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Since the early part of the twentieth century, the use of integral equations has developed into a range of tools for the study of partial differential equations. This includes the use of single- and double-layer potentials to treat classical boundary value problems. The aim of this book is to give a self-contained presentation of an asymptotic theory for eigenvalue problems using layer potential techniques with applications in the fields of inverse problems, band gap structures, and optimal design, in particular the optimal design of photonic and phononic crystals. Throughout this book, it is shown how powerful the layer potentials techniques are for solving not only boundary value problems but also eigenvalue problems if they are combined with the elegant theory of Gohberg and Sigal on meromorphic operator-valued functions. The general approach in this book is developed in detail for eigenvalue problems for the Laplacian and the Lame system in the following two situations: one under variation of domains or boundary conditions and the other due to the presence of inclusions. The book will be of interest to researchers and graduate students working in the fields of partial differential equations, integral equations, and inverse problems. Researchers in engineering and physics may also find this book helpful.
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• An outline Definitions and technicalities Turbulence Classifying homeomorphisms Infinite dimensional group representations A generalized Scott analysis GE groups The dark side Beyond Borel Looking ahead Ordinals Notation Bibliography Index.
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Actions of Polish groups are ubiquitous in mathematics. In certain branches of ergodic theory and functional analysis, one finds a systematic study of the group of measure-preserving transformations and the unitary group. In logic, the analysis of countable models intertwines with results concerning the actions of the infinite symmetric group. This text develops the theory of Polish group actions entirely from scratch, ultimately presenting a coherent theory of the resulting orbit equivalence classes that may allow complete classification by invariants of an indicated form.The book concludes with a criterion for an orbit equivalence relation classifiable by countable structures considered up to isomorphism. This self-contained volume offers a complete treatment of this active area of current research and develops a difficult general theory classifying a class of mathematical objects up to some relevant notion of isomorphism or equivalence. Greg Hjorth received the Carol Karp Prize for outstanding work on turbulence and countable Borel equivalence relations from the Association of Symbolic Logic.
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• Autonomous theory: Semigroups and global attractors Upper and lower semicontinuity Topological structural stability of attractors Neighborhood of a critical element Morse-Smale semigroups Non-autonomous theory: Non-autonomous dynamical systems and their attractors Upper and lower semicontinuity Topological structural stability Neighborhood of a global hyperbolic solution Non-autonomous Morse-Smale dynamical systems Bibliography List of figures Index.
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This book provides a comprehensive study of how attractors behave under perturbations for both autonomous and non-autonomous problems. Furthermore, the forward asymptotics of non-autonomous dynamical systems is presented here for the first time in a unified manner. When modelling real world phenomena imprecisions are unavoidable. On the other hand, it is paramount that mathematical models reflect the modelled phenomenon, in spite of unimportant neglectable influences discounted by simplifications, small errors introduced by empirical laws or measurements, among others. The authors deal with this issue by investigating the permanence of dynamical structures and continuity properties of the attractor. This is done in both the autonomous (time independent) and non-autonomous (time dependent) framework in four distinct levels of approximation: the upper semicontinuity, lower semicontinuity, topological structural stability and geometrical structural stability. This book is aimed at graduate students and researchers interested in dissipative dynamical systems and stability theory, and requires only a basic background in metric spaces, functional analysis and, for the applications, techniques of ordinary and partial differential equations.
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• Introduction. Dessins d'enfants: From polynomials through Belyi functions to weighted trees. Existence theorem. Recapitulation and perspective. Classification of unitrees. Computation of Davenport-Zannier pairs for unitrees. Primitive monodromy groups of weighted trees. Trees with primitive monodromy groups. A zoo of examples and constructions. Diophantine invariants. Enumeration. What remains to be done. Bibliography. Index.
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The French expression "dessins d'enfants'' means children's drawings. This term was coined by the great French mathematician Alexandre Grothendieck in order to denominate a method of pictorial representation of some highly interesting classes of polynomials and rational functions. The polynomials studied in this book take their origin in number theory. The authors show how, by drawing simple pictures, one can prove some long-standing conjectures and formulate new ones. The theory presented here touches upon many different fields of mathematics. The major part of the book is quite elementary and is easily accessible to an undergraduate student. The less elementary parts, such as Galois theory or group representations and their characters, would need a more profound knowledge of mathematics. The reader may either take the basic facts of these theories for granted or use our book as a motivation and a first approach to these subjects.
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• Introduction. Equivalence relations: The virtual realm. Turbulence. Nested sequences of models. Balanced extensions of the Solovay model: Balanced Suslin forcing. Simplicial complex forcings. Ultrafilter forcings. Other forcings. Preserving cardinalities. Uniformization. Locally countable structures. The Silver divide. The arity divide. Other combinatorics. Bibliography. Index.
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This book introduces a new research direction in set theory: the study of models of set theory with respect to their extensional overlap or disagreement. In Part I, the method is applied to isolate new distinctions between Borel equivalence relations. Part II contains applications to independence results in Zermelo-Fraenkel set theory without Axiom of Choice. The method makes it possible to classify in great detail various paradoxical objects obtained using the Axiom of Choice; the classifying criterion is a ZF-provable implication between the existence of such objects. The book considers a broad spectrum of objects from analysis, algebra, and combinatorics: ultrafilters, Hamel bases, transcendence bases, colorings of Borel graphs, discontinuous homomorphisms between Polish groups, and many more. The topic is nearly inexhaustible in its variety, and many directions invite further investigation.
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• Hopf algebras, Nichols algebras, braided monoidal categories, and quantized enveloping algebras: A quick introduction to Nichols algebras Basic Hopf algebra theory Braided monoidal categories Yetter-Drinfeld modules over Hopf algebras Gradings and filtrations Braided structures Nichols algebras Quantized enveloping algebras and generalizations Cartan graphs, Weyl groupoids, and root systems: Cartan graphs and Weyl groupoids The structure of Cartan graphs and root systems Cartan graphs of Lie superalgebras Weyl groupoids and root systems of Nichols algebras: A braided monoidal isomorphism of Yetter-Drinfeld modules Nichols systems, and semi-Cartan graph of Nichols algebras Right coideal subalgebras of Nichols systems, and Cartan graph of Nichols algebras Applications: Nichols algebras of diagonal type Nichols algebras of Cartan type Nichols algebras over non-abelian groups Bibliography Index of symbols Subject index.
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This book is an introduction to Hopf algebras in braided monoidal categories with applications to Hopf algebras in the usual sense. The main goal of the book is to present from scratch and with complete proofs the theory of Nichols algebras (or quantum symmetric algebras) and the surprising relationship between Nichols algebras and generalized root systems. In general, Nichols algebras are not classified by Cartan graphs and their root systems. However, extending partial results in the literature, the authors were able to associate a Cartan graph to a large class of Nichols algebras. This allows them to determine the structure of right coideal subalgebras of Nichols systems which generalize Nichols algebras. As applications of these results, the book contains a classification of right coideal subalgebras of quantum groups and of the small quantum groups, and a proof of the existence of PBW-bases that does not involve case by case considerations. The authors also include short chapter summaries at the beginning of each chapter and historical notes at the end of each chapter. The theory of Cartan graphs, Weyl groupoids, and generalized root systems appears here for the first time in a book form. Hence, the book serves as an introduction to the modern classification theory of pointed Hopf algebras for advanced graduate students and researchers working in categorial aspects and classification theory of Hopf algebras and their generalization.
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• Curves with many points. I: Modular curves Class field theory Curves with many points. II Infinite global fields Decoding: Some examples Sphere packings Codes from multidimensional varieties Applications Appendix: Some basic facts from Volume 1 Bibliography List of names Index.
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Algebraic Geometry Codes: Advanced Chapters is devoted to the theory of algebraic geometry codes, a subject related to several domains of mathematics. On one hand, it involves such classical areas as algebraic geometry and number theory; on the other, it is connected to information transmission theory, combinatorics, finite geometries, dense packings, and so on. The book gives a unique perspective on the subject. Whereas most books on coding theory start with elementary concepts and then develop them in the framework of coding theory itself within, this book systematically presents meaningful and important connections of coding theory with algebraic geometry and number theory. Among many topics treated in the book, the following should be mentioned: curves with many points over finite fields, class field theory, asymptotic theory of global fields, decoding, sphere packing, codes from multi-dimensional varieties, and applications of algebraic geometry codes. The book is the natural continuation of Algebraic Geometric Codes: Basic Notions by the same authors. The concise exposition of the first volume is included as an appendix.
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• The Dirichlet space Foundations: Geometry and analysis on the disk Hilbert spaces of holomorphic functions Intermezzo: Hardy spaces Carleson measures Analysis on trees The Pick property Interpolation The Dirichlet space Selected topics: Onto interpolation Boundary values Alternative norms and applications Shift operators and invariant subspaces Invariant subspaces of the Dirichlet shift Bilinear forms on $\mathcal{D}$ Besov spaces on the ball: Besov spaces on balls and trees Interpolating sequences Spaces on trees Corona theorems for Besov spaces in $\mathbb{C}^n$ Some functional analysis Schur's test Bibliography Index.
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The study of the classical Dirichlet space is one of the central topics on the intersection of the theory of holomorphic functions and functional analysis. It was introduced about100 years ago and continues to be an area of active current research. The theory is related to such important themes as multipliers, reproducing kernels, and Besov spaces, among others. The authors present the theory of the Dirichlet space and related spaces starting with classical results and including some quite recent achievements like Dirichlet-type spaces of functions in several complex variables and the corona problem. The first part of this book is an introduction to the function theory and operator theory of the classical Dirichlet space, a space of holomorphic functions on the unit disk defined by a smoothness criterion. The Dirichlet space is also a Hilbert space with a reproducing kernel, and is the model for the dyadic Dirichlet space, a sequence space defined on the dyadic tree. These various viewpoints are used to study a range of topics including the Pick property, multipliers, Carleson measures, boundary values, zero sets, interpolating sequences, the local Dirichlet integral, shift invariant subspaces, and Hankel forms. Recurring themes include analogies, sometimes weak and sometimes strong, with the classical Hardy space; and the analogy with the dyadic Dirichlet space. The final chapters of the book focus on Besov spaces of holomorphic functions on the complex unit ball, a class of Banach spaces generalizing the Dirichlet space. Additional techniques are developed to work with the nonisotropic complex geometry, including a useful invariant definition of local oscillation and a sophisticated variation on the dyadic Dirichlet space. Descriptions are obtained of multipliers, Carleson measures, interpolating sequences, and multiplier interpolating sequences; $\overline\partial$ estimates are obtained to prove corona theorems.
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• Preface
• Introducton
• Chapter 1. Nonassociative algebras
• Chapter 2. General facts on Lie algebras
• Chapter 3. Absolute zero divisors
• Chapter 4. Jordan elements
• Chapter 5. Von Neumann regular elements
• Chapter 6. Extremal elements
• Chapter 7. A characterization of strong primeness
• Chapter 8. From Lie algebras to Jordan algebras
• Chapter 9. The Kostrikin radical
• Chapter 10. Algebraic Lie algebras and local finiteness
• Chapter 11. From Lie algebras to Jordan pairs
• Chapter 12. An Artinian theory for Lie algebras
• Chapter 13. Inner ideal structure of Lie algebras
• Chapter 14. Classical infinite-dimensional Lie algebras
• Chapter 15. Classical Banach-Lie algebras.

This book explores applications of Jordan theory to the theory of Lie algebras. It begins with the general theory of nonassociative algebras and of Lie algebras and then focuses on properties of Jordan elements of special types. Then it proceeds to the core of the book, in which the author explains how properties of the Jordan algebra attached to a Jordan element of a Lie algebra can be used to reveal properties of the Lie algebra itself. One of the special features of this book is that it carefully explains Zelmanov's seminal results on infinite-dimensional Lie algebras from this point of view. The book is suitable for advanced graduate students and researchers who are interested in learning how Jordan algebras can be used as a powerful tool to understand Lie algebras, including infinite-dimensional Lie algebras. Although the book is on an advanced and rather specialized topic, it spends some time developing necessary introductory material, includes exercises for the reader, and is accessible to a student who has finished their basic graduate courses in algebra and has some familiarity with Lie algebras in an abstract algebraic setting.
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• JB$^*$-triples in dual Banach spaces From bounded domains to symmetric Banach manifolds: Analytic manifolds and their automorphism groups Uniform manifolds and their automorphism groups The semigroup $\mathcal{O}_c(X)$ of holomorphic contractions Manifolds with a compatible invariant metrics Manifolds with a compatible tangent norm Symmetric normed manifolds J$^*$-triples and their related Lie algebras The J$^*$-triple associated with a symmetric manifold The symmetric manifold associated with a J$^*$-triple Finite Rank J$^*$-triples and JH$^*$-triples: Algebraic study of J$^*$-triples Atomic J$^*$-triples and JH$^*$-triples From symmetric Banach manifolds to JB$^*$-triples: Spectral properties and bounded J$^*$-triples The Riemann mapping theorem for JB$^*$-triples The category of JB$^*$-triples Automorphisms of bounded symmetric domains Tripotents in JB$^*$-triples Functional calculus in a JB$^*$-triple. Applications Automorphisms of Banach-Grassmann manifolds Symmetric Grassmann manifolds over Hilbert spaces Affine structure of the unit ball in a JB$^*$-triple JB$^*$-triples in dual Banach spaces or JBW$^*$-triples: Structure theory for JBW$^*$-triples and their preduals Facial structure in JBW$^*$-triples and in JB$^*$-triples The strong and strong* topologies in JBW$^*$-triples Derivations of JB$^*$-triples Some results on functional analysis Lists of symbols and their meanings Bibliography Index.
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This book is a systematic account of the impressive developments in the theory of symmetric manifolds achieved over the past 50 years. It contains detailed and friendly, but rigorous, proofs of the key results in the theory. Milestones are the study of the group of holomomorphic automorphisms of bounded domains in a complex Banach space (Vigue and Upmeier in the late 1970s), Kaup's theorem on the equivalence of the categories of symmetric Banach manifolds and that of hermitian Jordan triple systems, and the culminating point in the process: the Riemann mapping theorem for complex Banach spaces (Kaup, 1982). This led to the introduction of wide classes of Banach spaces known as $\mathrm{JB}^*$-triples and $\mathrm{JBW}^*$-triples whose geometry has been thoroughly studied by several outstanding mathematicians in the late 1980s. The book presents a good example of fruitful interaction between different branches of mathematics, making it attractive for mathematicians interested in various fields such as algebra, differential geometry and, of course, complex and functional analysis. This book is published in cooperation with Real Sociedad Matematica Espanola.
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• Introduction Distributions and function spaces Spectral theory of nonselfadjoint operators Linear stability of NLS solitary waves Solitary waves of nonlinear Schrodinger equation Limiting absorption principle Carleman-Berthier-Georgescu estimates The Dirac matrices The Soler model Bi-frequency solitary waves Bifurcations of eigenvalues from the essential spectrum Nonrelativistic asymptotics of solitary waves Spectral stability in the nonrelativistic limit Bibliography Index List of symbols.
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This monograph gives a comprehensive treatment of spectral (linear) stability of weakly relativistic solitary waves in the nonlinear Dirac equation. It turns out that the instability is not an intrinsic property of the Dirac equation that is only resolved in the framework of the second quantization with the Dirac sea hypothesis. Whereas general results about the Dirac-Maxwell and similar equations are not yet available, we can consider the Dirac equation with scalar self-interaction, the model first introduced in 1938. In this book we show that in particular cases solitary waves in this model may be spectrally stable (no linear instability). This result is the first step towards proving asymptotic stability of solitary waves. The book presents the necessary overview of the functional analysis, spectral theory, and the existence and linear stability of solitary waves of the nonlinear Schrodinger equation. It also presents the necessary tools such as the limiting absorption principle and the Carleman estimates in the form applicable to the Dirac operator, and proves the general form of the Dirac-Pauli theorem. All of these results are used to prove the spectral stability of weakly relativistic solitary wave solutions of the nonlinear Dirac equation.
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• Differential equations and expressions: First order systems Quasi-differential expressions and equations The Lagrange identity and maximal and minimal operators Deficiency indices Symmetric, self-adjoint, and dissipative operators: Regular symmetric operators Singular symmetric operators Self-adjoint operators Self-adjoint and symmetric boundary conditions Solutions and spectrum Coefficients, the deficiency index, spectrum Dissipative operators Two-interval problems: Two-interval symmetric domains Two-interval symmetric domain characterization with maximal domain functions Other topics: Green's function and adjoint problems Notation Topics not covered and open problems Bibliography Index.
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In 1910 Herman Weyl published one of the most widely quoted papers of the 20th century in Analysis, which initiated the study of singular Sturm-Liouville problems. The work on the foundations of Quantum Mechanics in the 1920s and 1930s, including the proof of the spectral theorem for unbounded self-adjoint operators in Hilbert space by von Neumann and Stone, provided some of the motivation for the study of differential operators in Hilbert space with particular emphasis on self-adjoint operators and their spectrum. Since then the topic developed in several directions and many results and applications have been obtained. In this monograph the authors summarize some of these directions discussing self-adjoint, symmetric, and dissipative operators in Hilbert and Symplectic Geometry spaces. Part I of the book covers the theory of differential and quasi-differential expressions and equations, existence and uniqueness of solutions, continuous and differentiable dependence on initial data, adjoint expressions, the Lagrange Identity, minimal and maximal operators, etc. In Part II characterizations of the symmetric, self-adjoint, and dissipative boundary conditions are established. In particular, the authors prove the long standing Deficiency Index Conjecture. In Part III the symmetric and self-adjoint characterizations are extended to two-interval problems. These problems have solutions which have jump discontinuities in the interior of the underlying interval. These jumps may be infinite at singular interior points. Part IV is devoted to the construction of the regular Green's function. The construction presented differs from the usual one as found, for example, in the classical book by Coddington and Levinson.
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• Preface / by Bryden Cais
• Introduction / by Peter Scholze
• Adic spaces / by Jared Weinstein
• Sheaves, stacks, and shtukas / by Kiran S. Kedlaya
• The Hodge-Tate decomposition via perfectoid spaces / by Bhargav Bhatt
• Perfectoid Shimura varieties / by Ana Caraiani.

Introduced by Peter Scholze in 2011, perfectoid spaces are a bridge between geometry in characteristic 0 and characteristic $p$, and have been used to solve many important problems, including cases of the weight-monodromy conjecture and the association of Galois representations to torsion classes in cohomology. In recognition of the transformative impact perfectoid spaces have had on the field of arithmetic geometry, Scholze was awarded a Fields Medal in 2018. This book, originating from a series of lectures given at the 2017 Arizona Winter School on perfectoid spaces, provides a broad introduction to the subject. After an introduction with insight into the history and future of the subject by Peter Scholze, Jared Weinstein gives a user-friendly and utilitarian account of the theory of adic spaces. Kiran Kedlaya further develops the foundational material, studies vector bundles on Fargues-Fontaine curves, and introduces diamonds and shtukas over them with a view toward the local Langlands correspondence. Bhargav Bhatt explains the application of perfectoid spaces to comparison isomorphisms in $p$-adic Hodge theory. Finally, Ana Caraiani explains the application of perfectoid spaces to the construction of Galois representations associated to torsion classes in the cohomology of locally symmetric spaces for the general linear group. This book will be an invaluable asset for any graduate student or researcher interested in the theory of perfectoid spaces and their applications.
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• From groupoid to algebra
• Groupoid actions and equivalence
• Measure theory
• Proof of the equivalence theorem
• Basic representation theory
• The existence and uniqueness of Haar systems
• Unitary representations
• Renault's Disintegration Theorem
• Amenability for groupoids
• Measurewise amenability for groupoids
• Comments on simplicity
• Duals and topological vector spaces.

• Introduction / by John W. Morgan
• Notes on Kuranishi atlases / by Dusa McDuff
• Gromov-Witten theory via Kuranishi structures /rby Mohammad F. Tehrani and Kenji Fukaya
• Kuranishi spaces as a 2-category / by Dominic Joyce.

The method of using the moduli space of pseudo-holomorphic curves on a symplectic manifold was introduced by Mikhail Gromov in 1985. From the appearance of Gromov's original paper until today this approach has been the most important tool in global symplectic geometry. To produce numerical invariants of these manifolds using this method requires constructing a fundamental cycle associated with moduli spaces. This volume brings together three approaches to constructing the ``virtual'' fundamental cycle for the moduli space of pseudo-holomorphic curves. All approaches are based on the idea of local Kuranishi charts for the moduli space. Workers in the field will get a comprehensive understanding of the details of these constructions and the assumptions under which they can be made. These techniques and results will be essential in further applications of this approach to producing invariants of symplectic manifolds. This book is published in cooperation with Simons Center for Geometry and Physics.
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• Background: Simple groups and their properties Outline of the proof of the CFSG: Some main ideas Thompson factorization--and its failure: FF-methods Recognition theorms for simple groups Representation theory of simple groups Maximal subgroups and primitive representations Geometries for simple groups Some fusion techniques for classification problems Some applications close to finite group theory Some applications farther afield from finite groups Appendix A: Some supplementary notes to the text Appendix B: Further remarks on certain exercises Bibliography Index.
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Classification of Finite Simple Groups (CFSG) is a major project involving work by hundreds of researchers. The work was largely completed by about 1983, although final publication of the ``quasithin'' part was delayed until 2004. Since the 1980s, CFSG has had a huge influence on work in finite group theory and in many adjacent fields of mathematics. This book attempts to survey and sample a number of such topics from the very large and increasingly active research area of applications of CFSG. The book is based on the author's lectures at the September 2015 Venice Summer School on Finite Groups. With about 50 exercises from original lectures, it can serve as a second-year graduate course for students who have had first-year graduate algebra. It may be of particular interest to students looking for a dissertation topic around group theory. It can also be useful as an introduction and basic reference; in addition, it indicates fuller citations to the appropriate literature for readers who wish to go on to more detailed sources.
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• Preliminaries-- Basic theory of Fourier and Fourier-Stieltjes algebras-- Miscellaneous further topics-- Amenability properties of $A(G)$ and $B(G)$-- Multiplier algebras of Fourier algebras-- Spectral synthesis and ideal theory-- Extension and separation properties of positive definite functions-- Appendix-- Bibliography-- Index.
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The theory of the Fourier algebra lies at the crossroads of several areas of analysis. Its roots are in locally compact groups and group representations, but it requires a considerable amount of functional analysis, mainly Banach algebras. In recent years it has made a major connection to the subject of operator spaces, to the enrichment of both. In this book two leading experts provide a road map to roughly 50 years of research detailing the role that the Fourier and Fourier-Stieltjes algebras have played in not only helping to better understand the nature of locally compact groups, but also in building bridges between abstract harmonic analysis, Banach algebras, and operator algebras. All of the important topics have been included, which makes this book a comprehensive survey of the field as it currently exists. Since the book is, in part, aimed at graduate students, the authors offer complete and readable proofs of all results. The book will be well received by the community in abstract harmonic analysis and will be particularly useful for doctoral and postdoctoral mathematicians conducting research in this important and vibrant area.
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• Hilbert schemes of points on surfaces: Basic results on Hilbert schemes of points The nef cone and flip structure of $(\mathbb{P}^2)^{[n]}$ Hilbert schemes and infinite dimensional Lie algebras: Hilbert schemes and infinite dimensional Lie algebras Chern character operators Multiple $q$-zeta values and Hilbert schemes Lie algebras and incidence Hilbert schemes Cohomology rings of Hilbert schemes of points: The cohomology rings of Hilbert schemes of points on surfaces Ideals of the cohomology rings of Hilbert schemes Integral cohomology of Hilbert schemes The ring structure of $H^*_{\textrm{orb}}(X^{(n)})$ Equivariant cohomology of the Hilbert schemes of points: Equivariant cohomology of Hilbert schemes Hilbert/Gromov-Witten correspondence Gromov-Witten theory of the Hilbert schemes of points: Cosection localization for the Hilbert schemes of points Equivariant quantum operator of Okounkov-Pandharipande The genus-0 extremal Gromov-Witten invariants Ruan's Cohomological Crepant Resolution Conjecture Bibliography Index.
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Hilbert schemes, which parametrize subschemes in algebraic varieties, have been extensively studied in algebraic geometry for the last 50 years. The most interesting class of Hilbert schemes are schemes $X^{[n]}$ of collections of $n$ points (zero-dimensional subschemes) in a smooth algebraic surface $X$. Schemes $X^{[n]}$ turn out to be closely related to many areas of mathematics, such as algebraic combinatorics, integrable systems, representation theory, and mathematical physics, among others. This book surveys recent developments of the theory of Hilbert schemes of points on complex surfaces and its interplay with infinite dimensional Lie algebras. It starts with the basics of Hilbert schemes of points and presents in detail an example of Hilbert schemes of points on the projective plane. Then the author turns to the study of cohomology of $X^{[n]}$, including the construction of the action of infinite dimensional Lie algebras on this cohomology, the ring structure of cohomology, equivariant cohomology of $X^{[n]}$ and the Gromov-Witten correspondence. The last part of the book presents results about quantum cohomology of $X^{[n]}$ and related questions. The book is of interest to graduate students and researchers in algebraic geometry, representation theory, combinatorics, topology, number theory, and theoretical physics.
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• Introduction: Some motivating questions-- The Cauchy-Kovalevskaya theorem with estimates-- Remarks on the Cauchy-Kovalevskaya theorem-- Zerner's theorem-- The method of globalizing families-- Holmgren's uniqueness theorem-- The continuity method of F. John-- The Bony-Schapira theorem-- Applications of the Bony-Schapira theorem: Part I - Vekua hulls-- Applications of the Bony-Schapira theorem: Part II - Szego's theorem revisited-- The reflection principle-- The reflection principle (continued)-- Cauchy problems and the Schwarz potential conjecture-- The Schwarz potential conjecture for spheres-- Potential theory on ellipsoids: Part I - The mean value property-- Potential theory on ellipsoids: Part II - There is no gravity in the cavity-- Potential theory on ellipsoids: Part III - The Dirichlet problem-- Singularities encountered by the analytic continuation of solutions to the Dirichlet problem-- An introduction to J. Leray's principle on propagation of singularities through $\mathbb{C}^n$-- Global propagation of singularities in $\mathbb{C}^n$-- Quadrature domains and Laplacian growth-- Other varieties of quadrature domains-- Bibliography-- Index.
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Why do solutions of linear analytic PDE suddenly break down? What is the source of these mysterious singularities, and how do they propagate? Is there a mean value property for harmonic functions in ellipsoids similar to that for balls? Is there a reflection principle for harmonic functions in higher dimensions similar to the Schwarz reflection principle in the plane? How far outside of their natural domains can solutions of the Dirichlet problem be extended? Where do the continued solutions become singular and why? This book invites graduate students and young analysts to explore these and many other intriguing questions that lead to beautiful results illustrating a nice interplay between parts of modern analysis and themes in ``physical'' mathematics of the nineteenth century. To make the book accessible to a wide audience including students, the authors do not assume expertise in the theory of holomorphic PDE, and most of the book is accessible to anyone familiar with multivariable calculus and some basics in complex analysis and differential equations.
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• Introduction
• Part 1. Mathematical and computation tools: Generalized argument principle and Rouche's theorem Layer potentials Perturbations of cavities and resonators
• Part 2. Diffraction gratings and band-gap materials: Diffraction gratings Photonic band gaps Phononic band gaps
• Part 3. Sub-wavelength resonant structures and super-resolution: Plasmonic resonances for nanoparticles Imaging of small particles Super-resolution imaging
• Part 4. Metamaterials: Near-cloaking Anomalous resonance cloaking and shielding Plasmonic metasurfaces
• Part 5. Sub-wavelength phonics: Helmholtz resonator Minnaert resonances for bubbles Appendix A. Spectrum of self-adjoint operators Appendix B. Optimal control and level set representation Bibliography Index.
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The fields of photonics and phononics encompass the fundamental science of light and sound propagation and interactions in complex structures, as well as its technological applications. This book reviews new and fundamental mathematical tools, computational approaches, and inversion and optimal design methods to address challenging problems in photonics and phononics. An emphasis is placed on analyzing sub-wavelength resonators, super-focusing and super-resolution of electromagnetic and acoustic waves, photonic and phononic crystals, electromagnetic cloaking, and electromagnetic and elastic metamaterials and metasurfaces. Throughout this book, the authors demonstrate the power of layer potential techniques for solving challenging problems in photonics and phononics when they are combined with asymptotic analysis. This book might be of interest to researchers and graduate students working in the fields of applied and computational mathematics, partial differential equations, electromagnetic theory, elasticity, integral equations, and inverse and optimal design problems in photonics and phononics.
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