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This edited volume has a two-fold purpose. First, comprehensive survey articles provide a way for beginners to ease into the corresponding sub-fields. These are then supplemented by original works that give the more advanced readers a glimpse of the current research in geometric analysis and related PDEs. The book is of significant interest for researchers, including advanced Ph.D. students, working in geometric analysis. Readers who have a secondary interest in geometric analysis will benefit from the survey articles. The results included in this book will stimulate further advances in the subjects: geometric analysis, including complex differential geometry, symplectic geometry, PDEs with a geometric origin, and geometry related to topology. Contributions by Claudio Arezzo, Alberto Della Vedova, Werner Ballmann, Henrik Matthiesen, Panagiotis Polymerakis, Sun-Yung A. Chang, Zheng-Chao Han, Paul Yang, Tobias Holck Colding, William P. Minicozzi II, Panagiotis Dimakis, Richard Melrose, Akito Futaki, Hajime Ono, Jiyuan Han, Jeff A. Viaclovsky, Bruce Kleiner, John Lott, Sławomir Kołodziej, Ngoc Cuong Nguyen, Chi Li, Yuchen Liu, Chenyang Xu, YanYan Li, Luc Nguyen, Bo Wang, Shiguang Ma, Jie Qing, Xiaonan Ma, Sean Timothy Paul, Kyriakos Sergiou, Tristan Rivière, Yanir A. Rubinstein, Natasa Sesum, Jian Song, Jeffrey Streets, Neil S. Trudinger, Yu Yuan, Weiping Zhang, Xiaohua Zhu and Aleksey Zinger.

Global analysis (Mathematics). --- Manifolds (Mathematics). --- Global Analysis and Analysis on Manifolds. --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Geometric analysis. --- Geometric analysis PDEs (Geometric partial differential equations) --- Geometry --- Mathematical analysis

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This book covers recent advances in several important areas of geometric analysis including extremal eigenvalue problems, mini-max methods in minimal surfaces, CR geometry in dimension three, and the Ricci flow and Ricci limit spaces. An output of the CIME Summer School "Geometric Analysis" held in Cetraro in 2018, it offers a collection of lecture notes prepared by Ailana Fraser (UBC), André Neves (Chicago), Peter M. Topping (Warwick), and Paul C. Yang (Princeton). These notes will be a valuable asset for researchers and advanced graduate students in geometric analysis. .

Differential geometry. --- Mathematical analysis. --- Analysis (Mathematics). --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Differential Geometry. --- Analysis. --- Global Analysis and Analysis on Manifolds. --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- 517.1 Mathematical analysis --- Mathematical analysis --- Differential geometry --- Geometric analysis --- Geometry, Differential. --- Global analysis (Mathematics) --- Manifolds (Mathematics)

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The book consists of articles based on the XXXVIII Białowieża Workshop on Geometric Methods in Physics, 2019. The series of Białowieża workshops, attended by a community of experts at the crossroads of mathematics and physics, is a major annual event in the field. The works in this book, based on presentations given at the workshop, are previously unpublished, at the cutting edge of current research, typically grounded in geometry and analysis, with applications to classical and quantum physics. For the past eight years, the Białowieża Workshops have been complemented by a School on Geometry and Physics, comprising series of advanced lectures for graduate students and early-career researchers. The extended abstracts of the five lecture series that were given in the eighth school are included. The unique character of the Workshop-and-School series draws on the venue, a famous historical, cultural and environmental site in the Białowieża forest, a UNESCO World Heritage Centre in the east of Poland: lectures are given in the Nature and Forest Museum and local traditions are interwoven with the scientific activities. The chapter “Toeplitz Extensions in Noncommutative Topology and Mathematical Physics” is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.

Global analysis (Mathematics). --- Manifolds (Mathematics). --- Group theory. --- Special functions. --- Geometry. --- Global Analysis and Analysis on Manifolds. --- Group Theory and Generalizations. --- Special Functions. --- Mathematics --- Euclid's Elements --- Special functions --- Mathematical analysis --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Global analysis (Mathematics)

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This monograph collects cutting-edge results and techniques for solving nonlinear partial differential equations using critical points. Including many of the author’s own contributions, a range of proofs are conveniently collected here, Because the material is approached with rigor, this book will serve as an invaluable resource for exploring recent developments in this active area of research, as well as the numerous ways in which critical point theory can be applied. Different methods for finding critical points are presented in the first six chapters. The specific situations in which these methods are applicable is explained in detail. Focus then shifts toward the book’s main subject: applications to problems in mathematics and physics. These include topics such as Schrödinger equations, Hamiltonian systems, elliptic systems, nonlinear wave equations, nonlinear optics, semilinear PDEs, boundary value problems, and equations with multiple solutions. Readers will find this collection of applications convenient and thorough, with detailed proofs appearing throughout. Critical Point Theory will be ideal for graduate students and researchers interested in solving differential equations, and for those studying variational methods. An understanding of fundamental mathematical analysis is assumed. In particular, the basic properties of Hilbert and Banach spaces are used.

Critical point theory (Mathematical analysis) --- Calculus of variations --- Differential topology --- Global analysis (Mathematics) --- Mathematical optimization. --- Operator theory. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Mathematical analysis. --- Analysis (Mathematics). --- Optimization. --- Operator Theory. --- Global Analysis and Analysis on Manifolds. --- Analysis. --- 517.1 Mathematical analysis --- Mathematical analysis --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Functions of complex variables --- Geometry, Algebraic --- Functional analysis --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Manifolds (Mathematics)

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Gathering and updating results scattered in journal articles over thirty years, this self-contained monograph gives a comprehensive introduction to the subject. Its goal is to: - motivate and explain the method for general Lie groups, reducing the proof of deep results in invariant analysis to the verification of two formal Lie bracket identities related to the Campbell-Hausdorff formula (the "Kashiwara-Vergne conjecture"); - give a detailed proof of the conjecture for quadratic and solvable Lie algebras, which is relatively elementary; - extend the method to symmetric spaces; here an obstruction appears, embodied in a single remarkable object called an "e-function"; - explain the role of this function in invariant analysis on symmetric spaces, its relation to invariant differential operators, mean value operators and spherical functions; - give an explicit e-function for rank one spaces (the hyperbolic spaces); - construct an e-function for general symmetric spaces, in the spirit of Kashiwara and Vergne's original work for Lie groups. The book includes a complete rewriting of several articles by the author, updated and improved following Alekseev, Meinrenken and Torossian's recent proofs of the conjecture. The chapters are largely independent of each other. Some open problems are suggested to encourage future research. It is aimed at graduate students and researchers with a basic knowledge of Lie theory.

Mathematics. --- Nonassociative rings. --- Rings (Algebra). --- Harmonic analysis. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Differential geometry. --- Abstract Harmonic Analysis. --- Differential Geometry. --- Non-associative Rings and Algebras. --- Global Analysis and Analysis on Manifolds. --- Global differential geometry. --- Algebra. --- Global analysis. --- Mathematics --- Mathematical analysis --- Geometry, Differential --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Differential geometry --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Algebraic rings --- Ring theory --- Algebraic fields --- Rings (Algebra) --- Lie groups. --- Symmetric spaces. --- Spaces, Symmetric --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups