DOCSLIB.ORG

Geometric Analysis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Geometric Analysis IAS/PARK CITY MATHEMATICS SERIES Volume 22 Geometric Analysis Hubert L. Bray Greg Galloway Rafe Mazzeo Natasa Sesum Editors American Mathematical Society Institute for Advanced Study Geometric Analysis https://doi.org/10.1090//pcms/022 IAS/PARK CITY MATHEMATICS SERIES Volume 22 Geometric Analysis Hubert L. Bray Greg Galloway Rafe Mazzeo Natasa Sesum Editors American Mathematical Society Institute for Advanced Study Hubert Lewis Bray, Gregory J. Galloway, Rafe Mazzeo, and Natasa Sesum, Volume Editors IAS/Park City Mathematics Institute runs mathematics education programs that bring together high school mathematics teachers, researchers in mathematics and mathematics education, undergraduate mathematics faculty, graduate students, and undergraduates to participate in distinct but overlapping programs of research and education. This volume contains the lecture notes from the Graduate Summer School program 2010 Mathematics Subject Classification. Primary 53-06, 35-06, 83-06. Library of Congress Cataloging-in-Publication Data Geometric analysis / Hubert L. Bray, editor [and three others]. pages cm. — (IAS/Park City mathematics series ; volume 22) Includes bibliographical references. ISBN 978-1-4704-2313-1 (alk. paper) 1. Geometric analysis. 2. Mathematical analysis. I. Bray, Hubert L., editor. QA360.G455 2015 515.1—dc23 2015031562 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2016 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 22212019181716 Contents Preface xiii Gerhard Huisken Heat Diffusion in Geometry 1 Heat Diffusion in Geometry 3 1. Heat diffusion 3 2. Curve shortening 5 3. Mean curvature flow 8 4. Ricci flow 10 5. Towards surgery 12 Bibliography 13 Peter Topping Applications of Hamilton’s Compactness Theorem for Ricci Flow 15 Overview 17 Lecture 1. Ricci flow basics – existence and singularities 19 1.1. Initial PDE remarks 19 1.2. Basic Ricci flow theory 20 Lecture 2. Cheeger-Gromov convergence and Hamilton’s compactness theorem 23 2.1. Convergence and compactness of manifolds 23 2.2. Convergence and compactness of flows 25 Lecture 3. Applications to Singularity Analysis 27 3.1. The rescaled flows 27 3.2. Perelman’s no local collapsing theorem 28 Lecture 4. The case of compact surfaces – an alternative approach to the results of Hamilton and Chow 31 Lecture 5. The 2D case in general – Instantaneously complete Ricci flows 35 5.1. How to pose the Ricci flow in general 35 5.2. The existence and uniqueness theory 36 5.3. Asymptotics 38 5.4. Singularities not modelled on shrinking spheres 39 v vi CONTENTS Lecture 6. Contracting Cusp Ricci flows 41 Lecture 7. Subtleties of Hamilton’s compactness theorem 45 7.1. Intuition behind the construction 46 7.2. Fixing proofs requiring completeness in the extended form of Hamilton’s compactness theorem 47 Bibliography 49 Ben Weinkove The K¨ahler-Ricci flow on compact K¨ahler manifolds 51 Preface 53 Lecture 1. An Introduction to K¨ahler geometry 55 1.1. Complex manifolds 55 1.2. Vector fields, 1-forms, Hermitian metrics and tensors 56 1.3. K¨ahler metrics and covariant differentiation 59 1.4. Curvature 61 Lecture 2. The K¨ahler-Ricci flow and the K¨ahler cone 65 2.1. The K¨ahler-Ricci flow and simple examples 65 2.2. The K¨ahler cone and the first Chern class 66 2.3. Maximal existence time for the K¨ahler-Ricci flow 68 Lecture 3. The parabolic complex Monge-Amp`ere equation 71 3.1. Reduction to the complex Monge-Amp`ere equation 71 3.2. Estimates on ϕ andϕ ˙ 74 3.3. Estimate on the metric 76 3.4. Higher order estimates 78 Lecture 4. Convergence results 81 4.1. Negative first Chern class 81 4.2. Zero first Chern class 85 Lecture 5. The K¨ahler-Ricci flow on K¨ahler surfaces, and beyond 89 5.1. Riemann surfaces 89 5.2. K¨ahler surfaces, blowing up and Kodaira dimension 89 5.3. Behavior of the K¨ahler-Ricci flow on K¨ahler surfaces 91 5.4. Non-K¨ahler surfaces and the Chern-Ricci flow 95 Appendix A. Solutions to exercises 99 Bibliography 105 Steve Zelditch Park City lectures on Eigenfunctions 109 Section 1. Introduction 111 1.1. The eigenvalue problem on a compact Riemannian manifold 112 1.2. Nodal and critical point sets 114 CONTENTS vii 1.3. Motivation 115 Section 2. Results 116 2,1. Nodal hypersurface volumes for C∞ metrics 116 2.2. Nodal hypersurface volumes for real analytic (M,g) 117 2.3. Number of intersections of nodal sets with geodesics and number of nodal domains 117 2.4. Dynamics of the geodesic or billiard flow 119 2.5. Quantum ergodic restriction theorems and nodal intersections 120 2.6. Complexification of M and Grauert tubes 121 2.7. Equidistribution of nodal sets in the complex domain 122 2.8. Intersection of nodal sets and real analytic curves on surfaces 122 2.9. Lp norms of eigenfunctions 124 2.10. Quasi-modes 126 2.11. Format of these lectures and references to the literature 127 Section 3. Foundational results on nodal sets 128 3.1. Vanishing order and scaling near zeros 128 3.2. Proof of Proposition 1 129 3.3. A second proof 130 3.4. Rectifiability of the nodal set 130 m−1 ∞ Section 4. Lower bounds for H (Nλ)forC metrics 130 4.1. Proof of Lemma 4.4 132 4.2. Modifications 135 4.3. Lower bounds on L1 norms of eigenfunctions 135 4.4. Dong’s upper bound 135 4.5. Other level sets 136 4.6. Examples 137 Section 5. Quantum ergodic restriction theorem for Dirichlet or Neumann data 138 5.1. Quantum ergodic restriction theorems for Dirichlet data 138 5.2. Quantum ergodic restriction theorems for Cauchy data 139 Section 6. Counting intersections of nodal sets and geodesics 141 6.1. Kuznecov sum formula on surfaces 141 Section 7. Counting nodal domains 142 Section 8. Analytic continuation of eigenfuntions to the complex domain 144 8.1. Grauert tubes 144 8.2. Weak * limit problem for Husimi measures in the complex domain 145 8.3. Background on currents and PSH functions 146 8.4. Poincar´e-Lelong formula 146 8.5. Pluri-subharmonic functions and compactness 146 8.6. A general weak* limit problem for logarithms of Husimi functions 147 Section 9. Poisson operator and Szeg¨o operators on Grauert tubes 147 9.1. Poisson operator and analytic Continuation of eigenfunctions 147 9.2. Analytic continuation of the Poisson wave group 148 9.3. Complexified spectral projections 149 9.4. Poisson operator as a complex Fourier integral operator 149 viii CONTENTS 9.5. Toeplitz dynamical construction of the wave group 151 Section 10. Equidistribution of complex nodal sets of real ergodic eigenfunctions 151 10.1. Sketch of the proof 152 10.2. Growth properties of complexified eigenfunctions 154 10.3. Proof of Lemma 10.6 and Theorem 10.3 155 10.4. Proof of Lemma 10.8 156 10.5. Proof of Lemma 10.7 156 Section 11. Intersections of nodal sets and analytic curves on real analytic surfaces 157 11.1. Counting nodal lines which touch the boundary in analytic plane domains 158 11.2. Application to Pleijel’s conjecture 162 11.3. Equidistribution of intersections of nodal lines and geodesics on surfaces 163 11.4. Real zeros and complex analysis 166 Section 12. Lp norms of eigenfuncions 166 12.1. Generic upper bounds on Lp norms 166 12.2. Lower bounds on L1 norms 167 12.3. Riemannian manifolds with maximal eigenfunction growth 168 12.4. Theorem 9 169 12.5. Sketch of proof of Theorem 9 170 12.6. Size of the remainder at a self-focal point 172 12.7. Decomposition of the remainder into almost loop directions and far from loop directions 173 12.8. Points in M\TL 173 12.9. Perturbation theory of the remainder 173 12.10. Conclusions 174 Section 13. Appendix on the phase space and the geodesic flow 174 Section 14. Appendix: Wave equation and Hadamard parametrix 176 14.1. Hormander parametrix 177 14.2. Wave group: r2 − t2 178 14.3. Exact formula in spaces of constant curvature 179 14.4. Sn 180 14.5. Analytic continuation into the complex 181 Section 15. Appendix: Lagrangian distributions, quasi-modes and Fourier integral operators 181 15.1. Semi-classical Lagrangian distributions and Fourier integral operators 181 15.2. Homogeneous Fourier integral operators 183 15.3.
Load more

Recommended publications
  • Geometric Analysis of Shapes and Its Application to Medical Image Analysis
    Geometric Analysis of Shapes and Its application to Medical Image Analysis by Anirban Mukhopadhyay (Under the DIRECTION of Suchendra M. Bhandarkar) Abstract Geometric analysis of shapes plays an important role in the way the visual world is per- ceived by modern computers. To this end, low-level geometric features provide most obvious and important cues towards understanding the visual scene. A novel intrinsic geometric sur- face descriptor, termed as the Geodesic Field Estimate (GFE) is proposed. Also proposed is a parallel algorithm, well suited for implementation on Graphics Processing Units, for efficient computation of the shortest geodesic paths. Another low level geometric descriptor, termed as the Biharmonic Density Estimate, is proposed to provide an intrinsic geometric scale space signature for multiscale surface feature-based representation of deformable 3D shapes. The computer vision and graphics communities rely on mid-level geometric understanding as well to analyze a scene. Symmetry detection and partial shape matching play an important role as mid-level cues. A comprehensive framework for detection and characterization of partial intrinsic symmetry over 3D shapes is proposed. To identify prominent overlapping symmetric regions, the proposed framework is decoupled into Correspondence Space Voting followed by Transformation Space Mapping procedure. Moreover, a novel multi-criteria optimization framework for matching of partially visible shapes in multiple images using joint geometric embedding is also proposed. The ultimate goal of geometric shape analysis is to resolve high level applications of modern world. This dissertation has focused on three different application scenarios. In the first scenario, a novel approach for the analysis of the non-rigid Left Ventricular (LV) endocardial surface from Multi-Detector CT images, using a generalized isometry-invariant Bag-of-Features (BoF) descriptor, is proposed and implemented.
    [Show full text]
  • Mathematical Analysis of the Accordion Grating Illusion: a Differential Geometry Approach to Introduce the 3D Aperture Problem
    Neural Networks 24 (2011) 1093–1101 Contents lists available at SciVerse ScienceDirect Neural Networks journal homepage: www.elsevier.com/locate/neunet Mathematical analysis of the Accordion Grating illusion: A differential geometry approach to introduce the 3D aperture problem Arash Yazdanbakhsh a,b,∗, Simone Gori c,d a Cognitive and Neural Systems Department, Boston University, MA, 02215, United States b Neurobiology Department, Harvard Medical School, Boston, MA, 02115, United States c Universita' degli studi di Padova, Dipartimento di Psicologia Generale, Via Venezia 8, 35131 Padova, Italy d Developmental Neuropsychology Unit, Scientific Institute ``E. Medea'', Bosisio Parini, Lecco, Italy article info a b s t r a c t Keywords: When an observer moves towards a square-wave grating display, a non-rigid distortion of the pattern Visual illusion occurs in which the stripes bulge and expand perpendicularly to their orientation; these effects reverse Motion when the observer moves away. Such distortions present a new problem beyond the classical aperture Projection line Line of sight problem faced by visual motion detectors, one we describe as a 3D aperture problem as it incorporates Accordion Grating depth signals. We applied differential geometry to obtain a closed form solution to characterize the fluid Aperture problem distortion of the stripes. Our solution replicates the perceptual distortions and enabled us to design a Differential geometry nulling experiment to distinguish our 3D aperture solution from other candidate mechanisms (see Gori et al. (in this issue)). We suggest that our approach may generalize to other motion illusions visible in 2D displays. ' 2011 Elsevier Ltd. All rights reserved. 1. Introduction need for line-ends or intersections along the lines to explain the illusion.
    [Show full text]
  • Karen Keskulla Uhlenbeck
    2019 The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2019 to Karen Keskulla Uhlenbeck University of Texas at Austin “for her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.” Karen Keskulla Uhlenbeck is a founder of modern by earlier work of Morse, guarantees existence of Geometric Analysis. Her perspective has permeated minimisers of geometric functionals and is successful the field and led to some of the most dramatic in the case of 1-dimensional domains, such as advances in mathematics in the last 40 years. closed geodesics. Geometric analysis is a field of mathematics where Uhlenbeck realised that the condition of Palais— techniques of analysis and differential equations are Smale fails in the case of surfaces due to topological interwoven with the study of geometrical and reasons. The papers of Uhlenbeck, co-authored with topological problems. Specifically, one studies Sacks, on the energy functional for maps of surfaces objects such as curves, surfaces, connections and into a Riemannian manifold, have been extremely fields which are critical points of functionals influential and describe in detail what happens when representing geometric quantities such as energy the Palais-Smale condition is violated. A minimising and volume. For example, minimal surfaces are sequence of mappings converges outside a finite set critical points of the area and harmonic maps are of singular points and by using rescaling arguments, critical points of the Dirichlet energy. Uhlenbeck’s they describe the behaviour near the singularities major contributions include foundational results on as bubbles or instantons, which are the standard minimal surfaces and harmonic maps, Yang-Mills solutions of the minimising map from the 2-sphere to theory, and integrable systems.
    [Show full text]
  • Iasinstitute for Advanced Study
    IAInsti tSute for Advanced Study Faculty and Members 2012–2013 Contents Mission and History . 2 School of Historical Studies . 4 School of Mathematics . 21 School of Natural Sciences . 45 School of Social Science . 62 Program in Interdisciplinary Studies . 72 Director’s Visitors . 74 Artist-in-Residence Program . 75 Trustees and Officers of the Board and of the Corporation . 76 Administration . 78 Past Directors and Faculty . 80 Inde x . 81 Information contained herein is current as of September 24, 2012. Mission and History The Institute for Advanced Study is one of the world’s leading centers for theoretical research and intellectual inquiry. The Institute exists to encourage and support fundamental research in the sciences and human - ities—the original, often speculative thinking that produces advances in knowledge that change the way we understand the world. It provides for the mentoring of scholars by Faculty, and it offers all who work there the freedom to undertake research that will make significant contributions in any of the broad range of fields in the sciences and humanities studied at the Institute. Y R Founded in 1930 by Louis Bamberger and his sister Caroline Bamberger O Fuld, the Institute was established through the vision of founding T S Director Abraham Flexner. Past Faculty have included Albert Einstein, I H who arrived in 1933 and remained at the Institute until his death in 1955, and other distinguished scientists and scholars such as Kurt Gödel, George F. D N Kennan, Erwin Panofsky, Homer A. Thompson, John von Neumann, and A Hermann Weyl. N O Abraham Flexner was succeeded as Director in 1939 by Frank Aydelotte, I S followed by J.
    [Show full text]
  • A Mathematical Analysis of Student-Generated Sorting Algorithms Audrey Nasar
    The Mathematics Enthusiast Volume 16 Article 15 Number 1 Numbers 1, 2, & 3 2-2019 A Mathematical Analysis of Student-Generated Sorting Algorithms Audrey Nasar Let us know how access to this document benefits ouy . Follow this and additional works at: https://scholarworks.umt.edu/tme Recommended Citation Nasar, Audrey (2019) "A Mathematical Analysis of Student-Generated Sorting Algorithms," The Mathematics Enthusiast: Vol. 16 : No. 1 , Article 15. Available at: https://scholarworks.umt.edu/tme/vol16/iss1/15 This Article is brought to you for free and open access by ScholarWorks at University of Montana. It has been accepted for inclusion in The Mathematics Enthusiast by an authorized editor of ScholarWorks at University of Montana. For more information, please contact [email protected]. TME, vol. 16, nos.1, 2&3, p. 315 A Mathematical Analysis of student-generated sorting algorithms Audrey A. Nasar1 Borough of Manhattan Community College at the City University of New York Abstract: Sorting is a process we encounter very often in everyday life. Additionally it is a fundamental operation in computer science. Having been one of the first intensely studied problems in computer science, many different sorting algorithms have been developed and analyzed. Although algorithms are often taught as part of the computer science curriculum in the context of a programming language, the study of algorithms and algorithmic thinking, including the design, construction and analysis of algorithms, has pedagogical value in mathematics education. This paper will provide an introduction to computational complexity and efficiency, without the use of a programming language. It will also describe how these concepts can be incorporated into the existing high school or undergraduate mathematics curriculum through a mathematical analysis of student- generated sorting algorithms.
    [Show full text]
  • Arxiv:1912.01885V1 [Math.DG]
    DIRAC-HARMONIC MAPS WITH POTENTIAL VOLKER BRANDING Abstract. We study the influence of an additional scalar potential on various geometric and analytic properties of Dirac-harmonic maps. We will create a mathematical wishlist of the pos- sible benefits from inducing the potential term and point out that the latter cannot be achieved in general. Finally, we focus on several potentials that are motivated from supersymmetric quantum field theory. 1. Introduction and Results The supersymmetric nonlinear sigma model has received a lot of interest in modern quantum field theory, in particular in string theory, over the past decades. At the heart of this model is an action functional whose precise structure is fixed by the invariance under various symmetry operations. In the physics literature the model is most often formulated in the language of supergeome- try which is necessary to obtain the invariance of the action functional under supersymmetry transformations. For the physics background of the model we refer to [1] and [18, Chapter 3.4]. However, if one drops the invariance under supersymmetry transformations the resulting action functional can be investigated within the framework of geometric variational problems and many substantial results in this area of research could be achieved in the last years. To formulate this mathematical version of the supersymmetric nonlinear sigma model one fixes a Riemannian spin manifold (M, g), a second Riemannian manifold (N, h) and considers a map φ: M → N. The central ingredients in the resulting action functional are the Dirichlet energy for the map φ and the Dirac action for a spinor defined along the map φ.
    [Show full text]
  • The Orthosymplectic Lie Supergroup in Harmonic Analysis Kevin Coulembier
    The orthosymplectic Lie supergroup in harmonic analysis Kevin Coulembier Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Belgium Abstract. The study of harmonic analysis in superspace led to the Howe dual pair (O(m) × Sp(2n);sl2). This Howe dual pair is incomplete, which has several implications. These problems can be solved by introducing the orthosymplectic Lie supergroup OSp(mj2n). We introduce Lie supergroups in the supermanifold setting and show that everything is captured in the Harish-Chandra pair. We show the uniqueness of the supersphere integration as an orthosymplectically invariant linear functional. Finally we study the osp(mj2n)-representations of supersymmetric tensors for the natural module for osp(mj2n). Keywords: Howe dual pair, Lie supergroup, Lie superalgebra, harmonic analysis, supersphere PACS: 02.10.Ud, 20.30.Cj, 02.30.Px PRELIMINARIES Superspaces are spaces where one considers not only commuting but also anti-commuting variables. The 2n anti- commuting variables x`i generate the complex Grassmann algebra L2n. Definition 1. A supermanifold of dimension mj2n is a ringed space M = (M0;OM ), with M0 the underlying manifold of dimension m and OM the structure sheaf which is a sheaf of super-algebras with unity on M0. The sheaf OM satisfies the local triviality condition: there exists an open cover fUigi2I of M0 and isomorphisms of sheaves of super algebras T : Ui ! ¥ ⊗ L . i OM CUi 2n The sections of the structure sheaf are referred to as superfunctions on M . A morphism of supermanifolds ] ] F : M ! N is a morphism of ringed spaces (f;f ), f : M0 ! N0, f : ON ! f∗OM .
    [Show full text]
  • Hamilton's Ricci Flow
    The University of Melbourne, Department of Mathematics and Statistics Hamilton's Ricci Flow Nick Sheridan Supervisor: Associate Professor Craig Hodgson Second Reader: Professor Hyam Rubinstein Honours Thesis, November 2006. Abstract The aim of this project is to introduce the basics of Hamilton's Ricci Flow. The Ricci flow is a pde for evolving the metric tensor in a Riemannian manifold to make it \rounder", in the hope that one may draw topological conclusions from the existence of such \round" metrics. Indeed, the Ricci flow has recently been used to prove two very deep theorems in topology, namely the Geometrization and Poincar´eConjectures. We begin with a brief survey of the differential geometry that is needed in the Ricci flow, then proceed to introduce its basic properties and the basic techniques used to understand it, for example, proving existence and uniqueness and bounds on derivatives of curvature under the Ricci flow using the maximum principle. We use these results to prove the \original" Ricci flow theorem { the 1982 theorem of Richard Hamilton that closed 3-manifolds which admit metrics of strictly positive Ricci curvature are diffeomorphic to quotients of the round 3-sphere by finite groups of isometries acting freely. We conclude with a qualitative discussion of the ideas behind the proof of the Geometrization Conjecture using the Ricci flow. Most of the project is based on the book by Chow and Knopf [6], the notes by Peter Topping [28] (which have recently been made into a book, see [29]), the papers of Richard Hamilton (in particular [9]) and the lecture course on Geometric Evolution Equations presented by Ben Andrews at the 2006 ICE-EM Graduate School held at the University of Queensland.
    [Show full text]
  • Geometric Analysis and Integral Geometry
    598 Geometric Analysis and Integral Geometry AMS Special Session on Radon Transforms and Geometric Analysis in Honor of Sigurdur Helgason’s 85th Birthday January 4–7, 2012 Boston, MA Tufts University Workshop on Geometric Analysis on Euclidean and Homogeneous Spaces January 8–9, 2012 Medford, MA Eric Todd Quinto Fulton Gonzalez Jens Gerlach Christensen Editors American Mathematical Society Geometric Analysis and Integral Geometry AMS Special Session on Radon Transforms and Geometric Analysis in Honor of Sigurdur Helgason’s 85th Birthday January 4–7, 2012 Boston, MA Tufts University Workshop on Geometric Analysis on Euclidean and Homogeneous Spaces January 8–9, 2012 Medford, MA Eric Todd Quinto Fulton Gonzalez Jens Gerlach Christensen Editors 598 Geometric Analysis and Integral Geometry AMS Special Session on Radon Transforms and Geometric Analysis in Honor of Sigurdur Helgason’s 85th Birthday January 4–7, 2012 Boston, MA Tufts University Workshop on Geometric Analysis on Euclidean and Homogeneous Spaces January 8–9, 2012 Medford, MA Eric Todd Quinto Fulton Gonzalez Jens Gerlach Christensen Editors American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Martin J. Strauss 2010 Mathematics Subject Classification. Primary 22E30, 43A85, 44A12, 45Q05, 92C55; Secondary 22E46, 32L25, 35S30, 65R32. Library of Congress Cataloging-in-Publication Data AMS Special Session on Radon Transforms and Geometric Analysis (2012 : Boston, Mass.) Geometric analysis and integral geometry : AMS special session in honor of Sigurdur Helgason’s 85th birthday, radon transforms and geometric analysis, January 4-7, 2012, Boston, MA ; Tufts University Workshop on Geometric Analysis on Euclidean and Homogeneous Spaces, January 8-9, 2012, Medford, MA / Eric Todd Quinto, Fulton Gonzalez, Jens Gerlach Christensen, editors.
    [Show full text]
  • On P-Adic Mathematical Physics B
    ISSN 2070-0466, p-Adic Numbers, Ultrametric Analysis and Applications, 2009, Vol. 1, No. 1, pp. 1–17. c Pleiades Publishing, Ltd., 2009. REVIEW ARTICLES On p-Adic Mathematical Physics B. Dragovich1**, A. Yu. Khrennikov2***, S.V.Kozyrev3****,andI.V.Volovich3***** 1Institute of Physics, Pregrevica 118, 11080 Belgrade, Serbia 2Va¨xjo¨ University, Va¨xjo¨,Sweden 3 Steklov Mathematical Institute, ul. Gubkina 8, Moscow, 119991 Russia Received December 15, 2008 Abstract—A brief review of some selected topics in p-adic mathematical physics is presented. DOI: 10.1134/S2070046609010014 Key words: p-adic numbers, p-adic mathematical physics, complex systems, hierarchical structures, adeles, ultrametricity, string theory, quantum mechanics, quantum gravity, prob- ability, biological systems, cognitive science, genetic code, wavelets. 1. NUMBERS: RATIONAL, REAL, p-ADIC We present a brief review of some selected topics in p-adic mathematical physics. More details can be found in the references below and the other references are mainly contained therein. We hope that this brief introduction to some aspects of p-adic mathematical physics could be helpful for the readers of the first issue of the journal p-Adic Numbers, Ultrametric Analysis and Applications. The notion of numbers is basic not only in mathematics but also in physics and entire science. Most of modern science is based on mathematical analysis over real and complex numbers. However, it is turned out that for exploring complex hierarchical systems it is sometimes more fruitful to use analysis over p-adic numbers and ultrametric spaces. p-Adic numbers (see, e.g. [1]), introduced by Hensel, are widely used in mathematics: in number theory, algebraic geometry, representation theory, algebraic and arithmetical dynamics, and cryptography.
    [Show full text]
  • Perspectives on Geometric Analysis
    Surveys in Differential Geometry X Perspectives on geometric analysis Shing-Tung Yau This essay grew from a talk I gave on the occasion of the seventieth anniversary of the Chinese Mathematical Society. I dedicate the lecture to the memory of my teacher S.S. Chern who had passed away half a year before (December 2004). During my graduate studies, I was rather free in picking research topics. I[731] worked on fundamental groups of manifolds with non-positive curva- ture. But in the second year of my studies, I started to look into differential equations on manifolds. However, at that time, Chern was very much inter- ested in the work of Bott on holomorphic vector fields. Also he told me that I should work on Riemann hypothesis. (Weil had told him that it was time for the hypothesis to be settled.) While Chern did not express his opinions about my research on geometric analysis, he started to appreciate it a few years later. In fact, after Chern gave a course on Calabi’s works on affine geometry in 1972 at Berkeley, S.Y. Cheng told me about these inspiring lec- tures. By 1973, Cheng and I started to work on some problems mentioned in Chern’s lectures. We did not realize that the great geometers Pogorelov, Calabi and Nirenberg were also working on them. We were excited that we solved some of the conjectures of Calabi on improper affine spheres. But soon after we found out that Pogorelov [563] published his results right be- fore us by different arguments. Nevertheless our ideas are useful in handling other problems in affine geometry, and my knowledge about Monge-Amp`ere equations started to broaden in these years.
    [Show full text]
  • Recent Advances in Geometric Analysis
    Advanced Lectures in Mathematics Volume XI Recent Advances in Geometric Analysis Editors: Yng-Ing Lee, Chang-Shou Lin, and Mao-Pei Tsui International Press 浧䷘㟨十⒉䓗䯍 www.intlpress.com HIGHER EDUCATION PRESS Yng-Ing Lee Chang-Shou Lin National Taiwan University National Chung Cheng University Mao-Pei Tsui University of Toledo Copyright © 2010 by International Press, Somerville, Massachusetts, U.S.A., and by Higher Education Press, Beijing, China. This work is published and sold in China exclusively by Higher Education Press of China. No part of this work can be reproduced in any form, electronic or mechanical, recording, or by any information storage and data retrieval system, without prior approval from International Press. Requests for reproduction for scientific and/or educational purposes will normally be granted free of charge. In those cases where the author has retained copyright, requests for permission to use or reproduce any material should be addressed directly to the author. ISBN 978-1-57146-143-8 Typeset using the LaTeX system. Printed in the USA on acid-free paper. ADVANCED LECTURES IN MATHEMATICS Executive Editors Shing-Tung Yau Kefeng Liu Harvard University University of California at Los Angeles Zhejiang University Lizhen Ji Hangzhou, China University of Michigan, Ann Arbor Editorial Board Chongqing Cheng Tatsien Li Nanjing University Fudan University Nanjing, China Shanghai, China Zhong-Ci Shi Zhiying Wen Institute of Computational Mathematics Tsinghua University Chinese Academy of Sciences (CAS) Beijing, China Beijing, China Lo Yang Zhouping Xin Institute of Mathematics The Chinese University of Hong Kong Chinese Academy of Sciences (CAS) Hong Kong, China Beijing, China Weiping Zhang Xiangyu Zhou Nankai University Institute of Mathematics Tianjin, China Chinese Academy of Sciences (CAS) Beijing, China Xiping Zhu Zhongshan University Guangzhou, China Preface “2007 International Conference in Geometric Analysis” was held in Taiwan University from June 18th to 22nd, 2007.
    [Show full text]