| Lefschetz Pencils, Branched Covers and Symplectic Invariants | p. 1 |
| Introduction and Background | p. 1 |
| Symplectic Manifolds | p. 1 |
| Almost-Complex Structures | p. 3 |
| Pseudo-Holomorphic Curves and Gromov-Witten Invariants | p. 5 |
| Lagrangian Floer Homology | p. 6 |
| The Topology of Symplectic Four-Manifolds | p. 9 |
| Symplectic Lefschetz Fibrations | p. 10 |
| Fibrations and Monodromy | p. 10 |
| Approximately Holomorphic Geometry | p. 17 |
| Symplectic Branched Covers of <$>{op CP}^2<$> | p. 22 |
| Symplectic Branched Covers | p. 22 |
| Monodromy Invariants for Branched Covers of <$>{op CP}^2<$> | p. 26 |
| Fundamental Groups of Branch Curve Complements | p. 30 |
| Symplectic Isotopy and Non-Isotopy | p. 33 |
| Symplectic Surfaces from Symmetric Products | p. 35 |
| Symmetric Products | p. 35 |
| Taubes' Theorem | p. 39 |
| Fukaya Categories and Lefschetz Fibrations | p. 42 |
| Matching Paths and Lagrangian Spheres | p. 43 |
| Fukaya Categories of Vanishing Cycles | p. 44 |
| Applications to Mirror Symmetry | p. 48 |
| References | p. 50 |
| Differentiable and Deformation Type of Algebraic Surfaces, Real and Symplectic Structures | p. 55 |
| Introduction | p. 55 |
| Lecture 1: Projective and Kähler Manifolds, the Enriques Classification, Construction Techniques | p. 57 |
| Projective Manifolds, Kähler and Symplectic Structures | p. 57 |
| The Birational Equivalence of Algebraic Varieties | p. 63 |
| The Enriques Classification: An Outline | p. 65 |
| Some Constructions of Projective Varieties | p. 66 |
| Lecture 2: Surfaces of General Type and Their Canonical Models: Deformation Equivalence and Singularities | p. 70 |
| Rational Double Points | p. 70 |
| Canonical Models of Surfaces of General Type | p. 74 |
| Deformation Equivalence of Surfaces | p. 82 |
| Isolated Singularities, Simultaneous Resolution | p. 85 |
| Lecture 3: Deformation and Diffeomorphism, Canonical Symplectic Structure for Surfaces of General Type | p. 91 |
| Deformation Implies Diffeomorphism | p. 92 |
| Symplectic Approximations of Projective Varieties with Isolated Singularities | p. 93 |
| Canonical Symplectic Structure for Varieties with Ample Canonical Class and Canonical Symplectic Structure for Surfaces of General Type | p. 95 |
| Degenerations Preserving the Canonical Symplectic Structure | p. 96 |
| Lecture 4: Irrational Pencils, Orbifold Fundamental Groups, and Surfaces Isogenous to a Product | p. 98 |
| Theorem of Castelnuovo-De Franchis, Irrational Pencils and the Orbifold Fundamental Group | p. 99 |
| Varieties Isogenous to a Product | p. 105 |
| Complex Conjugation and Real Structures | p. 108 |
| Beauville Surfaces | p. 114 |
| Lecture 5: Lefschetz Pencils, Braid and Mapping Class Groups, and Diffeomorphism of ABC-Surfaces | p. 116 |
| Surgeries | p. 116 |
| Braid and Mapping Class Groups | p. 119 |
| Lefschetz Pencils and Lefschetz Fibrations | p. 125 |
| Simply Connected Algebraic Surfaces: Topology Versus Differential Topology | p. 130 |
| ABC Surfaces | p. 134 |
| Epilogue: Deformation, Diffeomorphism and Symplectomorphism Type of Surfaces of General Type | p. 140 |
| Deformations in the Large of ABC Surfaces | p. 141 |
| Manetti Surfaces | p. 145 |
| Deformation and Canonical Symplectomorphism | p. 152 |
| Braid Monodromy and Chisini' Problem | p. 154 |
| References | p. 159 |
| Smoothings of Singularities and Deformation Types of Surfaces | p. 169 |
| Introduction | p. 169 |
| Deformation Equivalence of Surfaces | p. 174 |
| Rational Double Points | p. 174 |
| Quotient Singularities | p. 178 |
| RDP-Deformation Equivalence | p. 181 |
| Relative Canonical Model | p. 182 |
| Automorphisms of Canonical Models | p. 183 |
| The Kodaira Spencer Map | p. 184 |
| Moduli Space for Canonical Surfaces | p. 187 |
| Gieseker's Theorem | p. 188 |
| Constructing Connected Components: Some Strategies | p. 189 |
| Outline of Proof of Gieseker Theorem | p. 190 |
| Smoothings of Normal Surface Singularities | p. 194 |
| The Link of an Isolated Singularity | p. 194 |
| The Milnor Fibre | p. 196 |
| <$>{op Q}<$>-Gorenstein Singularities and Smoothings | p. 197 |
| T-Deformation Equivalence of Surfaces | p. 201 |
| A Non Trivial Example of T-Deformation Equivalence | p. 203 |
| Double and Multidouble Covers of Normal Surfaces | p. 204 |
| Flat Abelian Covers | p. 204 |
| Flat Double Covers | p. 205 |
| Automorphisms of Generic Flat Double Covers | p. 207 |
| Example: Automorphisms of Simple Iterated Double Covers | p. 209 |
| Flat Multidouble Covers | p. 210 |
| Stability Criteria for Flat Double Covers | p. 213 |
| Restricted Natural Deformations of Double Covers | p. 214 |
| Openess of N (a, b,c) | p. 217 |
| RDP-Degenerations of Double Covers | p. 218 |
| RDP-Degenerations of <$>{op P}^1 times {op P}^1<$> | p. 221 |
| Proof of Theorem 6.1 | p. 222 |
| Moduli of Simple Iterated Double Covers | p. 223 |
| References | p. 225 |
| Lectures on Four-Dimensional Dehn Twists | p. 231 |
| Introduction | p. 231 |
| Definition and First Properties | p. 235 |
| Floer and Quantum Homology | p. 249 |
| Pseudo-Holomorphic Sections and Curvature | p. 259 |
| References | p. 265 |
| Lectures on Pseudo-Holomorphic Curves and the Symplectic Isotopy Problem | p. 269 |
| Introduction | p. 269 |
| Pseudo-Holomorphic Curves | p. 270 |
| Almost Complex and Symplectic Geometry | p. 270 |
| Basic Properties of Pseudo-Holomorphic Curves | p. 272 |
| Moduli Spaces | p. 273 |
| Applications | p. 276 |
| Pseudo-Analytic Inequalities | p. 279 |
| Unobstructedness I: Smooth and Nodal Curves | p. 281 |
| Preliminaries on the <$>overline {partial}<$>-Equation | p. 281 |
| The Normal <$>overline {partial}<$>-Operator | p. 282 |
| Immersed Curves | p. 286 |
| Smoothings of Nodal Curves | p. 287 |
| The Theorem of Micallef and White | p. 288 |
| Statement of Theorem | p. 288 |
| The Case of Tacnodes | p. 289 |
| The General Case | p. 291 |
| Unobstructedness II: The Integrable Case | p. 292 |
| Motivation | p. 292 |
| Special Covers | p. 292 |
| Description of the Deformation Space | p. 294 |
| The Holomorphic Normal Sheaf | p. 296 |
| Computation of the Linearization | p. 299 |
| A Vanishing Theorem | p. 300 |
| The Unobstructedness Theorem | p. 301 |
| Application to Symplectic Topology in Dimension Four | p. 302 |
| Monodromy Representations - Hurwitz Equivalence | p. 303 |
| Hyperelliptic Lefschetz Fibrations | p. 304 |
| Braid Monodromy and the Structure of Hyperelliptic Lefschetz Fibrations | p. 307 |
| Symplectic Noether-Horikawa Surfaces | p. 309 |
| The <$>{scr C}^0<$>-Compactness Theorem for Pseudo-Holomorphic Curves | p. 311 |
| Statement of Theorem and Conventions | p. 311 |
| The Monotonicity Formula for Pseudo-Holomorphic Maps | p. 312 |
| A Removable Singularities Theorem | p. 315 |
| Proof of the Theorem | p. 316 |
| Second Variation of the <$>overline {partial}_J<$>-Equation and Applications | p. 320 |
| Comparisons of First and Second Variations | p. 321 |
| Moduli Spaces of Pseudo-Holomorphic Curves with Prescribed Singularities | p. 323 |
| The Locus of Constant Deficiency | p. 324 |
| Second Variation at Ordinary Cusps | p. 328 |
| The Isotopy Theorem | p. 332 |
| Statement of Theorem and Discussion | p. 332 |
| Pseudo-Holomorphic Techniques for the Isotopy Problem | p. 333 |
| The Isotopy Lemma | p. 334 |
| Sketch of Proof | p. 336 |
| References | p. 339 |
| List of Participants | p. 343 |
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