Table of Contents

Part 1. Mathematical Preliminaries: Differential geometry Algebraic geometry Differential and algebraic topology Equivariant cohomology and fixed-point theorems Complex and Kahler geometry Calabi-Yau manifolds and their moduli Toric geometry for string theory Part 2. Physics Preliminaries: What is a QFT? QFT in $d=0$ QFT in dimension 1: Quantum mechanics Free quantum field theories 1 + 1 dimensions $mathcal{N} = (2,2)$ supersymmetry Non-linear sigma models and Landau-Ginzburg models Renormalization group flow Linear sigma models Chiral rings and topological field theory Chiral rings and the geometry of the vacuum bundle BPS solitons in $mathcal{N}=2$ Landau-Ginzburg theories D-branes Part 3. Mirror Symmetry: Physics Proof: Proof of mirror symmetry Part 4. Mirror Symmetry: Mathematics Proof: Introduction and overview Complex curves (non-singular and nodal) Moduli spaces of curves Moduli spaces $bar{mathcal M}_{g,n}(X,beta)$ of stable maps Cohomology classes on $bar{mathcal M}_{g,n}$ and ($bar{mathcal M})_{g,n}(X,beta)$ The virtual fundamental class, Gromov-Witten invariants, and descendant invariants Localization on the moduli space of maps The fundamental solution of the quantum differential equation The mirror conjecture for hypersurfaces I: The Fano case The mirror conjecture for hypersurfaces II: The Calabi-Yau case Part 5. Advanced Topics: Topological strings Topological strings and target space physics Mathematical formulation of Gopakumar-Vafa invariants Multiple covers, integrality, and Gopakumar-Vafa invariants Mirror symmetry at higher genus Some applications of mirror symmetry Aspects of mirror symmetry and D-branes More on the mathematics of D-branes: Bundles, derived categories and Lagrangians Boundary $mathcal{N}=2$ theories References Bibliography Index.