| Preface | p. v |
| Lectures on Gromov-Witten Invariants of Orbifolds | p. 1 |
| Introduction | p. 1 |
| What This Is | p. 1 |
| Introspection | p. 1 |
| Where Does All This Come From? | p. 2 |
| Acknowledgements | p. 2 |
| Gromov-Witten Theory | p. 2 |
| Kontsevich's Formula | p. 2 |
| Set-Up for a Streamlined Proof | p. 3 |
| The Space of Stable Maps | p. 7 |
| Natural Maps | p. 8 |
| Boundary of Moduli | p. 9 |
| Gromov-Witten Classes | p. 10 |
| The WDVV Equations | p. 11 |
| Proof of WDVV | p. 12 |
| About the General Case | p. 15 |
| Orbifolds/Stacks | p. 16 |
| Geometric Orbifolds | p. 16 |
| Moduli Stacks | p. 17 |
| Where Do Stacks Come Up? | p. 19 |
| Attributes of Orbifolds | p. 19 |
| Etale Gerbes | p. 20 |
| Twisted Stable Maps | p. 21 |
| Stable Maps to a Stack | p. 21 |
| Twisted Curves | p. 22 |
| Twisted Stable Maps | p. 23 |
| Transparency 25: The Stack of Twisted Stable Maps | p. 24 |
| Twisted Curves and Roots | p. 25 |
| Valuative Criterion for Properness | p. 27 |
| Gromov-Witten Classes | p. 29 |
| Contractions | p. 29 |
| Gluing and Rigidified Inertia | p. 29 |
| Evaluation Maps | p. 31 |
| The Boundary of Moduli | p. 32 |
| Orbifold Gromov-Witten Classes | p. 32 |
| Fundamental Classes | p. 34 |
| WDVV, Grading and Computations | p. 35 |
| The Formula | p. 35 |
| Quantum Cohomology and Its Grading | p. 36 |
| Grading the Rings | p. 38 |
| Examples | p. 38 |
| Other Work | p. 41 |
| Mirror Symmetry and the Crepant Resolution Conjecture | p. 42 |
| The Legend of String Cohomology: Two Letters of Maxim Kontsevich to Lev Borisov | p. 43 |
| The Legend of String Cohomology | p. 43 |
| The Archaeological Letters | p. 44 |
| References | p. 46 |
| Lectures on the Topological Vertex | p. 49 |
| Introduction and Overview | p. 49 |
| Chern-Simons Theory | p. 51 |
| Basic Ingredients | p. 51 |
| Perturbative Approach | p. 55 |
| Non-Perturbative Solution | p. 61 |
| Framing Dependence | p. 68 |
| The 1/N Expansion in Chern-Simons Theory | p. 70 |
| Topological Strings | p. 73 |
| Topological Strings and Gromov-Witten Invariants | p. 74 |
| Integrality Properties and Gopakumar-Vafa Invariants | p. 76 |
| Open Topological Strings | p. 77 |
| Toric Geometry and Calabi-Yau Threefolds | p. 79 |
| Non-Compact Calabi-Yau Geometries: An Introduction | p. 79 |
| Constructing Toric Calabi-Yau Manifolds | p. 81 |
| Examples of Closed String Amplitudes | p. 87 |
| The Topological Vertex | p. 89 |
| The Gopakumar-Vafa Duality | p. 89 |
| Framing of Topological Open String Amplitudes | p. 89 |
| Definition of the Topological Vertex | p. 91 |
| Gluing Rules | p. 93 |
| Explicit Expression for the Topological Vertex | p. 95 |
| Applications | p. 96 |
| Symmetric Polynomials | p. 99 |
| References | p. 100 |
| Floer Cohomology with Gerbes | p. 105 |
| Floer Cohomology | p. 106 |
| Newton's Second Law | p. 106 |
| The Hamiltonian Formalism | p. 107 |
| The Arnold Conjecture | p. 108 |
| Floer's Proof | p. 108 |
| Morse Theory | p. 109 |
| Bott-Morse Theory | p. 110 |
| Morse Theory on the Loop Space | p. 110 |
| Re-Interpretation #1: Sections of the Symplectic Mapping Torus | p. 112 |
| Re-Interpretation #2: Two Lagrangian Submanifolds | p. 113 |
| Product Structures | p. 114 |
| The Finite-Order Case | p. 115 |
| Givental's Philosophy | p. 115 |
| Gerbes | p. 117 |
| Definition of Stacks | p. 117 |
| Examples of Stacks | p. 118 |
| Morphisms and 2-Morphisms | p. 118 |
| Definition of Gerbes | p. 120 |
| The Gerbe of Liftings | p. 121 |
| The Lien of a Gerbe | p. 122 |
| Classification of Gerbes | p. 123 |
| Allowing the Base Space to Be a Stack | p. 123 |
| Definition of Orbifolds | p. 124 |
| Twisted Vector Bundles | p. 124 |
| Strominger-Yau-Zaslow | p. 125 |
| Orbifold Cohomology and Its Relatives | p. 126 |
| Cohomology of Sheaves on Stacks | p. 126 |
| The Inertia Stack | p. 127 |
| Orbifold Cohomology | p. 128 |
| Twisted Orbifold Cohomology | p. 129 |
| The Case of Discrete Torsion | p. 129 |
| The Fantechi-Gottsche Ring | p. 130 |
| Twisting the Fantechi-Gottsche Ring with Discrete Torsion | p. 131 |
| Twisting It with an Arbitrary Flat Unitary Gerbe | p. 131 |
| The Loop Space of an Orbifold | p. 132 |
| Addition of the Gerbe | p. 134 |
| The Non-Orbifold Case | p. 135 |
| The Equivariant Case | p. 135 |
| A Concluding Puzzle | p. 136 |
| Notes on the Literature | p. 137 |
| Notes to Lecture 1 | p. 137 |
| Notes to Lecture 2 | p. 139 |
| Notes to Lecture 3 | p. 140 |
| The Moduli Space of Curves and Gromov-Witten Theory | p. 143 |
| Introduction | p. 143 |
| The Moduli Space of Curves | p. 145 |
| Tautological Cohomology Classes on Moduli Spaces of Curves, and Their Structure | p. 154 |
| A Blunt Tool: Theorem * and Consequences | p. 173 |
| Stable Relative Maps to P[superscript 1] and Relative Virtual Localization | p. 177 |
| Applications of Relative Virtual Localization | p. 186 |
| Towards Faber's Intersection Number Conjecture 3.23 via Relative Virtual Localization | p. 190 |
| Conclusion | p. 194 |
| References | p. 194 |
| List of Participants | p. 199 |
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