Ideally, all infinite-type surfaces can be triangulated

in Bull. Lond. Math. Soc. (2022), 54(5), 2032--2040

Minimal Delaunay triangulations of hyperbolic surfaces

in 37th International Symposium on Computational Geometry (2021)

Measuring Pants

E-print/Working paper (2020)

We investigate the terms arising in an identity for hyperbolic surfaces proved by Luo and Tan, namely showing that they vary monotonically in terms of lengths and that they verify certain convexity ... [more ▼]

We investigate the terms arising in an identity for hyperbolic surfaces proved by Luo and Tan, namely showing that they vary monotonically in terms of lengths and that they verify certain convexity properties. Using these properties, we deduce two results. As a first application, we show how to deduce a theorem of Thurston which states, in particular for closed hyperbolic surfaces, that if a simple length spectrum "dominates" another, then in fact the two surfaces are isometric. As a second application, we show how to find upper bounds on the number of pairs of pants of bounded length that only depend on the boundary length and the topology of the surface. [less ▲]

Short closed geodesics with self-intersections

in Math. Proc. Cambridge Philos. Soc. (2020), 169(3), 623--638

The geometry of flip graphs and mapping class groups

in Trans. Amer. Math. Soc. (2019), 372(6), 3809--3844

Delaunay Triangulations of Points on Circles

E-print/Working paper (2018)

Delaunay triangulations of a point set in the Euclidean plane are ubiquitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon ... [more ▼]

Delaunay triangulations of a point set in the Euclidean plane are ubiquitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon as 4 or more points are concyclic but since it is not a generic situation, this difficulty is usually handled by using a (symbolic or explicit) perturbation. As an alternative, we propose to define a canonical triangulation for a set of concyclic points by using a max-min angle characterization of Delaunay triangulations. This point of view leads to a well defined and unique triangulation as long as there are no symmetric quadruples of points. This unique triangulation can be computed in quasi-linear time by a very simple algorithm. [less ▲]

Simultaneous Flips on Triangulated Surfaces

in Michigan Mathematical Journal (2018), 67(3), 451-464

We investigate a type of distance between triangulations on finite-type surfaces where one moves between triangulations by performing simultaneous flips. We consider triangulations up to homeomorphism ... [more ▼]

We investigate a type of distance between triangulations on finite-type surfaces where one moves between triangulations by performing simultaneous flips. We consider triangulations up to homeomorphism, and our main results are upper bounds on the distance between triangulations that only depend on the topology of the surface. [less ▲]

Interrogating surface length spectra and quantifying isospectrality

in MATHEMATISCHE ANNALEN (2018), 370(3-4), 1759-1787

This article is about inverse spectral problems for hyperbolic surfaces and in particular how length spectra relate to the geometry of the underlying surface. A quantitative answer is given to the ... [more ▼]

This article is about inverse spectral problems for hyperbolic surfaces and in particular how length spectra relate to the geometry of the underlying surface. A quantitative answer is given to the following: how many questions do you need to ask a length spectrum to determine it completely? In answering this, a quantitative upper bound is given on the number of isospectral but non-isometric surfaces of a given genus. [less ▲]

The maximum number of systoles for genus two Riemann surfaces with abelian differentials

E-print/Working paper (2017)

This article explores the length and number of systoles associated to holomorphic $1$-forms on surfaces. In particular, we show that up to homotopy, there are at most $10$ systolic loops on such a genus ... [more ▼]

This article explores the length and number of systoles associated to holomorphic $1$-forms on surfaces. In particular, we show that up to homotopy, there are at most $10$ systolic loops on such a genus two surface and that the bound is realized by a unique translation surface up to homothety. We also provide sharp upper bounds on the the number of homotopy classes of systoles for a holomorphic $1$-form with a single zero in terms of the genus. [less ▲]

Geometric filling curves on surfaces

in Bulletin of the London Mathematical Society (2017), 49(4), 660--669