Research
The following is a summary of my research output, loosely grouped by theme.
Periodic unstable homotopy theory, spectral Lie algebras, and Goodwillie calculus
Lie algebras and vn-periodic spaces, Annals of Mathematics 193.1 (2021): 223-301.
This paper proves that vn-periodic unstable homotopy theory is equivalent to the homotopy theory of Lie algebras in T(n)-local spectra, generalizing Quillen's rational homotopy theory.
A Whitehead theorem for periodic homotopy groups, with Barthel and Meier, Israel Journal of Mathematics 241.1 (2021): 1-16.
We prove that the collection of vn-periodic homotopy groups detects homotopy equivalences between simply-connected finite spaces.
Goodwillie approximations to higher categories, Memoirs of the American Mathematical Society Vol. 272, Nr. 1333 (2021).
This paper produces a tower of categories interpolating between stable and unstable homotopy theory by introducing the idea of a k-excisive approximation to a (higher) category. It includes various applications, among which a model for the homotopy theory of simply-connected spaces in terms of Tate coalgebras. The results of this paper also produce a description of the "Goodwillie tower" of vn-periodic unstable homotopy theory.
Lie algebra models for unstable homotopy theory, chapter in the Handbook of Homotopy Theory (2020), edited by Haynes Miller.
This is a survey describing the use of spectral Lie algebras as models for certain localizations of homotopy theory.
The vn-periodic Goodwillie tower on wedges and cofibres, with Brantner, Homology, Homotopy and Applications 22.1 (2020): 167-184.
We study the convergence of the Goodwillie tower in vn-periodic homotopy theory. Arone and Mahowald proved that the tower converges for spheres; we demonstrate that this is not the case for wedges of spheres and certain finite spaces of type n (e.g. Moore spaces).
Monadicity of the Bousfield-Kuhn functor, with Eldred, Mathew, and Meier, Proceedings of the American Mathematical Society 147.4 (2019): 1789-1796.
We show that the Bousfield-Kuhn functor exhibits vn-periodic unstable homotopy theory as a theory of algebras for a monad on the category of T(n)-local spectra.
The theory of higher categories and higher operads
Partition complexes and trees, with Moerdijk, to appear in Proceedings of the American Mathematical Society.
We provide a comparison between the partition complex of a finite set A and a certain category of trees with leaves labelled by A. This gives an elementary way to compare different versions of the bar construction of an operad appearing in the literature.
A short proof of the straightening theorem, with Hebestreit and Ruit.
This paper gives a streamlined proof of Lurie's straightening theorem.
Simplicial and dendroidal homotopy theory, with Moerdijk, Ergebnisse der Mathematik.
A book providing an introduction to the theory of topological/simplicial operads and ∞-operads, using dendroidal sets. We are keeping a list of errata here.
Two models for the homotopy theory of ∞-operads, with Chu and Haugseng, Journal of Topology 11 (2018): 856-872.
We prove that the homotopy theory of dendroidal Segal spaces is equivalent to that of Barwick's Segal operads. This missing link proves that all models for ∞-operads currently in the literature are equivalent, in particular those of dendroidal sets and Lurie's ∞-operads.
On the equivalence between Lurie's model and the dendroidal model for ∞-operads, with Hinich and Moerdijk, Advances in Mathematics 302 (2016): 869-1043.
The first paper providing a direct link between the two models in the title. It proves the equivalence of the title for the class of ∞-operads without constants.
Left fibrations and homotopy colimits II, with Moerdijk.
For a simplicial category A, we prove that the homotopy colimit functor gives an equivalence of homotopy theories from simplicial diagrams on A to simplicial sets over the nerve of A. This gives a new and quick proof of the straightening-unstraightening equivalence of Lurie.
Left fibrations and homotopy colimits I, with Moerdijk, Mathematische Zeitschrift 279.3-4 (2015): 723-744.
Gives a very straightforward proof of essentially the same statement as part II above, but in the case of an ordinary category A. The techniques developed here allow for efficient proofs of many useful basic facts in higher category theory, such as homotopy invariance of the covariant model structure, as well as ∞-categorical versions of Quillen's Theorems A and B.
This paper describes a consequence of the more general results of the paper below. It proves that the homotopy theory of dendroidal sets modelling ∞-operads admits a localization (the "covariant model structure") which is Quillen equivalent to that of E∞-spaces.
Essentially my MSc thesis written at Utrecht University. It extends the theory of left fibrations and coCartesian fibrations from simplicial sets to dendroidal sets. The main result is that the homotopy theory of coCartesian fibrations over an ∞-operad X is equivalent to the homotopy theory of algebras (valued in ∞-categories) over the simplicial operad associated to X.
Miscellaneous
The cotangent complex, extended abstract for the Oberwolfach Arbeitsgemeinschaft on Elliptic Cohomology (2019).
Some basic expository notes on the cotangent complex of a simplicial ring and of a commutative ring spectrum, written for a talk at Oberwolfach.
Ambidexterity, with Lurie, book chapter in "Topology and Field Theories", Contemporary Math. 613 (2014): 79-110.
These are notes from a lecture series by Jacob Lurie at Notre Dame. Ambidexterity is a duality phenomenon which can be exploited to construct generalized forms of Dijkgraaf-Witten topological field theories. These notes focus specifically on ambidexterity in K(n)-local homotopy theory.