(K(\mathbb Z_p, n+1)\)-bundle \(P\to X\), the \(P\)-twisted \(R_n\)-theory of \(X\), \(R_{n*}(X, P)\), is defined and there exist a “universal coefficient” isomorphism \[R_{n*}(X, P)\cong R_{n*}(P)\otimes _{R_{n*}(K(\mathbb Z_p, n+1))} \mathbb R_{n*}.\]

This extends an analogous result on twisted \(K\)-theory in the \(K(n)\)-local category.

Twisted K-theory is a cohomology theory whose cocycles are like vector bundles but with locally twisted transition functions. If we instead consider twisted vector bundles with a symmetry encoded by the action of a compact Lie group, the resulting theory is equivariant twisted K-theory. This subject has garnered much attention for its connections to conformal field theory and representations of loop groups. While twisted K-theory can be defined entirely in terms of the geometry of vector bundles, there is a homotopy-theoretic formulation using the language of parametrized spectra. In fact, from this point of view we can define twists of any multiplicative generalized cohomology theory, not just K-theory. The aim of this talk is to explain how this works, and then to propose a definition of equivariant twisted cohomology theories using a similar framework. The main ingredient is a structured approach to multiplicative homotopy theory that allows for the notion of a G-torsor where G is a grouplike A_∞ space.

Morava E-theory is an important cohomology theory in chromatic homotopy theory. Using work of Ando, Hopkins, and Strickland, Rezk described the algebraic structure found in the homotopy of K(n)-local commutative E-algebras via a monad on E_∗-modules that encodes all power operations. However, the construction does not see that the homotopy of a K(n)-local spectrum is L-complete (in the sense of Greenlees-May and Hovey-Strickland). We improve the construction to a monad on L-complete E_∗-modules, and discuss some applications.

We will compute the action of the Dyer-Lashof algebra on relative smash products using the multiplicative structure of the Künneth spectral sequence. We will then interpret such operations in terms of different possible E_∞-structures. We will end with an application of these computations to give a non-existence result for E_∞-complex orientations of certain ring spectra.

One question in homotopy theory is to construct and compute stable power operations on elliptic cohomology theories. As a particular instance, for E a Morava E-theory spectrum of height 2, its algebra of power operations has the structure of a graded twisted bialgebra satisfying a Frobenius congruence. For the K(1)-localization of E, its algebra of power operations has a single generator over the coefficient ring. We illustrate this structure and provide explicit formulas by doing calculations, at the prime 3, with moduli of elliptic curves.

The slice filtration is a filtration of equivariant spectra developed by Hill, Hopkins, and Ravenel in their solution to the Kervaire Invariant One Problem. After briefly discussing some properties of the filtration, I will present the slice towers of \(\Sigma^nH\mathbb{Z}\) where the group G is a cyclic p-group. I will highlight the patterns we see in this context by displaying a few specific examples.

The groups bo*bo have a been long understood by the work of Adams, Mahowald, Milgram, and others. The Adams perspective is based on numerical polynomials, wheras the Mahowald perspective is based on a splitting of bo*bo into the bo-homology of integral Brown–Gitler spectra.

Following a program initiated by Mahowald, we will discuss the analogous approach at odd primes. We will explain how to relate Mahowald's bo-Brown-Gitler spectrum approach to Laures' 2-variable modular forms, and a third geometric approach involving isogenies of elliptic curves.