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Interrogating surface length spectra and quantifying isospectrality Parlier, Hugo E-print/Working paper (2016) 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 ▲] Detailed reference viewed: 89 (2 UL)Geometric filling curves on surfaces ; Parlier, Hugo ; E-print/Working paper (2016) This note is about a type of quantitative density of closed geodesics on closed hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic that $\varepsilon ... [more ▼] This note is about a type of quantitative density of closed geodesics on closed hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic that $\varepsilon$-fills the surface. [less ▲] Detailed reference viewed: 114 (0 UL)Short closed geodesics with self-intersections ; Parlier, Hugo E-print/Working paper (2016) Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer $k$, we are interested in the set of all closed geodesics with at least $k$ (but possibly more) self ... [more ▼] Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer $k$, we are interested in the set of all closed geodesics with at least $k$ (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We prove that their intersection numbers are upper bounded by a universal linear function in $k$ (which holds for any hyperbolic surface). Moreover, in the presence of cusps, we get bounds which imply that the self-intersection numbers behave asymptotically like $k$ for growing $k$. [less ▲] Detailed reference viewed: 46 (1 UL)Distances in domino flip graphs Parlier, Hugo ; E-print/Working paper (2016) This article is about measuring and visualizing distances between domino tilings. Given two tilings of a simply connected square tiled surface, we're interested in the minimum number of flips between two ... [more ▼] This article is about measuring and visualizing distances between domino tilings. Given two tilings of a simply connected square tiled surface, we're interested in the minimum number of flips between two tilings. Given a certain shape, we're interested in computing the diameters of the flip graphs, meaning the maximal distance between any two of its tilings. Building on work of Thurston and others, we give geometric interpretations of distances which result in formulas for the diameters of the flip graphs of rectangles or Aztec diamonds. [less ▲] Detailed reference viewed: 45 (5 UL)Once punctured disks, non-convex polygons, and pointihedra Parlier, Hugo ; E-print/Working paper (2016) We explore several families of flip-graphs, all related to polygons or punctured polygons. In particular, we consider the topological flip-graphs of once-punctured polygons which, in turn, contain all ... [more ▼] We explore several families of flip-graphs, all related to polygons or punctured polygons. In particular, we consider the topological flip-graphs of once-punctured polygons which, in turn, contain all possible geometric flip-graphs of polygons with a marked point as embedded sub-graphs. Our main focus is on the geometric properties of these graphs and how they relate to one another. In particular, we show that the embeddings between them are strongly convex (or, said otherwise, totally geodesic). We also find bounds on the diameters of these graphs, sometimes using the strongly convex embeddings. Finally, we show how these graphs relate to different polytopes, namely type D associahedra and a family of secondary polytopes which we call pointihedra. [less ▲] Detailed reference viewed: 89 (0 UL)Chromatic numbers of hyperbolic surfaces Parlier, Hugo ; in Indiana University Mathematics Journal (2016), 65(4), 1401--1423 Detailed reference viewed: 143 (4 UL)Filling sets of curves on punctured surfaces ; Parlier, Hugo in New York Journal of Mathematics (2016), 22 Detailed reference viewed: 40 (2 UL)Relative shapes of thick subsets of moduli space ; Parlier, Hugo ; in American Journal of Mathematics (2016), 138(2), 473--498 Detailed reference viewed: 159 (5 UL) |
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