Papers
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
On tensor products of equivariant commutative operads (2024). [draft intro]
I'm near finishing this one. A somewhat casual presentation of the binary case (with numerous errors) can be found at my notes "You can construct G-commutative algebras one norm at a time," prepared for a casual talk at Zygotop.Weak indexing systems via transfer systems.
Many of the combinatorial results concerning \(\mathcal{N}_\infty\)-operads factor through the recognition that indexing systems only depend on their intersections with the orbit category, which are intrinsically characterized via transfer systems. This project is aimed towards reducing the combinatorics of unital weak indexing systems (i.e. the arity-supports of unital \(G\)-operads) in terms of transfer systems via some manageable auxiliary data.Canonical indexed tensor products of homotopical Mackey functors.
I've had to develop many of the rudiments of Gepner-Groth-Nikolaus for non-indexed tensor products in the equivariant setting for the above project; the indexed side is in progress now. The stability side of this is of course already done by Denis Nardin, so this is mostly about rounding up routine arguments and developing a basic understanding of how indexed tensors in \(\mathrm{Pr}^L_G\) interact with Segal conditions. The intended upshot is showing that HHR norms, localized Day convolutions, and box products are all the same and occur in much greater generality; this is a first step in showing that homotopical Tambara functors are the same thing as $G$-commutative ring spectra.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.An \(\infty\)-categorical argument for the tom Dieck splitting of stable \(\mathcal{T}\)-Mackey functors.
I hope to flesh out the content of a talk I gave at Zygotop in order to prove a tom Dieck splitting for Mackey functors parameterized by atomic orbital \(\infty\)-categories and valued in arbitrary stable categories. Finishing this is low-priority, since I don't have a use, as I haven't found a use for \(\mathcal{T}\)-stabilization of categories of coefficient systems outside of the setting of \(G\)-equivariant suspension spectra, where the result is well-known.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.