Geometric Analysis Around Scalar Curvatures

This volume contains three expanded lecture notes from the program Scalar Curvature in Manifold Topology and Conformal Geometry that was held at the Institute for Mathematical Sciences from 1 November to 31 December 2014. The first chapter surveys the recent developments on the fourth-order equations with negative exponent from geometric points of view such as positive mass theorem and uniqueness results. The next chapter deals with the recent important progress on several conjectures such as the existence of non-flat smooth hyper-surfaces and Serrin's over-determined problem. And the final chapter induces a new technique to handle the equation with critical index and the sign change coefficient as well as the negative index term. These topics will be of interest to those studying conformal geometry and geometric partial differential equations. Contents:Lectures on the Fourth-Order Q Curvature Equation (Fengbo Hang and Paul C Yang)An Introduction to the Finite and Infinite Dimensional Reduction Methods (Manuel del Pino and Juncheng Wei)Einstein Constraint Equations on Riemannian Manifolds (Quôc Anh Ngô) Readership: Advanced undergraduates, graduate students and researchers interested in the study of conformal geometry and geometric partial differential equations.

This book contains a clear exposition of two contemporary topics in modern differential geometry: distance geometric analysis on manifolds, in particular, comparison theory for distance functions in spaces which have well defined bounds on their curvature the application of the Lichnerowicz formula for Dirac operators to the study of Gromov's invariants to measure the K-theoretic size of a Riemannian manifold. It is intended for both graduate students and researchers.

A central area of study in Differential Geometry is the examination of the relationship between the purely algebraic properties of the Riemann curvature tensor and the underlying geometric properties of the manifold. In this book, the findings of numerous investigations in this field of study are reviewed and presented in a clear, coherent form, including the latest developments and proofs. Even though many authors have worked in this area in recent years, many fundamental questions still remain unanswered. Many studies begin by first working purely algebraically and then later progressing onto the geometric setting and it has been found that many questions in differential geometry can be phrased as problems involving the geometric realization of curvature. Curvature decompositions are central to all investigations in this area. The authors present numerous results including the Singer–Thorpe decomposition, the Bokan decomposition, the Nikcevic decomposition, the Tricerri–Vanhecke decomposition, the Gray–Hervella decomposition and the De Smedt decomposition. They then proceed to draw appropriate geometric conclusions from these decompositions. The book organizes, in one coherent volume, the results of research completed by many different investigators over the past 30 years. Complete proofs are given of results that are often only outlined in the original publications. Whereas the original results are usually in the positive definite (Riemannian setting), here the authors extend the results to the pseudo-Riemannian setting and then further, in a complex framework, to para-Hermitian geometry as well. In addition to that, new results are obtained as well, making this an ideal text for anyone wishing to further their knowledge of the science of curvature. Contents:Introduction and Statement of ResultsRepresentation TheoryConnections, Curvature, and Differential GeometryReal Affine GeometryAffine Kähler GeometryRiemannian GeometryComplex Riemannian Geometry Readership: Graduate students, researchers, mathematicians and physicist interested in the study of curvature. Keywords:Affine Geometry;Riemannian Geometry;Pseudo-Riemannian Geometry;Kähler Geometry;Para-Kähler Geometry;Hermitian Geometry;Para-Hermitian Geometry;Hyper-Hermitian Geometry;Curvature Tensor;Weyl Geometry;Curvature Decompositions;Almost Complex GeometryKey Features:There are no competing titlesComplete proofs are given that are often only sketched in other literature. Curvature decompositions are presented in parallel for many different structure groups. Full computations of spaces of quadratic invariants appear - this is central to the subject and missing in previously published literature. New results in the pseudo-Riemannian, pseudo-Hermitian and para-Hermitian contexts are included. Geometric realization results are clearly organized and discussedThe relevant “background material” concerning differential geometry and representation theory is introduced, developed and presented in detail, therefore the book is self-contained

"Volume I contains a long article by Misha Gromov based on his many years of involvement in this subject. It came from lectures delivered in Spring 2019 at IHES. There is some background given. Many topics in the field are presented, and many open problems are discussed. One intriguing point here is the crucial role played by two seemingly unrelated analytic means: index theory of Dirac operators and geometric measure theory. Very recently there have been some real breakthroughs in the field. Volume I has several survey articles written by people who were responsible for these results. For Volume II, many people in areas of mathematics and physics, whose work is somehow related to scalar curvature, were asked to write about this in any way thay pleased. This gives rise to a wonderful collection of articles, some with very broad and historical views, others which discussed specific fascinating subjects. These two books give a rich and powerful view of one of geometry's very appealing sides"--

The Proceedings of the ICM publishes the talks, by invited speakers, at the conference organized by the International Mathematical Union every 4 years. It covers several areas of Mathematics and it includes the Fields Medal and Nevanlinna, Gauss and Leelavati Prizes and the Chern Medal laudatios.

In the 25 years since their introduction, Higgs bundles have seen a surprising number of interactions within different areas of mathematics and physics. There is a recent surge of interest following Ngô Bau Châu's proof of the Fundamental Lemma and the work of Kapustin and Witten on the Geometric Langlands program. The program on The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles, was held at the Institute for Mathematical Sciences at the National University of Singapore during 2014. It hosted a number of lectures on recent topics of importance related to Higgs bundles, and it is the purpose of this volume to collect these lectures in a form accessible to graduate students and young researchers interested in learning more about this field.

This volume is to pique the interest of many researchers in the fields of infinite dimensional analysis and quantum probability. These fields have undergone increasingly significant developments and have found many new applications, in particular, to classical probability and to different branches of physics. These fields are rather wide and are of a strongly interdisciplinary nature. For such a purpose, we strove to bridge among these interdisciplinary fields in our Workshop on IDAQP and their Applications that was held at the Institute for Mathematical Sciences, National University of Singapore from 3–7 March 2014. Readers will find that this volume contains all the exciting contributions by well-known researchers in search of new directions in these fields. Contents: Extensions of Quantum Theory Canonically Associated to Classical Probability Measures (Luigi Accardi)Hida Distribution Construction of Indefinite Metric (ϕp)d (d ≥ 4) Quantum Field Theory (Sergio Albeverio and Minoru W Yoshida)A Mathematical Realization of von Neumann's Measurement Scheme (Masanari Asano, Masanori Ohya and Yuta Yamamori)On Random White Noise Processes with Memory for Time Series Analysis (Christopher C Bernido and M Victoria Carpio-Bernido)Self-Repelling (Fractional) Brownian Motion: Results and Open Questions (Jinky Bornales and Ludwig Streit)Normal Approximation for White Noise Functionals by Stein's Method and Hida Calculus (Louis H Y Chen, Yuh-Jia Lee and Hsin-Hung Shih)Sensitive Homology Searching Based on MTRAP Alignment (Toshihide Hara and Masanori Ohya)Some of the Future Directions of White Noise Theory (Takeyuki Hida)Local Statistics for Random Selfadjoint Operators (Peter D Hislop and Maddaly Krishna)Multiple Markov Properties of Gaussian Processes and Their Control (Win Win Htay)Quantum Stochastic Differential Equations Associated with Square of Annihilation and Creation Processes (Un Cig Ji and Kalyan B Sinha)Itô Formula for Generalized Real and Complex White Noise Functionals (Yuh-Jia Lee)Quasi Quantum Quadratic Operators of 𝕄2(ℂ) (Farrukh Mukhamedov)New Noise Depending on the Space Parameter and the Concept of Multiplicity (Si Si)A Hysteresis Effect on Optical Illusion and Non-Kolmogorovian Probability Theory (Masanari Asano, Andrei Khrennikov, Masanori Ohya and Yoshiharu Tanaka)Note on Entropy-Type Complexity of Communication Processes (Noboru Watanabe) Readership: Mathematicians, physicists, biologists, and information scientists as well as advanced undergraduates, and graduate students studying in these fields. All researchers interested in the study of Quantum Information and White Noise Theory. Keywords: White Noise Analysis;Quantum Information;Quantum Probability;Bioinformatics;Genes;Adaptive Dynamics;Entanglement;Quantum Entropy;Non-Kolmogorovian Probability;Infinite Dimensional AnalysisReview: Key Features: Mainly focused on quantum information theory and white noise analysis in line with the fields of infinite dimensional analysis and quantum probabilityWhite noise analysis is in a leading position of the analysis on modern stochastic analysis, and this volume contains contributions to the development of these new exciting directions

Mathemusical Conversations celebrates the understanding of music through mathematics, and the appreciation of mathematics through music. This volume is a compilation of the invited talks given at the Mathemusical Conversations workshop that took place in Singapore from 13–15 February 2015, organized by Elaine Chew in partnership with Gérard Assayag for the scientific program and with Bernard Lanskey for the artistic program. The contributors are world experts and leading scholars, writing on the intersection of music and mathematics. They also focus on performance and composition, two topics which are foundational both to the understanding of human creativity and to the creation of tomorrow's music technologies. This book is essential reading for researchers in both music and mathematics. It will also appeal more broadly to scholars, students, musicians, and anyone interested in new perspectives on the intimate relationship between these two universal human activities. Contents:Foreword by Series EditorsForeword by Workshop OrganizersMathemusical Engagement:Without Our Consent (Paul Schoenfield)Approaches to Musical Expression in Harmonix Video Games (Eran Egozy)Motion and Gravitation in the Musical Spheres (Elaine Chew)Mathemusical Creativity:Improvising in Creative Symbolic Interaction (Gérard Assayag)Music, Creativity, and Computers (Margaret A Boden)Tiling Canons as a Key to Approaching Open Mathematical Conjectures? (Moreno Andreatta)Shaping Performance:Musical Motives in Performance: A Study of Absolute Timing Patterns (Neta Spiro, Nicolas Gold and John Rink)Playing with Variables: Anticipating One Particular Performance of Bach's Goldberg Variations (Bernard Lanskey and Stephen Emmerson)The Informatics Philharmonic in the Indiana University Summer String Academy (Christopher Raphael)Educating the Mathemusical:Mathematical Thought and Empirical Approaches in Higher Education in Music (Jian Yang)Action and Symbol: An Essential Tension (Jeanne Bamberger)Educating the Mathemusical: Balancing the Equation (Don McLean)Geometries:Graph-theoretic and Geometric Models of Music (Richard Cohn)In Quest of Musical Vectors (Dmitri Tymoczko)A Topological Approach of Musical Relationships (Jean-Louis Giavitto and Antoine Spicher)List of Contributors Readership: Advanced secondary school students; post-secondary school students; and scientists, mathematicians, musicians and members of the public interested in the mathematical music sciences.