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Author : Yi Huang
Publisher : American Mathematical Society
Page : 128 pages
File Size : 35,23 MB
Release : 2023-06-22
Category : Mathematics
ISBN : 1470463121
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Author : R. C. Penner
Publisher : European Mathematical Society
Page : 388 pages
File Size : 20,92 MB
Release : 2012
Category : Teichmu ller spaces
ISBN : 9783037190753
There is an essentially ``tinker-toy'' model of a trivial bundle over the classical Teichmuller space of a punctured surface, called the decorated Teichmuller space, where the fiber over a point is the space of all tuples of horocycles, one about each puncture. This model leads to an extension of the classical mapping class groups called the Ptolemy groupoids and to certain matrix models solving related enumerative problems, each of which has proved useful both in mathematics and in theoretical physics. These spaces enjoy several related parametrizations leading to a rich and intricate algebro-geometric structure tied to the already elaborate combinatorial structure of the tinker-toy model. Indeed, the natural coordinates give the prototypical examples not only of cluster algebras but also of tropicalization. This interplay of combinatorics and coordinates admits further manifestations, for example, in a Lie theory for homeomorphisms of the circle, in the geometry underlying the Gauss product, in profinite and pronilpotent geometry, in the combinatorics underlying conformal and topological quantum field theories, and in the geometry and combinatorics of macromolecules. This volume gives the story a wider context of these decorated Teichmuller spaces as developed by the author over the last two decades in a series of papers, some of them in collaboration. Sometimes correcting errors or typos, sometimes simplifying proofs, and sometimes articulating more general formulations than the original research papers, this volume is self contained and requires little formal background. Based on a master's course at Aarhus University, it gives the first treatment of these works in monographic form.
Author : David Ruelle
Publisher : Cambridge University Press
Page : 198 pages
File Size : 28,93 MB
Release : 2004-11-25
Category : Science
ISBN : 9781139455282
Reissued in the Cambridge Mathematical Library this classic book outlines the theory of thermodynamic formalism which was developed to describe the properties of certain physical systems consisting of a large number of subunits. It is aimed at mathematicians interested in ergodic theory, topological dynamics, constructive quantum field theory, the study of certain differentiable dynamical systems, notably Anosov diffeomorphisms and flows. It is also of interest to theoretical physicists concerned with the conceptual basis of equilibrium statistical mechanics. The level of the presentation is generally advanced, the objective being to provide an efficient research tool and a text for use in graduate teaching. Background material on mathematics has been collected in appendices to help the reader. Extra material is given in the form of updates of problems that were open at the original time of writing and as a new preface specially written for this new edition by the author.
Author : Michel Brion
Publisher : Springer Science & Business Media
Page : 259 pages
File Size : 43,91 MB
Release : 2007-08-08
Category : Mathematics
ISBN : 0817644059
Systematically develops the theory of Frobenius splittings and covers all its major developments. Concise, efficient exposition unfolds from basic introductory material on Frobenius splittings—definitions, properties and examples—to cutting edge research.
Author : R. C. Penner
Publisher :
Page : 560 pages
File Size : 12,98 MB
Release : 2008
Category : Mathematics
ISBN :
This volume is dedicated to Shigeyuki Morita on the occasion of his 60th birthday. It consists of selected papers on recent trends and results in the study of various groups of diffeomorphisms, including mapping class groups, from the point of view of algebraic and differential topology, as well as dynamical ones involving foliations and symplectic or contact diffeomorphisms. Most of the authors were invited speakers or participants of the International Symposium on Groups of Diffeomorphisms 2006, which was held at the University of Tokyo (Komaba) in September 2006. The editors believe that the scope of this volume well reflects Morita's mathematical interests and hope this book inspires not only the specialists in these fields but also a wider audience of mathematicians.
Author : Michael Gekhtman
Publisher : American Mathematical Soc.
Page : 264 pages
File Size : 22,84 MB
Release : 2010
Category : Mathematics
ISBN : 0821849727
The first book devoted to cluster algebras, this work contains chapters on Poisson geometry and Schubert varieties; an introduction to cluster algebras and their main properties; and geometric aspects of the cluster algebra theory, in particular on its relations to Poisson geometry and to the theory of integrable systems.
Author : Robert Edward Bowen
Publisher : Springer
Page : 85 pages
File Size : 24,74 MB
Release : 2008-04-04
Category : Mathematics
ISBN : 3540776958
For this printing of R. Bowen's book, J.-R. Chazottes has retyped it in TeX for easier reading, thereby correcting typos and bibliographic details. From the Preface by D. Ruelle: "Rufus Bowen has left us a masterpiece of mathematical exposition... Here a number of results which were new at the time are presented in such a clear and lucid style that Bowen's monograph immediately became a classic. More than thirty years later, many new results have been proved in this area, but the volume is as useful as ever because it remains the best introduction to the basics of the ergodic theory of hyperbolic systems."
Author : François Labourie
Publisher :
Page : 152 pages
File Size : 37,80 MB
Release : 2013
Category : Mathematics
ISBN :
The subject of these notes is the character variety of representations of a surface group in a Lie group. The author emphasizes the various points of view (combinatorial, differential, and algebraic) and is interested in the description of its smooth points, symplectic structure, volume and connected components. He also shows how a three manifold bounded by the surface leaves a trace in this character variety. These notes were originally designed for students with only elementary knowledge of differential geometry and topology. In the first chapters, the author does not focus on the details of the differential geometric constructions and refers to classical textbooks, while in the more advanced chapters proofs occasionally are provided only for special cases where they convey the flavor of the general arguments. These notes might also be used by researchers entering this fast expanding field as motivation for further studies. The concluding paragraph of every chapter provides suggestions for further research.
Author : R. C. Penner
Publisher : Princeton University Press
Page : 236 pages
File Size : 47,16 MB
Release : 1992
Category : Mathematics
ISBN : 9780691025315
Measured geodesic laminations are a natural generalization of simple closed curves in surfaces, and they play a decisive role in various developments in two-and three-dimensional topology, geometry, and dynamical systems. This book presents a self-contained and comprehensive treatment of the rich combinatorial structure of the space of measured geodesic laminations in a fixed surface. Families of measured geodesic laminations are described by specifying a train track in the surface, and the space of measured geodesic laminations is analyzed by studying properties of train tracks in the surface. The material is developed from first principles, the techniques employed are essentially combinatorial, and only a minimal background is required on the part of the reader. Specifically, familiarity with elementary differential topology and hyperbolic geometry is assumed. The first chapter treats the basic theory of train tracks as discovered by W. P. Thurston, including recurrence, transverse recurrence, and the explicit construction of a measured geodesic lamination from a measured train track. The subsequent chapters develop certain material from R. C. Penner's thesis, including a natural equivalence relation on measured train tracks and standard models for the equivalence classes (which are used to analyze the topology and geometry of the space of measured geodesic laminations), a duality between transverse and tangential structures on a train track, and the explicit computation of the action of the mapping class group on the space of measured geodesic laminations in the surface.
Author : Barbara Schapira
Publisher :
Page : 281 pages
File Size : 14,82 MB
Release : 2015
Category : Curvature
ISBN : 9782856298183
With their origin in thermodynamics and symbolic dynamics, Gibbs measures are crucial tools to study the ergodic theory of the geodesic flow on negatively curved manifolds. We develop a framework (through Patterson-Sullivan densities) allowing us to get rid of compactness assumptions on the manifold, and prove many existence, uniqueness and finiteness results of Gibbs measures. We give many applications to the variational principle, the counting and equidistribution of orbit points and periods, the unique ergodicity of the strong unstable foliation and the classification of Gibbs densities on some Riemannian covers