Ricci Flow and Geometric Applications

Cetraro, Italy 2010

Author: Michel Boileau,Gerard Besson,Carlo Sinestrari,Gang Tian

Publisher: Springer

ISBN: 3319423517

Category: Mathematics

Page: 136

View: 749


Presenting some impressive recent achievements in differential geometry and topology, this volume focuses on results obtained using techniques based on Ricci flow. These ideas are at the core of the study of differentiable manifolds. Several very important open problems and conjectures come from this area and the techniques described herein are used to face and solve some of them. The book’s four chapters are based on lectures given by leading researchers in the field of geometric analysis and low-dimensional geometry/topology, respectively offering an introduction to: the differentiable sphere theorem (G. Besson), the geometrization of 3-manifolds (M. Boileau), the singularities of 3-dimensional Ricci flows (C. Sinestrari), and Kähler–Ricci flow (G. Tian). The lectures will be particularly valuable to young researchers interested in differential manifolds.

The Ricci Flow in Riemannian Geometry

A Complete Proof of the Differentiable 1/4-Pinching Sphere Theorem

Author: Ben Andrews,Christopher Hopper

Publisher: Springer

ISBN: 364216286X

Category: Mathematics

Page: 302

View: 2419


This book focuses on Hamilton's Ricci flow, beginning with a detailed discussion of the required aspects of differential geometry, progressing through existence and regularity theory, compactness theorems for Riemannian manifolds, and Perelman's noncollapsing results, and culminating in a detailed analysis of the evolution of curvature, where recent breakthroughs of Böhm and Wilking and Brendle and Schoen have led to a proof of the differentiable 1/4-pinching sphere theorem.

Modelling and Optimisation of Flows on Networks

Cetraro, Italy 2009, Editors: Benedetto Piccoli, Michel Rascle

Author: Luigi Ambrosio,Alberto Bressan,Dirk Helbing,Axel Klar,Enrique Zuazua

Publisher: Springer

ISBN: 3642321607

Category: Mathematics

Page: 497

View: 4811


In recent years flows in networks have attracted the interest of many researchers from different areas, e.g. applied mathematicians, engineers, physicists, economists. The main reason for this ubiquity is the wide and diverse range of applications, such as vehicular traffic, supply chains, blood flow, irrigation channels, data networks and others. This book presents an extensive set of notes by world leaders on the main mathematical techniques used to address such problems, together with investigations into specific applications. The main focus is on partial differential equations in networks, but ordinary differential equations and optimal transport are also included. Moreover, the modeling is completed by analysis, numerics, control and optimization of flows in networks. The book will be a valuable resource for every researcher or student interested in the subject.

Hamilton's Ricci Flow

Author: Bennett Chow,Peng Lu,Lei Ni

Publisher: American Mathematical Soc.

ISBN: 9780821883990

Category: Mathematics

Page: 608

View: 2379


Ricci flow is a powerful analytic method for studying the geometry and topology of manifolds. This book is an introduction to Ricci flow for graduate students and mathematicians interested in working in the subject. To this end, the first chapter is a review of the relevant basics of Riemannian geometry. For the benefit of the student, the text includes a number of exercises of varying difficulty. The book also provides brief introductions to some general methods of geometric analysis and other geometric flows. Comparisons are made between the Ricci flow and the linear heat equation, mean curvature flow, and other geometric evolution equations whenever possible. Several topics of Hamilton's program are covered, such as short time existence, Harnack inequalities, Ricci solitons, Perelman's no local collapsing theorem, singularity analysis, and ancient solutions. A major direction in Ricci flow, via Hamilton's and Perelman's works, is the use of Ricci flow as an approach to solving the Poincare conjecture and Thurston's geometrization conjecture.

Lecture Notes on Mean Curvature Flow

Author: Carlo Mantegazza

Publisher: Springer Science & Business Media

ISBN: 3034801459

Category: Mathematics

Page: 168

View: 4839


This book is an introduction to the subject of mean curvature flow of hypersurfaces with special emphasis on the analysis of singularities. This flow occurs in the description of the evolution of numerous physical models where the energy is given by the area of the interfaces. These notes provide a detailed discussion of the classical parametric approach (mainly developed by R. Hamilton and G. Huisken). They are well suited for a course at PhD/PostDoc level and can be useful for any researcher interested in a solid introduction to the technical issues of the field. All the proofs are carefully written, often simplified, and contain several comments. Moreover, the author revisited and organized a large amount of material scattered around in literature in the last 25 years.

Some Mathematical Models from Population Genetics

École d'Été de Probabilités de Saint-Flour XXXIX-2009

Author: Alison Etheridge

Publisher: Springer

ISBN: 3642166326

Category: Mathematics

Page: 119

View: 5773


This work reflects sixteen hours of lectures delivered by the author at the 2009 St Flour summer school in probability. It provides a rapid introduction to a range of mathematical models that have their origins in theoretical population genetics. The models fall into two classes: forwards in time models for the evolution of frequencies of different genetic types in a population; and backwards in time (coalescent) models that trace out the genealogical relationships between individuals in a sample from the population. Some, like the classical Wright-Fisher model, date right back to the origins of the subject. Others, like the multiple merger coalescents or the spatial Lambda-Fleming-Viot process are much more recent. All share a rich mathematical structure. Biological terms are explained, the models are carefully motivated and tools for their study are presented systematically.

Splitting Methods for Partial Differential Equations with Rough Solutions

Analysis and MATLAB Programs

Author: Helge Holden

Publisher: European Mathematical Society

ISBN: 9783037190784

Category: Mathematics

Page: 226

View: 5727


Operator splitting (or the fractional steps method) is a very common tool to analyze nonlinear partial differential equations both numerically and analytically. By applying operator splitting to a complicated model one can often split it into simpler problems that can be analyzed separately. In this book one studies operator splitting for a family of nonlinear evolution equations, including hyperbolic conservation laws and degenerate convection-diffusion equations. Common for these equations is the prevalence of rough, or non-smooth, solutions, e.g., shocks. Rigorous analysis is presented, showing that both semi-discrete and fully discrete splitting methods converge. For conservation laws, sharp error estimates are provided and for convection-diffusion equations one discusses a priori and a posteriori correction of entropy errors introduced by the splitting. Numerical methods include finite difference and finite volume methods as well as front tacking. The theory is illustrated by numerous examples. There is a dedicated web page that provides MATLAB codes for many of the examples. The book is suitable for graduate students and researchers in pure and applied mathematics, physics, and engineering.

Ricci Flow and the Poincaré Conjecture

Author: John W. Morgan,G. Tian

Publisher: American Mathematical Soc.

ISBN: 9780821843284

Category: Mathematics

Page: 521

View: 770


For over 100 years the Poincare Conjecture, which proposes a topological characterization of the 3-sphere, has been the central question in topology. Since its formulation, it has been repeatedly attacked, without success, using various topological methods. Its importance and difficulty were highlighted when it was chosen as one of the Clay Mathematics Institute's seven Millennium Prize Problems. In 2002 and 2003 Grigory Perelman posted three preprints showing how to use geometric arguments, in particular the Ricci flow as introduced and studied by Hamilton, to establish the Poincare Conjecture in the affirmative. This book provides full details of a complete proof of the Poincare Conjecture following Perelman's three preprints. After a lengthy introduction that outlines the entire argument, the book is divided into four parts. The first part reviews necessary results from Riemannian geometry and Ricci flow, including much of Hamilton's work. The second part starts with Perelman's length function, which is used to establish crucial non-collapsing theorems. Then it discusses the classification of non-collapsed, ancient solutions to the Ricci flow equation. The third part concerns the existence of Ricci flow with surgery for all positive time and an analysis of the topological and geometric changes introduced by surgery. The last part follows Perelman's third preprint to prove that when the initial Riemannian 3-manifold has finite fundamental group, Ricci flow with surgery becomes extinct after finite time. The proofs of the Poincare Conjecture and the closely related 3-dimensional spherical space-form conjecture are then immediate. The existence of Ricci flow with surgery has application to 3-manifolds far beyond the Poincare Conjecture. It forms the heart of the proof via Ricci flow of Thurston's Geometrization Conjecture. Thurston's Geometrization Conjecture, which classifies all compact 3-manifolds, will be the subject of a follow-up article. The organization of the material in this book differs from that given by Perelman. From the beginning the authors present all analytic and geometric arguments in the context of Ricci flow with surgery. In addition, the fourth part is a much-expanded version of Perelman's third preprint; it gives the first complete and detailed proof of the finite-time extinction theorem. With the large amount of background material that is presented and the detailed versions of the central arguments, this book is suitable for all mathematicians from advanced graduate students to specialists in geometry and topology. Clay Mathematics Institute Monograph Series The Clay Mathematics Institute Monograph Series publishes selected expositions of recent developments, both in emerging areas and in older subjects transformed by new insights or unifying ideas.

The Grid

Blueprint for a New Computing Infrastructure

Author: Ian Foster,Carl Kesselman

Publisher: Morgan Kaufmann

ISBN: 1558609334

Category: Computers

Page: 748

View: 5783


"The Grid" is an emerging infrastructure that will fundamentally change the way people think about and use computing. The editors reveal the revolutionary impact of large-scale resource sharing and virtualization within science and industry, and the intimate relationships between organization and resource sharing structures.

Moduli Spaces of Riemannian Metrics

Author: Wilderich Tuschmann,David J. Wraith

Publisher: Springer

ISBN: 3034809484

Category: Mathematics

Page: 123

View: 9649


This book studies certain spaces of Riemannian metrics on both compact and non-compact manifolds. These spaces are defined by various sign-based curvature conditions, with special attention paid to positive scalar curvature and non-negative sectional curvature, though we also consider positive Ricci and non-positive sectional curvature. If we form the quotient of such a space of metrics under the action of the diffeomorphism group (or possibly a subgroup) we obtain a moduli space. Understanding the topology of both the original space of metrics and the corresponding moduli space form the central theme of this book. For example, what can be said about the connectedness or the various homotopy groups of such spaces? We explore the major results in the area, but provide sufficient background so that a non-expert with a grounding in Riemannian geometry can access this growing area of research.

On the Power of Fuzzy Markup Language

Author: Giovanni Acampora,Vincenzo Loia,Chang-Shing Lee,Mei-Hui Wang

Publisher: Springer

ISBN: 3642354882

Category: Computers

Page: 188

View: 9001


One of the most successful methodology that arose from the worldwide diffusion of Fuzzy Logic is Fuzzy Control. After the first attempts dated in the seventies, this methodology has been widely exploited for controlling many industrial components and systems. At the same time, and very independently from Fuzzy Logic or Fuzzy Control, the birth of the Web has impacted upon almost all aspects of computing discipline. Evolution of Web, Web2.0 and Web 3.0 has been making scenarios of ubiquitous computing much more feasible; consequently information technology has been thoroughly integrated into everyday objects and activities. What happens when Fuzzy Logic meets Web technology? Interesting results might come out, as you will discover in this book. Fuzzy Mark-up Language is a son of this synergistic view, where some technological issues of Web are re-interpreted taking into account the transparent notion of Fuzzy Control, as discussed here. The concept of a Fuzzy Control that is conceived and modeled in terms of a native web wisdom represents another step towards the last picture of Pervasive Web Intelligence.

Mathematical Models in the Manufacturing of Glass

C.I.M.E. Summer School, Montecatini Terme, Italy 2008

Author: Angiolo Farina,Axel Klar,Robert M.M. Mattheij,Andro Mikeli ́c,Norbert Siedow

Publisher: Springer

ISBN: 3642159672

Category: Technology & Engineering

Page: 227

View: 2982


This volume presents a review of advanced technological problems in the glass industry and of the mathematics involved. It is amazing that such a seemingly small research area is extremely rich and calls for an impressively large variety of mathematical methods, including numerical simulations of considerable complexity. The problems treated here are very typical of the field of glass manufacturing and cover a large spectrum of complementary subjects: injection molding by various techniques, radiative heat transfer in glass, nonisothermal flows and fibre spinning. The book can certainly be useful not only to applied mathematicians, but also to physicists and engineers, who can find in it an overview of the most advanced models and methods.

Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications

Cetraro, Italy 2011, Editors: Paola Loreti, Nicoletta Anna Tchou

Author: Yves Achdou,Guy Barles,Hitoshi Ishii,Grigory L. Litvinov

Publisher: Springer

ISBN: 3642364330

Category: Mathematics

Page: 301

View: 1823


These Lecture Notes contain the material relative to the courses given at the CIME summer school held in Cetraro, Italy from August 29 to September 3, 2011. The topic was "Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications". The courses dealt mostly with the following subjects: first order and second order Hamilton-Jacobi-Bellman equations, properties of viscosity solutions, asymptotic behaviors, mean field games, approximation and numerical methods, idempotent analysis. The content of the courses ranged from an introduction to viscosity solutions to quite advanced topics, at the cutting edge of research in the field. We believe that they opened perspectives on new and delicate issues. These lecture notes contain four contributions by Yves Achdou (Finite Difference Methods for Mean Field Games), Guy Barles (An Introduction to the Theory of Viscosity Solutions for First-order Hamilton-Jacobi Equations and Applications), Hitoshi Ishii (A Short Introduction to Viscosity Solutions and the Large Time Behavior of Solutions of Hamilton-Jacobi Equations) and Grigory Litvinov (Idempotent/Tropical Analysis, the Hamilton-Jacobi and Bellman Equations).

Differential Geometry and Continuum Mechanics

Author: Gui-Qiang Chen,Michael Grinfeld,R. J. Knops

Publisher: Springer

ISBN: 331918573X

Category: Mathematics

Page: 387

View: 402


This book examines the exciting interface between differential geometry and continuum mechanics, now recognised as being of increasing technological significance. Topics discussed include isometric embeddings in differential geometry and the relation with microstructure in nonlinear elasticity, the use of manifolds in the description of microstructure in continuum mechanics, experimental measurement of microstructure, defects, dislocations, surface energies, and nematic liquid crystals. Compensated compactness in partial differential equations is also treated. The volume is intended for specialists and non-specialists in pure and applied geometry, continuum mechanics, theoretical physics, materials and engineering sciences, and partial differential equations. It will also be of interest to postdoctoral scientists and advanced postgraduate research students. These proceedings include revised written versions of the majority of papers presented by leading experts at the ICMS Edinburgh Workshop on Differential Geometry and Continuum Mechanics held in June 2013. All papers have been peer reviewed.

The Porous Medium Equation

Mathematical Theory

Author: Juan Luis Vazquez

Publisher: Oxford University Press

ISBN: 0198569033

Category: Mathematics

Page: 624

View: 1504


Aimed at research students and academics in mathematics and engineering, as well as engineering specialists, this book provides a systematic and comprehensive presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation.

Carbon Dioxide Recovery and Utilization

Author: M. Aresta

Publisher: Springer Science & Business Media

ISBN: 9401702454

Category: Science

Page: 408

View: 8979


Carbon Dioxide Recovery and Utilization is a complete and informative resource on the carbon dioxide sources and market at the European Union level, with reference to the world situation. The book covers the following themes: - Sources of carbon dioxide and their purity, - Market of carbon dioxide and its uses, - Separation techniques of carbon dioxide from flue gases, - Analysis of the potential of each technique and application, - Basic science and technology of supercritical CO2, - Reactions in supercritical CO2 and its use as reactive solvent, - Utilization of CO2 in the synthesis of chemicals with low energy input, - Conversion of CO2 into fuels: existing techniques, - Dry reforming of methane, - Assessment of the use of carbon dioxide for the synthesis of methanol. This book is unique in providing integrated information and a perspective on innovative technologies for the use of carbon dioxide. The book is suitable for use as a textbook for courses in chemical engineering and chemistry. It is also of great interest as a general reference for those involved with technologies for avoiding carbon dioxide production and for economists. This is an invaluable reference for specialists on synthetic chemistry, gas separation, supercritical fluids, carbon dioxide marketing, renewable energy and sustainable development. In addition, it will be useful for those working in the chemical industry and for policy makers for carbon dioxide mitigation, innovative technologies, carbon recycling, and power generation.

Lie Groups and Geometric Aspects of Isometric Actions

Author: Marcos M. Alexandrino,Renato G. Bettiol

Publisher: Springer

ISBN: 3319166131

Category: Mathematics

Page: 213

View: 9237


This book provides quick access to the theory of Lie groups and isometric actions on smooth manifolds, using a concise geometric approach. After a gentle introduction to the subject, some of its recent applications to active research areas are explored, keeping a constant connection with the basic material. The topics discussed include polar actions, singular Riemannian foliations, cohomogeneity one actions, and positively curved manifolds with many symmetries. This book stems from the experience gathered by the authors in several lectures along the years and was designed to be as self-contained as possible. It is intended for advanced undergraduates, graduate students and young researchers in geometry and can be used for a one-semester course or independent study.