Research
My research interests lie in algebraic topology, where one uses tools from algebra to study properties (invariants) of topological spaces. I am particularly interested in stable homotopy theory with an equivariant flavour, functor calculus, and how these relate to one another. More recent projects involve (unstable) chromatic homotopy theory, model (2-)categories, and higher structures.
Publications
Comparing orthogonal calculus and calculus with Reality. To appear in Mathematische Zeitschrift (2024).
We show that there exists a suitable 𝐶_2-fixed points functor from calculus with Reality to the orthogonal calculus of Weiss which recovers orthogonal calculus “up to a shift” in an analogous way with the recovery of real topological 𝐾 -theory from Atiyah’s 𝐾 -theory with Reality via appropriate 𝐶_2-fixed points.
The localization of orthogonal calculus with respect to homology To appear in Algebraic and Geometry Topology (2024).
We construct a version of Weiss’ orthogonal calculus which only depends on the local homotopy type of the functor involved. Our theory specialises to homological localizations and nullifications at a based space. We give a variety of applications including a reformulation of the Telescope Conjecture in terms of our local orthogonal calculus and a calculus version of Postnikov sections.
Recovering unitary calculus from calculus with Reality Journal of Pure and Applied Algebra 227 (2023), no. 12.
We show that unitary functor calculus can be completely recovered from the unitary functor calculus with reality, in analogy to how complex topological K–theory is completely recovered from K–theory with reality via forgetting the C_2–action.
Unitary calculus: model categories and convergence. J. Homotopy Relat. Struct. 17 (2022), no. 3, 419--462.
We construct the unitary analogue of orthogonal calculus developed by Weiss, utilising model categories to give a clear description of the intricacies in the equivariance and homotopy theory involved.
We construct a calculus of functors in the spirit of orthogonal calculus, which is designed to study “functors with reality” such as the Real classifying space functor, BU_R(−).
Comparing the orthogonal and unitary functor calculi. Homology, Homotopy and Applications, 23 (2021), no.2, 227-256.
We construct functors between orthogonal and unitary calculi from the complexification-realification adjunction between real and complex inner product spaces. These allow for movement between the versions of calculi, and comparisons between the Taylor towers produced by both calculi.
Preprints
Symplectic Weiss calculi joint with M. Carr (2024)
We provide two candidates for symplectic Weiss calculus based on two different, but closely related, collections of groups. In the case of the non-compact symplectic groups we show that the calculus deformation retracts onto unitary calculus as a corollary of the fact that Weiss calculus only depends on the homotopy type of the groupoid core of the diagram category. In the case of the compact symplectic groups we provide a comparison with the other known versions of Weiss calculus and classify certain stably trivial quaternion vector bundles.
The Goodwillie calculus of polyhedral products joint with G. Boyde (2024)
We describe the Goodwillie calculus of polyhedral products in the case that the fat wedge filtration on the associated real moment-angle complex is trivial. As a corollary we show that the Goodwillie calculus of these polyhedral products converges integrally and diverges in vh-periodic homotopy unless the simplicial complex is a full simplex.
Expository Writing
The following are some expository notes. All mistakes are mine.
We use the Eilenberg–Moore spectral sequence to calculate the second derivative of BO(−) in orthogonal calculus, expanding on the proof provided by Weiss.
We use the language of model categories to construct the Postnikov tower and connective cover of (orthogonal) spectra.
Thesis
We construct new versions of orthogonal calculus, a unitary version which considers complex vector spaces, and a calculus with reality, an extension of the unitary calculus which takes into account the complex conjugation action on the complex vector spaces. We then compare all of these versions to discuss a unified approach to Weiss calculus.