Research

I am broadly interested in smooth 4-manifold topology, particularly when there is a finite group action. I am also learning a lot about Khovanov homology as a tool to study 4-manifolds and related ideas, and I am really enjoying the associated homological algebra. I hope to apply Khovanov homology to cobordism and concordance problems for knots.

Selected Works:

[Or go to the full list of publications and preprints, coming soon!]

Equivariant trisections for group actions on four-manifolds

With Jeffrey Meier

Preprint (2025)

We lay the foundations for a theory of trisections for 4-manifolds X equipped with a smooth action of a finite group G. We begin by giving a solid definition, and show that such a trisection always exists for a given G-action on X. We show that an equivariant trisection allows you to 'see' the G-action via a G-action on a trisection diagram.
We then develop the technology some: we show that equivariant trisections behave nicely with respect to the quotient by the group action, we elaborate a large quantity of examples, and we classify examples of low trisection genus. Analogues of these results are also developed for G-invariant surfaces via a notion of equivariant bridge trisections.

We conclude by discussing several directions for future research. (arXiv, published in Groups, Geometry, and Dynamics)

An equivariant Laudenbach-Poénaru theorem

With Jeffrey Meier

Preprint (2025)

We prove an analogue of the classical Laudenbach-Poénaru theorem in the presence of a finite group G acting on all manifolds involved. We first identify the correct analogue to a 1-handlebody in the equivariant setting, a class of G-actions on 1-handlebodies which we call linearly parted actions, and then directly manipulate the resulting equivariant handle decompositions. In the case where G is the trivial group, this gives a new proof of the classical theorem with interesting technical features. With an eye towards equivariant trisections, we also prove a generalization to the case where the 1-handlebody contains an invariant set of properly embedded, equivariantly unknotted disks. (arXiv, to appear in Michigan Mathematical Journal)

Two sufficient conditions for a polyhedron to be (locally) Rupert

Preprint (2022)

In my first-ever research effort, we analyze an interesting property of polyhedra: a polyhedron P is Rupert if one can take two identical copies of P and bore a hole straight through the first, keeping it in one piece, which is large enough to pass the other straight through. Equivalently, P is Rupert if there are two planar projections of P so that one is contained in the interior of the other; see the image for such projections for the cube. We give two general sufficient conditions for when such a passage exists in which the two projections are arbitrarily similar. (arXiv)

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