LEADER 04746nam 2200541 450 001 9910595045203321 005 20231110223542.0 010 $a3-031-08705-4 035 $a(MiAaPQ)EBC7098176 035 $a(Au-PeEL)EBL7098176 035 $a(CKB)24866051600041 035 $a(PPN)264953967 035 $a(EXLCZ)9924866051600041 100 $a20230220d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aMirzakhani's curve counting and geodesic currents /$fViveka Erlandsson, Juan Souto, and Hugo Parlier 210 1$aCham, Switzerland :$cSpringer International Publishing,$d[2022] 210 4$d©2022 215 $a1 online resource (233 pages) 225 1 $aProgress in Mathematics ;$vv.345 311 08$aPrint version: Erlandsson, Viveka Mirzakhani's Curve Counting and Geodesic Currents Cham : Springer International Publishing AG,c2022 9783031087042 327 $aIntro -- Preface -- Acknowledgments -- Contents -- Notation -- 1 Introduction -- Some Things that the Reader Will Not Find in This Book -- What the Reader Will Find Here -- A Comment on the Way This Book Is Written -- 2 Read Me -- Curves, Multicurves, and Arcs -- Intersection Number -- Geodesic Representatives -- Random Geodesics. Or Length Versus Intersections -- Space of Geodesics -- Geodesics Projecting into a Fixed Compact Set -- Laminations -- Hausdorff Limits -- Mapping Class Group -- Shortening Curves -- 3 Geodesic Currents -- Definition and Examples -- Curves as Currents -- Weighted Multicurves -- Measured Laminations -- The Space of Geodesic Currents -- Interesting Subsets of C(?) -- Intersection Form -- Characterization of Measured Laminations -- Filling Currents -- The Mapping Class Group Action -- Mapping Class Group Action on the Set of Filling Currents -- Comments (The Liouville Current) -- 4 Train Tracks -- Train Tracks as Smoothly Embedded Complexes -- Almost Geodesic Train Tracks -- Band Complexes and Train Tracks -- Curves and Laminations Carried by Train Tracks -- Weights and Recurrence -- Switch Equations -- Thurston Measure -- First Counting Result -- Masur's Ergodicity Theorem -- An Example: The Case of the Once Punctured Torus -- Comments -- Piecewise Linear Structure -- Symplectic Structure -- Weil-Petersson Volume Form -- Other Interesting Measures on ML(S) -- 5 Radallas -- Terminology -- The Game of Cars -- Almost Geodesic Radallas -- Finding Almost Geodesic Radallas -- Generic Curves -- Comments: Expected Angle of Self-Intersection -- 6 Subconvergence of Measures -- Idea of the Proof of Theorem 6.1 -- Local Map -- Global Map -- New Measures -- Comments -- 7 Approximating the Thurston Measure -- Dribbling -- Radallas Carrying the Curves in ??0 -- Reducing to a Linear Algebra Problem -- Proof of Proposition 7.2. 327 $aComments -- 8 The Main Theorem -- Notation and the Constants -- The Theorem -- The Fundamental Domain -- The Meat -- The Not Ker-Invariant Case -- Comments -- 9 Counting Curves -- Main Curve Counting Theorem -- Frequencies and Relative Frequencies -- Functions on the Space of Currents -- 10 Counting Square-Tiled Surfaces -- Square-Tiled Surfaces -- Recovering Square-Tiled Surface Structures from Multicurves -- Square-Tiled Surfaces of Given Type -- Counting -- Comments -- 11 Statistics of Simple Curves -- Credit to Kasra -- Hard Data -- A Formula for c(?0) for Simple Curves -- Punctured Spheres -- Separating Curves in Closed Surfaces -- Non-Separating Curves in Closed Surfaces -- Combinatorial Computation -- Changing Weights -- Comments -- 12 Smörgåsbord -- Random Pants Decompositions -- Lattice Point Counting in Teichmüller Space -- Genericty of pseudo-Anosov Elements -- A Radon Measures -- Point-Set Topology -- Weak-*-Topology -- Sequential Compactness -- B Computing Thurston Volumes -- Punctured Spheres -- Separating Curves in Closed Surfaces -- Non-separating Curves in Closed Surfaces -- References -- Index. 410 0$aProgress in Mathematics 606 $aCurves, Algebraic 606 $aCurves, Algebraic$xData processing 606 $aCorbes$2thub 606 $aCorbes algebraiques$2thub 608 $aLlibres electrònics$2thub 615 0$aCurves, Algebraic. 615 0$aCurves, Algebraic$xData processing. 615 7$aCorbes 615 7$aCorbes algebraiques 676 $a516.352 700 $aErlandsson$b Viveka$01258335 702 $aClement$b Juan Souto 702 $aParlier$b Hugo 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910595045203321 996 $aMirzakhani's curve counting and geodesic currents$93027979 997 $aUNINA