Lagrangian Floer Theory and Its Deformations : An Introduction to Filtered Fukaya Category

• Autore: Oh Yong-Geun
• Pubblicazione: Singapore : , : Springer, , 2024; ©2024
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource (426 pages)
• Nota di contenuto: Intro -- Preface -- Contents -- Introduction -- Acknowledgements -- Conventions -- Notations -- List of Figures -- 1 Based Loop Space and upper A Subscript normal infinityAinfty Space -- 1.1 Based Loop Space and Stasheff Polytope -- 1.2 Rooted Ribbon Trees -- 1.3 Stasheff Polytopes and upper A Subscript nAn Space -- 1.4 Two Realizations of Stasheff Polytopes upper K Subscript nKn -- 1.4.1 Metric Ribbon Trees -- 1.4.2 Moduli Space of Bordered Stable Curves -- 1.4.3 Configuration Space of upper S Superscript 1S1 and ModifyingAbove script upper M With quotation dash Subscript k plus 1overlinemathcalMk+1 -- 1.4.4 Duality Between the Cell Structures of ModifyingAbove script upper M With quotation dash Subscript k plus 1overlinemathcalMk+1 and ModifyingAbove upper G r With quotation dash Subscript k plus 1overlineGrk+1 -- 1.5 The Based Loop Space Is an upper A Subscript normal infinityAinfty Space -- 2 upper A Subscript normal infinityAinfty Algebras and Modules: Unfiltered Case -- 2.1 Definition of upper A Subscript normal infinityAinfty Algebra -- 2.1.1 Definition of upper A Subscript normal infinityAinfty Structure Maps German m Subscript kmathfrakmk -- 2.1.2 Coalgebra and Bar Complex -- 2.2 Massey Product and upper A Subscript normal infinityAinfty Algebra -- 2.2.1 Higher-Order Linking of Borromean Ring -- 2.2.2 upper A Subscript normal infinityAinfty-algebra Interpretation of Massey Product -- 2.3 Hochschild Complex of upper A Subscript normal infinityAinfty Algebras -- 2.4 upper A Subscript normal infinityAinfty Homomorphisms -- 2.5 Right upper A Subscript normal infinityAinfty Modules -- 2.6 upper A Subscript upper KAK Modules and upper A Subscript upper KAK Homomorphisms -- 2.7 Hochschild Cohomology and Whitehead Theorem -- 2.7.1 Hochschild Cohomology of upper A Subscript normal infinityAinfty Homomorphisms. ; 2.7.2 Hochschild Cohomology and upper A Subscript normal infinityAinfty Whitehead Theorem -- 2.8 Obstruction Theory and upper A Subscript normal infinityAinfty Whitehead Theorem -- 2.8.1 upper A Subscript upper K plus 1AK+1-obstruction Class upper O Subscript upper K plus 1 Baseline left parenthesis psi right parenthesisOK+1(ψ) -- 2.8.2 Wrap-Up of the Proof of Whitehead Theorem -- 2.9 upper A Subscript normal infinityAinfty Bimodules -- 3 Obstruction-Deformation Theory of Filtered upper A Subscript normal infinityAinfty Bimodules -- 3.1 Gapped Filtered upper A Subscript normal infinityAinfty Algebras and Homomorphisms -- 3.1.1 Universal Novikov Ring -- 3.1.2 Energy Filtration and Floer-Novikov Monoids -- 3.1.3 Filtered upper A Subscript normal infinityAinfty Homomorphism -- 3.1.4 Filtered upper A Subscript normal infinityAinfty Bimodules and Homomorphisms -- 3.2 Homological Perturbation Theory and Canonical Model -- 3.2.1 Unfiltered Cases: Statement -- 3.2.2 Unfiltered Cases: Proof -- 3.2.3 Canonical Model: Filtered Cases -- 3.3 Bounding Cochains and Deformations of upper A Subscript normal infinityAinfty Algebra -- 3.3.1 The (Strict) Bounding Cochains -- 3.3.2 The Weak Bounding Cochains, Gauge Equivalence and Potential Function -- 3.4 Boundary Deformations of upper A Subscript normal infinityAinfty Bimodules -- 3.4.1 The left parenthesis upper G 0 comma upper G 1 right parenthesis(G0,G1)-sets of Monoid Pair left parenthesis upper G 0 comma upper G 1 right parenthesis(G0,G1) -- 3.4.2 Deformations of Filtered upper A Subscript normal infinityAinfty Bimodules -- 3.5 Deformations of Filtered upper A Subscript normal infinityAinfty Bimodule Homomorphisms -- 3.5.1 The Case of upper A Subscript normal infinityAinfty Algebra Homomorphisms -- 3.5.2 The Case of Filtered Bimodule Homomorphisms -- 4 Symplectic Geometry and Hamiltonian Dynamics. ; 4.1 Definition of Symplectic Manifolds -- 4.2 Symplectic Linear Algebra -- 4.2.1 Lagrangian Grassmanian -- 4.2.2 Arnold Stratification -- 4.3 Lagrangian Submanifolds -- 4.3.1 Basic Definitions -- 4.3.2 Calculus of Lagrangian Submanifolds -- 4.4 Hamiltonian Diffeomorphisms and Hamiltonian Calculus -- 4.4.1 Hofer's Geometry of Ham left parenthesis upper M comma omega right parenthesisHam(M,ω) -- 4.4.2 Family of Hamiltonian Diffeomorphisms -- 4.5 Hamiltonian Displacement of Lagrangian Submanifolds -- Chapter 5 Analysis of Pseudoholomorphic Curves and Bordered Stable Maps -- 5.1 Almost Complex Manifolds and Hermitian Metric -- upper A Subscript normal infinity . -- De nition 5.1.1 -- upper S squared -- ]) Let -- is called an almost complex structure on a manifold -- . -- since the bracket is skew symmetric. Note that -- . -- . -- upper S squared -- . -- upper A Subscript normal infinity -- . -- . -- upper A Subscript normal infinity . -- as generalization of classical Cauchy-Riemann equation to a system thereof on an almost complex manifold. -- upper S squared -- Observation When -- upper A Subscript normal infinity -- Theorem 5.1.7 (Nijenhuis-Woolf [ -- A Riemannian metric -- 5.2 Pseudoholomorphic Curves and Their Boundary Value Problem -- . -- be given, and consider a 2-dimensional surface -- is called the fundamental two-form of -- upper S squared -- left parenthesis upper L 0 comma upper L 1 right parenthesis -- . -- . -- . -- . -- . -- , the exponential map -- . -- . -- is a diffeomorphism. -- is a contractible in nite- dimensional (Frechet) manifold (in the weak -- . -- . -- 5.1.7 -- not necessarily satisfying the integrability condition. -- . -- (1) -- orientable? -- . -- For the study of transversality, one should study the deformation problem of the moduli space under the change of. ; For the study of compactness, it is essential to obtain a uniform bound for the derivative -- upper S squared -- left parenthesis upper L 0 comma upper L 1 right parenthesis -- . Once such a bound is achieved, the Ascoli-Arzela theorem can be applied for the uniform convergence. Once this convergence is achieved, the deriva-tive convergence can be obtained by the a priori estimates based on the ellipticity of the -- . -- . -- (5.2.3) -- . -- . -- left parenthesis upper L 0 comma upper L 1 right parenthesis -- . -- . -- , -- Exercise 5.2.12 Consider the dilatation -- upper S squared -- -norm, is a borderline case because a domain -- upper A Subscript normal infinity -- . -- or even the symplectic area -- below, proves the following boundary analog to Corollary -- for -- upper S squared -- is a nonlinear elliptic boundary value problem for any such -- . -- . -- . -- . -- , when there is no non-constant -- . -- . -- . -- carries an a priori energy bound -- . -- . -- upper A Subscript normal infinity -- . -- . -- on -- . -- is a Euclidean vector bundle with inner product -- . -- . -- The following is proved in almost complex geometry. (See [ -- Here we denote by -- upper S squared -- . -- the trace Laplacian and -- . -- (5.2.11) -- . -- upper S squared -- . -- . -- . -- Remark 5.2.28 We have used the canonical connection associated to -- . -- . -- ] for complete details applied to more complicated equation of contact instantons.) -- upper S squared -- . -- ). If the connection -- 5.2.4 Local a Priori Elliptic Estimates -- . -- .) -- We use the (linear) ellipticity of the operator -- To proceed with the higher regularity estimate, we choose an isothermal coordinate -- upper S squared -- , we obtain the equation -- associated to the complex coordinate -- . -- (with respect to -- . -- . --. ; left parenthesis upper L 0 comma upper L 1 right parenthesis -- ], [ -- 2. -- An important application of the above boundary regularity result is the following removable boundary singularity theorem [ -- contained in a compact subset -- . -- , -- upper S squared -- left parenthesis upper L 0 comma upper L 1 right parenthesis -- Theorem 5.2.37 (Removable singularity theorem) Let -- . -- . -- has no sub- sequence -- . -- is -- is uniformly bounded. -- of pseudo-holomorphic maps, when the domain complex structure is xed, the genus 0 case is the most nontrivial to study. Because of this, we will focus on the case and assume -- :For -- . -- Let -- upper A Subscript normal infinity -- 's arising in Theorem -- considered as a density converges to a density -- upper A Subscript normal infinity -- . -- as -- . -- upper S squared -- left parenthesis upper L 0 comma upper L 1 right parenthesis -- upper A Subscript normal infinity -- -holomorphic curves should involve singular curves involving sphere and disc components. -- . -- upper A Subscript normal infinity -- , i.e., -- be a convergent sequence in -- ], Parker-Wolfson [ -- ].) However, bearing intuitive Gromov's original statements in mind is also useful and often enough for the application to various problems in symplectic topology, especially in the beginning stage of learning Gromov's com- pacti cation and its symplectic topology applications. -- . -- 5.3 Bordered Stable Maps -- upper A Subscript normal infinity -- . Then for any sequence of -- upper S squared -- so that the following hold: -- -holomorphic maps -- . -- . -- . -- We start with the de nition of real algebraic curves (aka symmetric Riemann surfaces). -- . Although the case -- upper S squared --. ; will not be needed explicitly in this book except in our mentioning of the Cardy relation with regard to the bulk-boundary map (in the paragraph after De nition.
• Titolo autorizzato: Lagrangian Floer Theory and Its Deformations
• ISBN: 9789819717989; 9789819717972
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910865256703321
• Lo trovi qui: Univ. Federico II
• Serie: KIAS Springer Series in Mathematics Series