Publications and preprints
- Derived Koszul duality and TQ-homology completion of structured ring spectra, (joint with John E. Harper), arXiv:1502.06944.
Working in the context of symmetric spectra, we consider any higher algebraic structures that can be described as algebras over an operad O. We prove that the fundamental adjunction comparing O-algebra spectra with coalgebra spectra over the associated comonad K, via topological Quillen homology (or TQ-homology), can be turned into an equivalence of homotopy theories by replacing O-algebras with the full subcategory of 0-connected O-algebras. This resolves in the affirmative the 0-connected case of a conjecture of Francis-Gaitsgory.
This derived Koszul duality result can be thought of as the spectral algebra analog of the fundamental work of Quillen and Sullivan on the rational homotopy theory of spaces, and the subsequent p-adic and integral work of Goerss and Mandell on cochains and homotopy type---the following are corollaries of our main result: (i) 0-connected O-algebra spectra are weakly equivalent if and only if their TQ-homology spectra are weakly equivalent as derived K-coalgebras, and (ii) if a K-coalgebra spectrum is 0-connected and cofibrant, then it comes from the TQ-homology spectrum of an O-algebra. We construct the spectral algebra analog of the unstable Adams spectral sequence that starts from the TQ-homology groups of an O-algebra X, and prove that it converges strongly to the homotopy groups of X when X is 0-connected.
- Manifolds, K-theory and the calculus of functors, (joint with Greg Arone), submitted, arxiv:1410.1809.
The Taylor tower of a functor from based spaces to spectra can be classified according to the action of a certain comonad on the collection of derivatives of the functor. We describe various equivalent conditions under which this action can be lifted to the structure of a module over the Koszul dual of the little L-discs operad. In particular, we show that this is the case when the functor is a left Kan extension from a certain category of `pointed framed L-manifolds' and pointed framed embeddings. As an application we prove that the Taylor tower of Waldhausen's algebraic K-theory of spaces functor is classified by an action of the Koszul dual of the little 3-discs operad.
- Cross-effects and the classification of Taylor towers, (joint with Greg Arone), submitted, arxiv:1404.1417.
We show that the partially-stabilized cross-effects of a functor from either based spaces of spectra, to spectra, admit the structure of a module over a certain operad: the Koszul dual of the little discs operad in the first case, and a desuspension of the commutative operad in the second case. The derivatives inherit a limiting structure as a module over a 'pro-operad'. We show that the Taylor tower of the functor can be recovered from this structure.
- Higher homotopy excision and Blakers-Massey theorems for structured ring spectra, (joint with John E. Harper), submitted, arXiv:1402.4775.
Working in the context of symmetric spectra, we prove higher homotopy excision and higher Blakers–Massey theorems, and their duals, for algebras and left modules over operads in the category of modules over a commutative ring spectrum (e.g., structured ring spectra). We also prove analogous results for algebras and left modules over operads in unbounded chain complexes.
- A classification of Taylor towers for functors of spaces and spectra, (joint with Greg Arone), Advances in Mathematics 272 (2015), 471-552.
We describe new structure on the Goodwillie derivatives of a functor, and we show how the full Taylor tower of the functor can be recovered from this structure. This new structure takes the form of a coalgebra over a certain comonad which we construct, and whose precise nature depends on the source and target categories of the functor in question. The Taylor tower can be recovered from standard cosimplicial cobar constructions on the coalgebra formed by the derivatives. We get from this an equivalence between the homotopy category of polynomial functors and that of bounded coalgebras over this comonad. For functors with values in the category of spectra, we give a rather explicit description of the associated comonads and their coalgebras. In particular, for functors from based spaces to spectra we interpret this new structure as that of a divided power right module over the operad formed by the derivatives of the identity on based spaces.
- Coalgebraic models for combinatorial model categories, (joint with Emily Riehl), Homology, Homotopy and Applications 16(2) (2014), 171-184.
We show that the category of algebraically cofibrant objects in a combinatorial and simplicial model category A has a model structure that is left-induced from that on A. In particular it follows that any presentable model category is Quillen equivalent (via a single Quillen equivalence) to one in which all objects are cofibrant.
- Apollonian circle packings of the half-plane, (joint with John Doyle), Journal of Combinatorics 3 (2012), 1-49.
We classify Apollonian circle packings of a half-plane up to Euclidean similarity and describe the relationship between such packings and continued fractions.
- A note on the composition product of symmetric sequences, Journal of Homotopy and Related Structure 7 (2012), 237-254.
We consider the composition product of symmetric sequences in the case where the underlying symmetric monoidal structure does not commute with coproducts. Even though this composition product is not a monoidal structure on symmetric sequences, it has enough structure, namely that of a `normal oplax' monoidal product, to be able to define monoids (which are then operads on the underlying category) and make a bar construction. The main benefit of this work is in the dual setting, where it allows us to define a cobar construction for cooperads.
- Bar-cobar duality for operads in stable homotopy theory, Journal of Topology 5 (2012), 39-80.
We describe a bar-cobar duality for operads of spectra. In particular, we construct a Quillen equivalence between the projective model structure on the category of operads, and a new model for the homotopy theory of cooperads of spectra.
- Operads and chain rules for calculus of functors, (joint with Greg Arone), Astérisque 338 (2011), 158 pages.
We study the structure possessed by the Goodwillie derivatives of a pointed homotopy functor of based topological spaces. These derivatives naturally form a bimodule over the operad consisting of the derivatives of the identity functor. We then use these bimodule structures to give a chain rule for higher derivatives in the calculus of functors, extending that of Klein and Rognes. This chain rule expresses the derivatives of FG as a derived composition product of the derivatives of F and G over the derivatives of the identity. There are two main ingredients in our proofs. Firstly, we construct new models for the Goodwillie derivatives of functors of spectra. These models allow for natural composition maps that yield operad and module structures. Then, we use a cosimplicial cobar construction to transfer this structure to functors of topological spaces. A form of Koszul duality for operads of spectra plays a key role in this.
- A chain rule for Goodwillie derivatives of functors from spectra to spectra, Trans. Amer. Math. Soc. 362 (2010), 399-426.
We prove a chain rule for the Goodwillie calculus of functors from spectra to spectra. We show that the (higher) derivatives of a composite functor FG at a base object X are given by taking the composition product (in the sense of symmetric sequences) of the derivatives of F at G(X) with the derivatives of G at X. We also consider the question of finding Pn(FG), and give an explicit formula for this when F is homogeneous.
- Bar constructions for topological operads and the Goodwillie derivatives of the identity, Geom. Topol. 9 (2005), 833-933.
We describe a cooperad structure on the simplicial bar construction on a reduced operad of based spaces or spectra and, dually, an operad structure on the cobar construction on a cooperad. We also show that if the homology of the original operad (respectively, cooperad) is Koszul, then the homology of the bar (respectively, cobar) construction is the Koszul dual. We use our results to construct an operad structure on the partition poset models for the Goodwillie derivatives of the identity functor on based spaces and show that this induces the ‘Lie’ operad structure on the homology groups of these derivatives. We also extend the bar construction to modules over operads (and, dually, to comodules over cooperads) and show that a based space naturally gives rise to a left module over the operad formed by the derivatives of the identity.
- Koszul duality for modules and comodules over operads of spectra
There are two versions of Koszul duality between modules over an operad of spectra and comodules over the Koszul dual operad. In one version, we have a Quillen equivalence between the category of P-modules and KP-comodules where the left adjoint is a model for the derived indecomposables of a P-module, and the right adjoint is a model for the derived primitives of a KP-comodule. In the second version, we have a Quillen adjunction (but not typically an equivalence) in which the left adjoint models the derived indecomposables of a KP-comodule and the right adjoint the derived primitives of a P-module. The monad and comonad associated to this adjunction encode structures which we think of as those of a divided power KP-comodule and divided power P-module, respectively.
- Algebras over the Taylor tower of the identity
The goal of this project is to construct monad models for the terms in the Taylor tower of the identity functor (on based spaces) and to compare algebras for these monads with coalgebras over the terms in the Taylor tower of the functor Sigma^infty Omega^infty.
- Localized Taylor towers (joint with Greg Arone)
This project is to extend Kuhn's results on the splitting of Taylor towers of functors of K(n)-local spectra to the unstable case. We do not expect split Taylor towers in this case, but we do expect the Taylor towers to be determined by the Goodwillie derivatives together with the appropriate bimodule structure. The idea is to apply Kuhn's theorem on vanishing Tate cohomology to the best approximation to a functor determined by the derivatives.
- Completion of operadic algebras with respect to Quillen homology (joint with John E. Harper)
The goal of this project is to give connectivity conditions under which an algebra A over an operad P (in spectra or chain complexes) can be recovered from its Quillen homology. This is part of the foundation of a Koszul duality for relating algebras over P to coalgebras over the bar construction BP.
- The Taylor tower and Goodwillie derivatives of algebraic K-theory (joint with Andrew Blumberg)
This is an attempt to calculate some parts of the Taylor tower and Goodwillie derivatives of algebraic K-theory viewed as a functor from associative (or commutative) S-algebras to spectra. We use the framework developed in my work with Greg Arone and use results of Basterra-Mandell on the stabilization of categories of S-algebras, and of Lindenstrauss-McCarthy on Taylor towers of other algebraic K-theory functors.
- Geometric self-duality for the little discs operads (joint with Benoit Fresse and Paolo Salvatore)
Getzler and Jones showed that the homology of the little n-discs operad is self-Koszul dual, up to suspension. This project is an attempt to realize that duality geometrically using my work on bar-cobar duality for operads of spectra.
- Calculus of functors and configuration spaces
This is a summary of a talk given at the Conference on Pure and Applied Topology on the Isle of Skye in June 2005. We describe a relationship between right modules over the operad formed by the Goodwillie derivatives of the identity functor, and configuration spaces on manifolds. The missing step in turning this project into a paper is a proper understanding of the self-Koszul-duality of the Fulton-MacPherson operads of compactified configuration spaces.
- Cross-effects of homotopy functors and spaces of trees
This paper describes models for the cross-effects of homotopy functors given in terms of spaces of trees. These link closely to the methods of the paper "Bar constructions for topological operads and the Goodwillie derivatives of the identity", but I do not have any useful applications for these models.
- Homotopy operations on simplicial algebras over an operad
This was my first attempt as a graduate student to understand what operations exist on the homotopy groups of a simplicial algebra over an operad, in a sense generalizing the Cartan-Bousfield-Dwyer operations on simplicial commutative algebras. It contains some partial results, but not a full description.