What's Next?: The Mathematical Legacy of William P. Thurston (AMS-205) (Annals of Mathematics Studies Book 363)

Bill Thurston made fundamental contributions to topology, geometry, and dynamical systems. But beyond these specific accomplishments he introduced new ways of thinking about and of seeing mathematics that have had a profound influence on the entire mathematical community. He discovered connections between disciplines that led to the creation of entirely new fields. In June of 2014 mathematicians from a broad spectrum of areas came together to describe recent advances and explore future directions motivated by Thurston's transformative ideas. The conference was called "What's Next? The Mathematical Legacy of Bill Thurston." This volume collects works by the speakers at the conference. We selected talks and papers to reflect the breadth of Bill's contributions to mathematics and his intensely geometric style. This includes two papers in dynamics based on Bill's unfinished work ("Degree-d-Invariant Laminations" and "On Balanced Planar Graphs, Following W. Thurston"), as well as a paper by Misha Gromov ("Morse Spectra"), continuing the long tradition of Gromov and Thurston building off of each other. The title of the contribution by Delecroix and Zorich, "Cries and Whispers in Wind-Tree Forests," exhibits another of Bill's refrains: look for ways to gain geometric intuition through senses other than sight. This geometric refrain continues through all of the papers, up to the last contribution by Dennis Sullivan with recollections of Bill's life, first published in the Notices of the AMS. This volume would not have been possible without the efforts of many people, including the conference organizers ( William Thurston (1946–2012) was one of the great mathematicians of the twentieth century. He was a visionary whose extraordinary ideas revolutionized a broad range of areas of mathematics, from foliations, contact structures, and Teichmüller theory to automorphisms of surfaces, hyperbolic geometry, geometrization of 3-manifolds, geometric group theory, and rational maps. In addition, he discovered connections between disciplines that led to astonishing breakthroughs in mathematical understanding as well as the creation of entirely new fields. His far-reaching questions and conjectures led to enormous progress by other researchers. In What's Next? , many of today's leading mathematicians describe recent advances and future directions inspired by Thurston's transformative ideas. This book brings together papers delivered by his colleagues and former students at "What's Next? The Mathematical Legacy of Bill Thurston," a conference held in June 2014 at Cornell University. It discusses Thurston's fundamental contributions to topology, geometry, and dynamical systems and includes many deep and original contributions to the field. Incisive and wide-ranging, the book explores how he introduced new ways of thinking about and doing mathematics—innovations that have had a profound and lasting impact on the mathematical community as a whole—and also features two papers based on Thurston's unfinished work in dynamics. William Thurston (1946–2012) was one of the great mathematicians of the twentieth century. He was a visionary whose extraordinary ideas revolutionized a broad range of mathematical fields, from foliations, contact structures, and Teichmüller theory to automorphisms of surfaces, hyperbolic geometry, geometrization of 3-manifolds, geometric group theory, and rational maps. In addition, he discovered connections between disciplines that led to astonishing breakthroughs in mathematical understanding as well as the creation of entirely new fields. His far-reaching questions and conjectures led to enormous progress by other researchers. What's Next? brings together many of today's leading mathematicians to describe recent advances and future directions inspired by Thurston's transformative ideas. Including valuable insights from his colleagues and former students, What's Next? discusses Thurston's fundamental contributions to topology, geometry, and dynamical systems and includes many deep and original contributions to the field. This incisive and wide-ranging book also explores how he introduced new ways of thinking about and doing mathematics, innovations that have had a profound and lasting impact on the mathematical community as a whole. Certifying the Thurston norm via SL(2,C)-twisted homology / Ian Agol and Nathan M. Dunfield -- The profinite completion of 3-manifold groups, fiberedness and the Thurston norm / Michel Boileau and Stefan Friedl -- Profinite rigidity, fibering, and the figure-eight knot / M.R. Bridson and A.W. Reid -- Coxeter groups and random groups / Danny Calegari -- Cries and whispers in wind-tree forests / Vincent Delecroix and Anton Zorich -- Groups of homeomorphisms of one-manifolds, I : actions of nonlinear groups / Benson Farb and John Franks -- Morse spectra, homology measures, spaces of cycles and parametric packing problems / Misha Gromov -- Right-angled hexagon tilings of the hyperbolic plane / R. Kenyon -- On balanced planar graphs, following W. Thurston / Sarah Koch and Tan Lei -- Construction of subsurfaces via good pants / Yi Liu and Vladimir Markovic -- Dehn surgery on knots in S3 producing nil Seifert fibered spaces / Yi Ni and Xingru Zhang -- Degree-d-invariant laminations / William P. Thurston, Hyungryul Baik, Gao Yan, John H. Hubbard, Kathryn A. Lindsey, Tan Lei, and Dylan P. Thurston -- An invitation to coherent groups / Daniel T. Wise -- A decade of Thurston stories / Dennis Sullivan "In June of 2014, a conference was held in William Thurston's honor called "What's Next? the mathematical legacy of Bill Thurston." This volume includes papers from conference presenters and solicited papers by prominent scholars. The papers are based on Thurston's fundamental contributions to topology, geometry, and dynamical systems. Beyond these specific accomplishments, they discuss how he introduced new ways of thinking about and of seeing mathematics that have had a profound influence on the entire mathematical community - Thurston discovered connections between disciplines that led to the creation of entirely new fields. This book brings together mathematicians from a broad spectrum of areas to describe recent advances and explore future directions motivated by Thurston's transformative ideas"-- Provided by publisher