Papers
Equivariant operads, symmetric sequences, and Boardman-Vogt tensor products (2025). [arxiv] [v1.5]
The purpose of this paper is to set the stage to study the homotopy theory of \(G\)-operads and their Boardman-Vogt tensor products. The important constructions are the underlying \(G\)-symmetric sequence and the Boardman-Vogt tensor product of \(G\)-operads. The important facts are the following:- The underlying \(G\)-symmetric sequence is monadic.
- There is a well-behaved localizing subcategory of \(G\)-\(d\)-operads (whose structure spaces are \((d-1)\)-truncated), compatible with all of the constructions.
- The \(\mathcal{O}\)-\(G\)-coefficient systems functor \(\mathcal{O} \mapsto \mathrm{Alg}_{\mathcal{O}}(\mathrm{Coeff}^G\mathcal{C})\) detects equivalences on the base change of the underlying \(G\)-symmetric sequence to \(\mathcal{C}\); in particular, evaluation on \(n\)-truncated \(G\)-spaces detects \(n\)-equivalences of \(G\)-operads and evaluation on \(G\)-spaces is conservative.
- The equivariant operadic nerve intertwines everything in sight, so it has a conservative right-derived functor; moreover, it's only the coherences that make things annoying, so it's easy to verify that it's an equivalence on one-color \(G\)-1-operads.
- The Boardman-Vogt tensor product works as you expect it to and intertwines with the Day convolution structure on \(G\)-symmetric monoidal \(\infty\)-categories via the envelope.
Orbital categories and weak indexing systems (2024). [arxiv] [v1.5]
This paper is an exposé on weak indexing systems, the combinatorics behind weak \(\mathcal{N}_\infty\) operads. The main point here is to show that, even though it's somewhat more complicated, you can run some version almost all of the homotopical combinatorics you know and love in the more general setting of subterminal \(G\)-operads, perhaps under mild unitality assumptions.This is not yet submitted: click here for v1.5.
Lower Bounds on Volumes of Hyperbolic 3-Manifolds via Decomposition (2021) with Colin Adams et al. [arxiv]
Lower bounds on the volumes of hyperbolic link complements are given via a new construction: a bracelet link is a link in \(S^3\) decomposed as a cycle of interconnected tangles, and it is proved that a bracelet link of \(2n\) tangles, such that each individual tangle may be replicated into a hyperbolic bracelet link of \(2n\) copies of the tangle, is hyperbolic, with volume at least the average of the replicated links.
This replication is generalized to arbitrary 3-manifolds via a construction called starbursts, which separate the manifold into pieces, which have a well defined \(2n\)-replicant; if the resulting pieces from removing a regular neighborhood of a starburst have hyperbolic replicants, then the 3-manifold is hyperbolic, with volume at least the average of the volumes of the replicants.
Applications are presented to hyperbolicity of links in thickened surfaces and in the solid torus.
Augmented cellular alternating links in thickened surfaces are hyperbolic (2021) with Colin Adams et al. [arxiv] [EJM]
Work of Colin Adams concerning hyperbolicity of generalized augmented alternating links in the 3-sphere is extended to hyperbolicity of such links in \(I\)-bundles over a surface other than the Klein bottle or \(\mathbf{RP}^2.\) This is used to prove hyperbolicity class of links in thickened orientable surfaces called rubber band links , which are generated by graphs. Both lower and upper bounds are provided for the volumes of rubber band links, both depending linearly on the number of edges in the graph.In-progress projects to look out for
On tensor products of equivariant commutative operads (2025). [draft]
Our main theorem constructs a unique equivalence \(\mathcal{N}_{I \infty} \otimes \mathcal{N}_{J \infty} \simeq \mathcal{N}_{I \vee J \infty}\). It'll come out soon, I promise.An Eckmann-Hilton argument in equivariant higher algebra (2025). [draft]
If you're familiar with Schlank-Yanovski, this works similarly; whereas connectivity of a space is a function on the orbit category, connectivity of a unital \(G\)-operad is most naturally viewed as a function on the poset of unital weak indexing systems (meaning minimum connectivity of \(I\)-admissible \(H\)-sets), and it turns out that connectivity function yields the obvious analog of Schlank-Yanovski's lower bound on Boardman-Vogt tensor products. We acquire probably the most _algebraic_ intrinsic characterization you can get for (almost unital weak) \(\mathcal{N}_\infty\)-operads: they are the targets of \(G\)-\(\infty\)-categorical Eckmann-Hilton arguments, or equivalently, they are the smashing localizations on (almost unital) \(G\)-operads.Connectivity of equivariant configuration spaces and \(\mathbb{E}_V\)-algebraic Wirthmüller maps (2025). [draft]
The results in this draft keep getting stronger, and the dependencies heavier; as of right now, the _point_ is that you can lift the Fadell-Neuwirth fibration to equivariant configurations, reducing connectivity statements for configurations in \(G\)-manifolds to connectivity and dimension statements in their strata with fixed isotropy. In the case of orthogonal representations, this itself is completely determined by the dimensions of various fixed point spaces; surprisingly, this means that the collection of arities \(S\) for which \(\mathbb{E}_V(S)\) is \(n\)-connected is closed under self-indexed coproducts and restruction, i.e. it's a _unital weak indexing system_. As a consequence, an easy dimension-counting condition completely classifies the connectivity of the fibers of semiadditive norm maps in the \(G\)-category of \(\mathbb{E}_V\)-\(G\)-spaces, quantifying how close these things are to \(G\)-semiadditivity (and hence how close \(\mathbb{E}_V\)-algebras are to weak \(\mathcal{N}_\infty\)-algebras).Canonical indexed tensor products of homotopical Mackey functors.
This is ongoing work with Bastiaan Cnossen, Tobias Lenz, and Sil Linskens.Homotopical equivariant Dunn additivity.
I'm working on writing up a proof that \(\mathbb{E}_V \otimes \mathbb{E}_W \simeq \mathbb{E}_{V \oplus W}\).Topics I intend to dive into within the next few years
A characterization of modules and cotangent complexes over equivariant operad algebras.
I hope to relitigate Higher Algebra chapter 7 in the setting of \(G\)-operads; the aim is to lift the characterization of cotangent complexes from Basterra-Mandell in the setting \(G = C_2\) using the (dihedral) Bar construction of e.g. Knoll-Gerhardt-Hill, in preparation for computations in Real deformation theory.The multiplication on \(\mathrm{BP}_{\mathbb{R}}\) and \(\mathrm{BP}_{\mathbb{R}} \langle n \rangle\).
I hope to construct canonical lifts of \(\mathbb{E}_{2n}-\mathrm{MU}\)-algebra structures on \(\mathrm{BP}\) to \(\mathbb{E}_{n \rho}-\mathrm{MU}_{\mathbb{R}}\)-algebra structures on \(\mathrm{BP}_{\mathbb{R}}\) using the cellularity results announced in Hill-Hopkins as well as the above conjectures concerning change of group functors applied to equivaraint operadic cotangent complexes.
After doing so, I hope to add \(\mathbb{R}\) and \(C_2\), as well as replace \(3\) with \(2 \rho - 1\), in section 2 of Hahn-Wilson and claim the new result as my own.
Infinite loop space theory for equivariant associative operads.
Rubin constructed an associative version of \(N_\infty\) operads; these are the family of \(\mathbb{E}_1\)-containing subfunctors of an equivariant associative operad. Tensor products of these are more complicated, as they are not tensor closed (indeed, \(\mathbb{E_2}\) is not a subfunctor of the equivariant associative operad, as it is not discrete)-- I hope to say something about this using techniques reminescent of Schlank-Yanovski and Guillou-May; in particular, I conjecture that the $k$-fold tensor product of \(\mathrm{As}(I)\) can be interpreted as having grouplike algebras which are the \(k\)-fold deloopings for the maps in the transfer system associated to \(I\), in the sense of limits of \(S\)-indexed cubes, and when \(k = \infty\), this computes \(\mathcal{N}_{I \infty}.\)
The aim of this is to supplant the \(\mathbb{E}_V\) family with a family whose tensor-indecomposables are of a combinatorial nature, bringing the explicit computational ease of the case \(G=C_2\) to arbitrary groups. This is low priority, since my computational examples of interest currently only have \(C_2\) actions.
Closing the gap between \(G\)-\(\infty\)-operads and genuine \(G\)-operads.
What remains in the equivariant version of the program to identify all models of operads with each other is an equivariant lift of Chu-Haugseng-Heuts; I hope to use the general machinery of algebraic patterns to add a \(G\) everywhere in the above paper.