**Research**

**Research Interests**

My research is in geometry and dynamical systems, particularly looking at how dynamics can be used to solve geometric problems. My interests include:

- Riemannian and pseudo-Riemannian geometry
- geometric dynamical systems — geodesic flows, frame flows...
- symmetric and homogeneous spaces
- homogeneous dynamics
- rigidity theory — see Ralf's "An Invitation to Rigidity Theory" for an idea of what this is.
- Lie Groups
- singular spaces (CAT(-1) and CAT(0) metrics and the like)

I have done specific work on rank-rigidity; shrinking target problems with Jon Chaika; marked length spectrum problems on my own and with Jean-François Lafont; Hausdorff-measure regularity of CAT(κ) surfaces with Jean-François Lafont; the one-sided ergodic Hilbert transform with Joanna Furno; the specification property with Jean-François Lafont and Dan Thompson; the geodesic structure of nonpositively curved manifolds with Jean-François Lafont, Dan Thompson and Ben McReynolds; and Hilbert geometries with Ilesanmi Adeboye and Harrison Bray.

Current projects include some further work with Jean-François Lafont and Dan Thompson.

**Students**

- Cameron Bishop (Ph.D. student)
- Noelle Sawyer (Ph.D. student)
- Narin Luangrath and Melissa Mischell (summer undergraduate researchers, 2015)
- Fanying Chen (summer undergraduate researcher, 2016)
- Jeanne Li and Alejandra Marcelino (summer undergraduate researchers, 2017)

**Preprints and Publications**

(joint w. Jean-François Lafont and Dan Thompson)

*Strong symbolic dynamics for geodesic flow on CAT(-1) spaces and other metric Anosov flows*(preprint).

We prove that the geodesic flow on a compact, locally CAT(-1) space has a strong Markov coding. That is, it can be coded by a suspension flow over a shift of finite type with a Hölder roof function. This gives a number of strong results about the dynamics of such flows, including Bernoullicity of the measure of maximal entropy. We also prove this strong coding for the geodesic flow associated to an Anosov representation.(joint w. Ilesanmi Adeboye and Harrison Bray)

*Entropy rigidity and Hilbert volume*, to appear in*Discrete and Continuous Dynamical Systems*(preprint).

We adapt the entropy rigidity result of Besson, Courtois and Gallot (following ideas of Boland and Newberger) to compact quotients of strictly convex real projective manifolds which admit a hyperbolic structure in dimension at least 3. We obtain a similar (though slightly weaker) entropy rigidity statement. Using this and some facts about the Blaschke metric, we prove that the ratio of the Hilbert volume to the hyperbolic volume is uniformly bounded below (by a constant depending only on volume), and that if the projective structure is deformed so that topological entropy of the geodesic flow goes to zero, the volume must go to infinity. We also prove the latter result in dimension 2.(joint w. Jean-François Lafont, D. Ben McReynolds, and Dan Thompson)

*Fat flats in rank-one manifolds,*to appear in Michigan Mathematical Journal. (preprint).

We construct some rank-one manifolds containing twisted 'fat geodesics' -- geodesics which have a uniform flat neighborhood -- but which still contain only a countable collection of closed geodesics. We examine the dynamics of the geodesic flow in such spaces. The paper also contains a closing lemma for 'fat*k*-flats' which proves that for*k*>1, having a uniform flat neighborhood of a*k*-flat implies that there are uncountably many closed*k*-flats.(joint w. Jean-François Lafont and Dan Thompson)

*The weak specification property for geodesic flows on CAT(-1) spaces*, to appear in Groups, Geometry, and Dynamics (preprint).

We prove a weakened, but still quite useful, version of the specification property for geodesic flows on CAT(-1) spaces. Geodesic flows on compact negatively curved manifolds are an important example of the usual specification property and it can be used to prove many strong results on the dynamics of such flows. With the weak specification property we recover a number of those results in the CAT(-1) setting, including uniqueness of the measure of maximal entropy.

(joint w. Jean-François Lafont)

*On the Hausdorff dimension of CAT(κ) surfaces*. Analysis and Geometry in Metric Spaces**4**(2016), 266-277.

We show that a compact surface equipped with a CAT(κ) metric has Hausdorff dimension 2 and discuss some connections between this regularity results and the dynamics of the geodesic flow. We also discuss entropy rigidity for CAT(-1) manifolds of higher dimension.

(joint w. Joanna Furno)

*Everywhere divergence of the one-sided ergodic Hilbert transform for circle rotations by Liouville numbers,*New York Journal of Mathematics**23**(2017), 273-294.

We prove some results on ergodic Hilbert transforms of a certain class of functions -- basically mean-zero indicator functions for a finite collection of intervals. We give a connection between everywhere divergence of the transform and Liouville numbers, and construct some Liouville numbers for which the transform converges everywhere.

*Marked length spectrum rigidity in nonpositive curvature with singularities,*to appear in Indiana University Mathematics Journal. (preprint).

Proves marked length spectrum rigidity for surfaces with metrics which are nonpositively curved and may have some cone singularities. The angle around each singularity should be >2π. The requirements on one metric may be softened to `non conjugate points' with the addition of an additional hypothesis which may always be true. The proof consists of combining the work of several previous authors on MLS rigidity for surfaces.

(joint w. Jean-François Lafont)

*Marked length rigidity for Fuchsian buildings,*to appear in Ergodic Theory and Dynamical Systems. (preprint).

We prove marked length spectrum rigidity for compact quotients of Fuchsian buildings under some negative curvature assumptions.

(joint w. Jean-François Lafont)

*Marked length rigidity for one-dimensional spaces*, to appear in Journal of Topology and Analysis. (preprint).

We prove marked length spectrum rigidity for geodesic metric spaces with topological dimension 1. These include the easy case of graphs, but also more exotic spaces like Hawaiian earrings and Laakso spaces.

*Compact Clifford-Klein forms -- Geometry, Topology and Dynamics.*In Geometry, Topology and Dynamics in Negative Curvature (Bangalore, 2010), London Math Society Lecture Notes 425. Eds. C.S. Aravinda, F.T. Farrell, and J.-F. Lafont. Cambridge University Press, 2016. pp. 110-145.

A survey article on the question of which homogeneous spaces have compact quotients. The focus here is on non-Riemannian homogeneous spaces.

(joint w. Jon Chaika)

*A quantitative shrinking target result on Sturmian sequences for rotations*, Discrete and Continuous Dynamical Systems**38**(2018), no.10, 5189-5204.

We prove a quantitative shrinking target result for Sturmian sequences derived from circle rotations. You can think of this result as giving some information about the asymptotics of the measure of points whose orbit coding up to step*n*does not determine the coding at step*n+1*(for a special and naturally derived coding). We prove a weak asymptotic and show why a stronger asymptotic fails.

(joint w. Jon Chaika)

*Quantitative shrinking target properties for rotations and interval exchanges*, to appear in Israel Journal of Mathematics. (preprint).

We prove several results on shrinking target problems for rotations and interval exchanges.

*2-Frame flow dynamics and hyperbolic rank-rigidity in nonpositive curvature*, Journal of Modern Dynamics**2**(2008), no. 4, 719-740.

Presents a rank-rigidity result achieved by looking at dynamics of the frame flow.

*Consequences of ergodic frame flow for rank rigidity in negative curvature*.

This is an earlier version of the paper above with results for negative curvature only. It is not planned for publication but may be of interest if you only want the negative curvature proof, which is considerably simpler. Only in dimensions 7 & 8 that is it really necessary to prove anything substantial here (see note after Thm 1 in "2-Frame flow...").

(joint w. Matt Darnall)

*Lengths of finite dimensional representations of PBW algebras*, Linear Algebra and its Applications**395**(2005), 175-181.

This is from a summer research program at Temple University directed by Ed Letzter.

My thesis, "Hyperbolic rank-rigidity and frame flow." The content of (12.) above, but with more detail.