My research is in topology and geometry—specifically, surgery theory and geometric group theory. A few of my favorite things include: aspherical manifolds, rational homotopy types of manifolds, group actions on manifolds, quantified versions of classical invariants.

You may also be interested in seeing a list of my publications or reading my research statement.

Work in progress

Projective planes Joint work with Zhixu Su

The Hirzebruch $L$-polynomial is one place where number theory very strongly interacts with high-dimensional topology [MR339202]. Recall that the Hirzebruch signature theorem relates the signature of a smooth closed oriented manifold $M^{4k}$ to $\sum_I L_I p_I(M)$. Unfortunately, the naïve method to compute coefficients $L_I$ of the Hirzebruch $L$-polynomial is much too slow for applications; Zhixu Su and I have discovered a recursive method which is fast enough to compute many coefficients.

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The no-three-in-line problem on a torus Joint work with Andrew Groot, Deven Pandya, Bart Snapp

Fr a group $G$, let $T(G)$ denote the cardinality of the largest subset $S \subset G$ so that no three elements of $S$ are in the same coset of a cyclic subgroup. Undergraduates Andrew Groot and Deven Pandya, advised by myself and my colleague Bart Snapp, considered the case $G = \mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$, and showed that \begin{align*} T(\mathbb{Z}_p \times \mathbb{Z}_{p^2}) &= 2p, \\ T(\mathbb{Z}_p \times \mathbb{Z}_{pq}) &= p+1. \end{align*} This problem can also be formulated as a Gröbner basis question; after doing so, we computed $T(\mathbb{Z}_m \times \mathbb{Z}_n)$ for $2 \leq m \leq 7$ and $2 \leq n \leq 19$.

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L-class computations

Computation of the total $L$-class would also solve some manifold recognition problems. Here is an example of such a problem and my proposed approach. Brehm and Kühnel in [MR1180457] exhibit a few different combinatorial 15-vertex triangulations of an 8-manifold “like” the quaternionic projective plane $\mathbb{H}P^8$. One of these examples $X^8$ is especially symmetric, and likely PL homeomorphic to $\mathbb{H}P^8$.

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Previous projects

Aspherical manifolds that cannot be triangulated Joint work with Michael W. Davis, Jean-François Lafont

Kirby and Siebenmann showed that there are manifolds that do not admit PL structures [MR242166], and yet the possibility remained that all manifolds could be triangulated, meaning that for every manifold $M$, there is a simplicial complex $K$ so that the geometric realization of $K$ is homeomorphic to $M$, but of course the simplicial complex $K$ is not a PL triangulation, meaning the links are not spheres.

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Finiteness properties

A combination of Bestvina–Brady Morse theory and an acyclic reflection group trick produces a torsion-free finitely presented $\mathbb{Q}$-Poincaré duality group which is not the fundamental group of an aspherical closed ANR $\mathbb{Q}$-homology manifold.

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Bounded homotopy theory Joint work with Crichton Ogle

In controlled topology, notions like homotopy are refined to include a condition on their size, measured via a reference map to a metric space. There are different versions of controlled topology in current use, including bounded control and continuous control. But there are other versions of control that are worth considering.

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Ph.D. Thesis

A group $G$ is a Poincaré duality group if its classifying space $BG$ satisfies Poincaré duality; examples include fundamental groups of aspherical manifolds. C. T. C. Wall asked whether a Poincaré duality group is necessarily the fundamental group of an aspherical manifold, and M. Davis in [MR1747535] broadened Wall's question to $R$-homology manifolds:

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Cusp size bounds from singular surfaces in hyperbolic 3-manifolds Joint work with Colin Adams, Adam Colestock, William Gillam, Eric Katerman

Singular maps of surfaces into a hyperbolic 3-manifold are utilized to find upper bounds on meridian length, $\ell$-curve length and maximal cusp volume for the manifold. This allows a proof of the fact that there exist hyperbolic knots with arbitrarily small cusp density and that every closed orientable 3-manifold contains a knot whose complement is hyperbolic with maximal cusp volume less than or equal to 9.

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Cleanliness of geodesics in hyperbolic 3-manifolds Joint work with Colin Adams, Adam Colestock, William Gillam, Eric Katerman

Imagine you are inside a cusped hyperbolic 3-manifold, and you have situated yourself at infinity, and now you are looking down the cusp. You see a pattern of horoballs within the fundamental parallelogram for the cusp subgroup fixing $\infty$. Imagining that horoballs are opaque, how many are visible? Do finitely many horoballs suffice to cover the fundamental parallelogram? This question was first asked by Darren Long, and this paper answers it.

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