Research

The Thurston metric on Teichmüller space, first introduced by W.P. Thurston is an asymmetric metric on Teichmüller space. This metric is geodesic, but geodesics are far from unique. In this thesis, we show that in the once-punctured torus, and in the four-times punctured sphere, geodesics stay a uniformly-bounded distance from each other. In other words, we show that the width of the geodesic envelope, E(X,Y) between any pair of points X,Y ∈ T(S) (where S is the once-punctured torus of the four-times punctured sphere) is bounded uniformly. To do this, we first identify extremal geodesics in Env(X,Y), and show that these correspond to stretch vectors, proving a conjecture of Y. Huang, K. Ohshika, and A. Papadopoulos,. We then compute Fenchel-Nielsen twisting along these paths, and use these computations, along with estimates on earthquake path lengths, to prove the main theorem.

In this article, we construct a new simplicial complex for infinite-type surfaces, which we call the grand arc graph. We show that if the end space of a surface has at least three different self-similar equivalence classes of maximal ends, then the grand arc graph is infinite-diameter and δ-hyperbolic. We also show that the mapping class group acts on the grand arc graph by isometries and that the action is quasi-continuous, which is a coarse relaxation of a continuous action. When the surface has stable maximal ends, we also show that this action has finitely many orbits. 

In this paper, we study the relationship between the mapping class group of an infinite-type surface and the simultaneous flip graph, a variant of the flip graph for infinite-type surfaces defined by Fossas and Parlier. We show that the extended mapping class group is isomorphic to a proper subgroup of the automorphism group of the flip graph, unlike in the finite-type case. This shows that Ivanov's metaconjecture, which states that any "sufficiently rich" object associated to a finite-type surface has the extended mapping class group as its automorphism group, does not extend to simultaneous flip graphs of infinite-type surfaces.

In political redistricting, the compactness of a district is used as a quantitative proxy for its fairness. Several well-established, yet competing, notions of geographic compactness are commonly used to evaluate the shapes of regions, including the Polsby-Popper score, the convex hull score, and the Reock score, and these scores are used to compare two or more districts or plans. In this paper, we prove mathematically that any map projection from the sphere to the plane reverses the ordering of the scores of some pair of regions for all three of these scores. We evaluate these results empirically on United States congressional districts and demonstrate that this order-reversal does occur in practice with respect to commonly-used projections.  Furthermore, the Reock score ordering in particular appears to be quite sensitive to the choice of map projection.

We introduce a notion of Ricci curvature for Cayley graphs that can be thought of as "medium-scale" because it is neither infinitesimal nor asymptotic, but based on a chosen finite radius parameter. We argue that it gives the foundation for a definition of Ricci curvature well adapted to geometric group theory, beginning by observing that the sign can easily be characterized in terms of conjugation in the group. With this conjugation curvature κ, abelian groups are identically flat, and in the other direction we show that κ≡0 implies the group is virtually abelian. Beyond that, κ captures known curvature phenomena in right-angled Artin groups (including free groups) and nilpotent groups, and has a strong relationship to other group-theoretic notions like growth rate and dead ends. We study dependence on generators and behavior under embeddings, and close with directions for further development and study. 

Let Dn be the n-punctured disk. We prove that a family of essential simple arcs starting and ending at the boundary and pairwise intersecting at most twice is of size at most n+1 choose 3. On the way, we also show that any nontrivial square complex homeomorphic to a disk whose hyperplanes are simple arcs intersecting at most twice must have a corner or a spur.