Papers
Orbital categories and weak indexing systems (2024). [arxiv]
This paper is an exposé on weak indexing systems, the combinatorics behind weak \(\mathcal{N}_\infty\) operads.Lower bounds on volumes of hyperbolic link complements in 3-manifolds (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
Equivariant operads, symmetric sequences, and the Boardman-Vogt tensor product (2024). [draft]
At some point was the first half of "On tensor products of equivariant commutative operads." I'm hoping to put it out in October.On tensor products of equivariant commutative operads (2024). [draft]
100 pages is too long, so I'm splitting this into two papers. This title corresponds with the second half of the split, which should be out in November.On connectivity of spaces of equivariant configurations (2024). [draft]
This paper is what it says on the tin; given \(V\) a real orthogonal \(G\)-representation (which may have no fixed points), we give sharp bounds on connectivity of spaces of \(H\)-equivariant configurations of finite \(H\)-sets in it; the upshot is that when \(V\) is large (e.g. it is a \(d\)-fold direct sum), \(\mathbb{E}_V\)-algebras canonically lift to \(AV\)-commutative algebras, and hence they are modelled by incomplete Mackey functors or bi-incomplete Tambara functors in a number of cases. I'm hoping to put this out in December.Canonical indexed tensor products of homotopical Mackey functors.
This is ongoing work with Bastiaan Cnossen, Tobias Lenz, and Sil Linskens.Stable additivity of the \(\mathbb{E}_V\) family of \(G\)-operads.
I hope to leverage the closedness of the equivariant BV tensor product to reduce stable additivity of \(\mathbb{E}_V\) operads to checking on algebras in pointed connected \(G\)-spaces, which are well-described using loop space theory.Topics I intend to dive into within the next few years
The equivariant \(\infty\)-cateogical Eckmann-Hilton argument
I think you can run an analogue of Schlank-Yanovski, with \(\mathrm{Comm}_G\) replaced with an almost essentially unital weak \(\mathcal{N}_\infty\)$-operad, so in particular, a unital one. In particular, this has some corollaries along the lines of equivariant Dunn additivity (and should not be very hard), so I'll probably try to work this one out soon after the connectivity paper goes up.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.