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Author: Richard Evan Schwartz
Publisher: American Mathematical Soc.
Published: 2011
Total Pages: 330
ISBN-13: 0821853686
DOWNLOAD EBOOKThe goal of the book is to present a tapestry of ideas from various areas of mathematics in a clear and rigorous yet informal and friendly way. Prerequisites include undergraduate courses in real analysis and in linear algebra, and some knowledge of complex analysis. --from publisher description.
Author: A. B. Katok
Publisher: American Mathematical Soc.
Published: 2008
Total Pages: 307
ISBN-13: 0821846795
DOWNLOAD EBOOKSurfaces are among the most common and easily visualized mathematical objects, and their study brings into focus fundamental ideas, concepts, and methods from geometry, topology, complex analysis, Morse theory, and group theory. This book introduces many of the principal actors - the round sphere, flat torus, Mobius strip, and Klein bottle.
Author: John Stillwell
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 225
ISBN-13: 1461209293
DOWNLOAD EBOOKThe geometry of surfaces is an ideal starting point for learning geometry, for, among other reasons, the theory of surfaces of constant curvature has maximal connectivity with the rest of mathematics. This text provides the student with the knowledge of a geometry of greater scope than the classical geometry taught today, which is no longer an adequate basis for mathematics or physics, both of which are becoming increasingly geometric. It includes exercises and informal discussions.
Author: Bertrand Eynard
Publisher: Springer Science & Business Media
Published: 2016-03-21
Total Pages: 427
ISBN-13: 3764387971
DOWNLOAD EBOOKThe problem of enumerating maps (a map is a set of polygonal "countries" on a world of a certain topology, not necessarily the plane or the sphere) is an important problem in mathematics and physics, and it has many applications ranging from statistical physics, geometry, particle physics, telecommunications, biology, ... etc. This problem has been studied by many communities of researchers, mostly combinatorists, probabilists, and physicists. Since 1978, physicists have invented a method called "matrix models" to address that problem, and many results have been obtained. Besides, another important problem in mathematics and physics (in particular string theory), is to count Riemann surfaces. Riemann surfaces of a given topology are parametrized by a finite number of real parameters (called moduli), and the moduli space is a finite dimensional compact manifold or orbifold of complicated topology. The number of Riemann surfaces is the volume of that moduli space. Mor e generally, an important problem in algebraic geometry is to characterize the moduli spaces, by computing not only their volumes, but also other characteristic numbers called intersection numbers. Witten's conjecture (which was first proved by Kontsevich), was the assertion that Riemann surfaces can be obtained as limits of polygonal surfaces (maps), made of a very large number of very small polygons. In other words, the number of maps in a certain limit, should give the intersection numbers of moduli spaces. In this book, we show how that limit takes place. The goal of this book is to explain the "matrix model" method, to show the main results obtained with it, and to compare it with methods used in combinatorics (bijective proofs, Tutte's equations), or algebraic geometry (Mirzakhani's recursions). The book intends to be self-contained and accessible to graduate students, and provides comprehensive proofs, several examples, and give s the general formula for the enumeration of maps on surfaces of any topology. In the end, the link with more general topics such as algebraic geometry, string theory, is discussed, and in particular a proof of the Witten-Kontsevich conjecture is provided.
Author: Jayadev S. Athreya
Publisher: American Mathematical Society
Published: 2024-04-19
Total Pages: 195
ISBN-13: 1470476770
DOWNLOAD EBOOKThis textbook offers an accessible introduction to translation surfaces. Building on modest prerequisites, the authors focus on the fundamentals behind big ideas in the field: ergodic properties of translation flows, counting problems for saddle connections, and associated renormalization techniques. Proofs that go beyond the introductory nature of the book are deftly omitted, allowing readers to develop essential tools and motivation before delving into the literature. Beginning with the fundamental example of the flat torus, the book goes on to establish the three equivalent definitions of translation surface. An introduction to the moduli space of translation surfaces follows, leading into a study of the dynamics and ergodic theory associated to a translation surface. Counting problems and group actions come to the fore in the latter chapters, giving a broad overview of progress in the 40 years since the ergodicity of the Teichmüller geodesic flow was proven. Exercises are included throughout, inviting readers to actively explore and extend the theory along the way. Translation Surfaces invites readers into this exciting area, providing an accessible entry point from the perspectives of dynamics, ergodicity, and measure theory. Suitable for a one- or two-semester graduate course, it assumes a background in complex analysis, measure theory, and manifolds, while some familiarity with Riemann surfaces and ergodic theory would be beneficial.
Author: William Griswold Smith
Publisher:
Published: 1912
Total Pages: 250
ISBN-13:
DOWNLOAD EBOOKAuthor: Jacques Jupille
Publisher: Springer
Published: 2015-02-09
Total Pages: 472
ISBN-13: 3319143670
DOWNLOAD EBOOKThis book presents the basics and characterization of defects at oxide surfaces. It provides a state-of-the-art review of the field, containing information to the various types of surface defects, describes analytical methods to study defects, their chemical activity and the catalytic reactivity of oxides. Numerical simulations of defective structures complete the picture developed. Defects on planar surfaces form the focus of much of the book, although the investigation of powder samples also form an important part. The experimental study of planar surfaces opens the possibility of applying the large armoury of techniques that have been developed over the last half-century to study surfaces in ultra-high vacuum. This enables the acquisition of atomic level data under well-controlled conditions, providing a stringent test of theoretical methods. The latter can then be more reliably applied to systems such as nanoparticles for which accurate methods of characterization of structure and electronic properties have yet to be developed. The book gives guidance to tailor oxide surfaces by controlling the nature and concentration of defects. The importance of defects in the physics and chemistry of metal oxide surfaces is presented in this book together with the prominent role of oxides in common life. The book contains contributions from leaders in the field. It serves as a reference for experts and beginners in the field.
Author: Franklin Tao
Publisher: John Wiley & Sons
Published: 2012-04-10
Total Pages: 456
ISBN-13: 0470562943
DOWNLOAD EBOOKThis book presents both fundamental knowledge and latest achievements of this rapidly growing field in the last decade. It presents a complete and concise picture of the the state-of-the-art in the field, encompassing the most active international research groups in the world. Led by contributions from leading global research groups, the book discusses the functionalization of semiconductor surface. Dry organic reactions in vacuum and wet organic chemistry in solution are two major categories of strategies for functionalization that will be described. The growth of multilayer-molecular architectures on the formed organic monolayers will be documented. The immobilization of biomolecules such as DNA on organic layers chemically attached to semiconductor surfaces will be introduced. The patterning of complex structures of organic layers and metallic nanoclusters toward sensing techniques will be presented as well.
Author: Clara Grima
Publisher: Springer Science & Business Media
Published: 2001-11-30
Total Pages: 212
ISBN-13: 9781402002021
DOWNLOAD EBOOKIn the last thirty years Computational Geometry has emerged as a new discipline from the field of design and analysis of algorithms. That dis cipline studies geometric problems from a computational point of view, and it has attracted enormous research interest. But that interest is mostly concerned with Euclidean Geometry (mainly the plane or Eu clidean 3-dimensional space). Of course, there are some important rea sons for this occurrence since the first applieations and the bases of all developments are in the plane or in 3-dimensional space. But, we can find also some exceptions, and so Voronoi diagrams on the sphere, cylin der, the cone, and the torus have been considered previously, and there are manY works on triangulations on the sphere and other surfaces. The exceptions mentioned in the last paragraph have appeared to try to answer some quest ions which arise in the growing list of areas in which the results of Computational Geometry are applicable, since, in practiee, many situations in those areas lead to problems of Com putational Geometry on surfaces (probably the sphere and the cylinder are the most common examples). We can mention here some specific areas in which these situations happen as engineering, computer aided design, manufacturing, geographie information systems, operations re search, roboties, computer graphics, solid modeling, etc.
Author: F. Cossec
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 409
ISBN-13: 1461236967
DOWNLOAD EBOOKThis is the first of two volumes representing the current state of knowledge about Enriques surfaces which occupy one of the classes in the classification of algebraic surfaces. Recent improvements in our understanding of algebraic surfaces over fields of positive characteristic allowed us to approach the subject from a completely geometric point of view although heavily relying on algebraic methods. Some of the techniques presented in this book can be applied to the study of algebraic surfaces of other types. We hope that it will make this book of particular interest to a wider range of research mathematicians and graduate students. Acknowledgements. The undertaking of this project was made possible by the support of several institutions. Our mutual cooperation began at the University of Warwick and the Max Planck Institute of Mathematics in 1982/83. Most of the work in this volume was done during the visit of the first author at the University of Michigan in 1984-1986. The second author was supported during all these years by grants from the National Science Foundation.
