Introduction 1 Based Loop Space and A∞ Space 1.1 Based Loop Space and Stasheff Polytope 1.2 Rooted Ribbon Trees 1.3 Stasheff Polytopes and An Space 1.4 Two Realizations of Stasheff Polytopes Kn 1.4.1 Metric ribbon Trees 1.4.

2 Moduli Space of Bordered Stable Curves 1.4.3 Configuration Space of S1 and Mk+1 1.4.4 Duality Between the Cell Structures of Mk+1 and Grk+1 1.5 The Based Loop Space is an A∞ space 2 A∞ Algebras and Modules: Unfiltered Case 2.1 Definition of A∞ Algebra 2.1.

1 Definition of A∞ Structure Maps mk 2.1.2 Coalgebra and Bar Complex 2.2 Massey Product and A∞ Algebra 2.2.1 Higher-order Linking of Borromean Ring 2.2.2 A∞-algebra Interpretation of Massey Product 2.

3 Hochschild Complex of A∞ Algebras 2.4 A∞ Homomorphisms 2.5 Right A∞ Modules 2.6 AK Modules and AK Homomorphisms 2.7 Hochschild Cohomology and Whitehead Theorem 2.7.1 Hochschild Cohomology of A∞ Homomorphisms 2.7.

2 Hochschild Cohomology and A∞ Whitehead Theorem 2.8 Obstruction Theory and A∞ Whitehead Theorem 2.8.1 AK+1-obstruction Class OK+1(ψ) 2.8.2 Wrap-up of the Proof of Whitehead Theorem 2.9 A∞ Bimodules 3 Obstruction-Deformation Theory of Filtered A∞ Bimodules 3.1 Gapped Filtered A∞ Algebras and Homomorphisms 3.

1.1 Universal Novikov Ring 3.1.2 Energy Filtration and Floer-Novikov Monoids 3.1.3 Filtered A∞ Homomorphism 3.1.4 Filtered A∞ 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 A∞ 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 A∞ Bimodules 3.4.1 The (G0,G1)-sets of Monoid Pair (G0,G1) 3.4.

2 Deformations of Filtered A∞ Bimodules 3.5 Deformations of Filtered A∞ Bimodule Homomorphisms 3.5.1 The Case of A∞ 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(M,ω) 4.4.2 Family of Hamiltonian Diffeomorphisms 4.5 Hamiltonian Displacement of Lagrangian Submanifolds 5 Analysis of Pseudoholomorphic Curves and Bordered Stable Maps 5.1 Almost Complex Manifolds and Hermitian Metric 5.

2 Pseudoholomorphic Curves and Their Boundary Value Problem 5.2.1 Energy Estimates (aka Global W1,2-bounds) 5.2.2 Totally Real Boundary Condition Versus Lagrangian Boundary Condition 5.2.3 Weitzenb¨ock Formula and W2,2-bounds 5.2.

4 Local a Priori Elliptic Estimates 5.2.5 Sachs-Uhlenbeck''s Rescaling and Gromov''s Weak Convergence 5.3 Bordered Stable Maps 5.3.1 Moduli Space of Prestable Curves 5.3.2 Moduli Space of Bordered Stable Maps of Genus 5.

3.3 Maslov Index and the Dimension of Moduli Space 5.3.4 Orientation of the Moduli Space of Bordered Stable Maps 5.4 Lagrangian Submanifolds and Filtered A∞ Algebras 5.4.1 Moduli Spaces as Correspondences 5.4.

2 Definition of Fukaya Algebra as a Filtered A∞ Algebra 5.5 Definition and Properties of the Fukaya Algebra 5.6 Example: on S2 5.6.1 Calculation of m0 5.6.2 Calculation of m1 5.6.

3 Calculation of m2 5.7 Geometry of Lagrangian Pairs (L0,L1) 5.7.1 The Novikov Covering and the Action Functional 5.7.2 Floer Trajectory Moduli Spaces 5.7.3 Coherent Orientations of Floer Trajectory Moduli Spaces 5.

8 Filtered A∞ Bimodules and Floer Cohomology 5.9 Floer Continuation Map Associated to Moving Lagrangian Boundary 5.9.1 Definition of Moduli Space for Floer Continuation Map 5.9.2 Energy Estimate for Moving Boundary Condition 6 Critical Points of Potential Functions and Floer Cohomology 6.1 BRST Anomaly and Holomorphic Discs 6.2 Weak Unobstructedness of Monotone Lagrangian Submanifolds 6.

3 Twisting of Floer Cohomology by Local Systems 6.3.1 Construction of Floer Cohomology of L with Local Systems 6.3.2 Holonomy-Twisted Canonical Model and Potential Functions 6.4 Lagrangian Floer Theory on Toric Manifolds 6.4.1 Geometry of Compact Toric Manifolds 6.

4.2 Structure Theorem of Moduli Spaces 6.4.3 Sphere Bubbles and Tn-equivariant Kuranishi Perturbations 6.4.4 One-Point Open Gromov-Witten Invariants; Toric Case 6.4.5 Forgetful Map Compatibility and Tn-Equivariance 6.

4.6 Tn-invariant Canonical Model on H(L(u),Λ0) xii Contents 6.4.7 Evaluation of Potential Function for the Toric Case 6.4.8 Gauge Equivalence on Maurer-Cartan Moduli Space and its Linear Sector 6.4.9 Relationship with the Landau-Ginzburg B-model 6.

4.10 Extending Potential Functions over Λ0Λ+: Algebraic versus Geometric 6.4.11 Critical Points of Wu and Nondisplaceable Fiber L(u) 6.5 Calculation of Potential Functions: Examples 6.5.1 The Case of S2 6.5.

2 The Case of CPn 6.5.3 The Case of S2 × S2 6.5.4 The Case of Hirzerbruch Surface F2(α) 7 Filtered Fukaya Category and its Bulk Deformations 7.1 Definitions of A∞ Categories, Functors and Modules 7.1.1 Unfiltered A∞ Categories 7.

1.2 Curved Filtered A∞ Categories 7.1.3 Strictification of Curved Filtered A∞ Category 7.1.4 A∞ Functors 7.1.5 A∞ Bimodules 7.

2 Construction of Filtered Fukaya Category 7.2.1 Description of the Cocycle Problems to Solve 7.2.2 Off-shell Geometry I: Abstract Index 7.2.3 Off-shell Geometry II: Polygonal Maslov Index 7.2.

4 On-shell Geometry I: Coherent Orientations of the Moduli Spaces 7.2.5 On-Shell Geometry II: Products - Structure maps mk, k ≥ 2 7.2.6 Relationship with Graded Lagrangian Submanifolds 7.2.7 Strictification 7.3 Bulk Deformations 7.

3.1 Bulk-Boundary Map (aka Closed-Open Map) 7.3.2 Bulk Deformations of Filtered Fukaya Category 7.3.3 Bulk Deformations on Toric Manifolds 7.4 Potential Functions with Bulk and Their Applications 7.4.

1 Bulk-Deformed Potential Functions 7.4.2 Continuum Family of Nondisplaceable Tori in S2 × S2 7.4.3 Gelfand-Cetlin Systems and Their Bulk Deformations 7.4.4 Mak and Smith''s Work on Lagrangian Links in S2.