Front Matter
 Part 3 Mathematical Physics, Algebraic
Geometry and Other Topics
 The Coherent-Constructible Correspondence
and Homological Mirror Symmetry for Toric Varieties
  Bohan Fang, Chiu-Chu Melissa Liu, David Treumann and Eric Zaslow
  1 Introduction
  2 Mirror symmetry for toric manifolds
  3 T-duality
  4 Microlocalization
  5 Coherent-constructible correspondence
  6 Examples
  References
 Superspace: a Comfortably Vast Algebraic Variety
  T. Hubsch
  1 Introduction
  2 Off-shell worldline supermultiplets
  3 Superspace, by construction
  4 The comfortably vast superspace
  References
 A Report on the Yau-Zaslow Formula
  Naichung Conan Leung
  1 Yau-Zaslow formula and its generalizations
  2 Yau-Zaslow approach
  3 Matching method
  4 Degeneration method
  5 Calabi-Yau threefold method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
  6 Conclusions
  References
 Hermitian-Yang-Mills Connections on Kahler Manifolds
  Jun Li
  1 Introduction
  2 Donaldson-Uhlenbeck-Yau theorem
  3 Hermitian-Yang-Mills connections on curves
  4 Hermitian-Yang-Mills connections on surfaces
  5 HYM connections on high dimensional varieties
  6 Concluding remark
  References
 Additivity and Relative Kodaira Dimensions
  Tian-Jun Li and Weiyi Zhang
  1 Introduction
  2 Kodaira Dimensions and ber bundles
  3 Embedded symplectic surfaces and relative Kod. dim. in dim. 4
  4 Relative Kod. dim. in dim. 2 and brations over a surface
  References
 Descendent Integrals and Tautological Rings of
Moduli Spaces of Curves
  Kefeng Liu and Hao Xu
  1 Introduction
  2 Intersection numbers and the Witten-Kontsevich theorem
  3 The n-point function
  4 Hodge integrals
  5 Higher Weil-Petersson volumes
  6 Faber’s conjecture on tautological rings
  7 Dimension of tautological rings
  8 Gromov-Witten invariants
  9 Witten’s r-spin numbers
  References
 A General Voronoi Summation Formula for GL(n; Z)
  Stephen D. Miller and Wilfried Schmid
  1 Introduction
  2 Automorphic Distributions
  3 Vanishing to innite order
  4 Classical proof of the formula
  5 Adelic proof of the formula
  References
 Geometry of Holomorphic Isometries and Related Maps
between Bounded Domains
  Ngaiming Mok
  1 Examples of holomorphic isometries
  2 Analytic continuation of germs of holomorphic isometries
  3 Holomorphic isometries of the Poincar e disk into
bounded symmetric domains
  4 Measure-preserving algebraic correspondences on irreducible
bounded symmetric domains
  5 Open problems
  References
 Abundance Conjecture
  Yum-Tong Siu
  0 Introduction
  1 Curvature current and dichotomy
  2 Gelfond-Schneider’s technique of algebraic values of solutions
of algebraically de ned di erential equations
  3 Final step of the case of zero numerical Kodaira dimension
  4 Numerically trivial foliations and brations for
canonical line bundle
  5 Curvature of zeroth direct image of relative canonical and
pluricanonical bundle
  6 Strict positivity of direct image of relative pluricanonical
bundle along numerically trivial bers in the base of
numerically trivial bration
  7 Technique of Nevanlinna’s rst main theorem for proof of
compactness of leaves of foliation
  References
 Sasaki-Einstein Geometry
  James Sparks
  1 Sasakian geometry
  2 Constructions of Sasaki-Einstein manifolds
  3 Obstructions
  4 Sasaki-Einstein manifolds in string theory
  References
 A Simple Proof of the Chiral Gravity Conjecture
  Andrew Strominger
 Geometry of Grand Unication
  Cumrun Vafa
  1 Introduction
  2 Standard model and gauge symmetry breaking
  3 Flavors and hierarchy
  4 Unication of gauge groups
  5 String theory, forces, matter, and interactions
  6 F-theory vacua
  7 Applications to particle physics
  8 Further issues
  References
 Quantum Invariance Under Flop Transitions
  Chin-Lung Wang
  1 Introduction
  ops: Genus zero theory
  3 Calabi-Yau ops
  References
 The Problem Of Gauge Theory
  Edward Witten
  1 Yang-Mills equations
  2 Classical phase space
  3 Quantization
  4 Nonperturbative approach
  5 Breaking of conformal invariance and the mass gap
  References
 Part 4 Appendices
 Shing-Tung Yau, a Manifold Man of Mathematics
  Lizhen Ji and Kefeng Liu
  1 Childhood and early school education
  2 Middle school and college
  3 Graduate school
  4 Professional career
  5 Major contributions to mathematics
  6 Visits to China
  7 Research centers and mathematics institutes
  8 ICCM
  9 Conferences and popular mathematics programs
  10 Mathematics and Chinese literature
  11 Family, friends and students
  12 Summary
  References
 Perspectives on Geometric Analysis
  Shing-Tung Yau
  1 History and contributors of the subject
  2 Construction of functions in geometry
  3 Mappings between manifolds and rigidity
of geometric structures
  4 Submanifolds dened by variational principles
  5 Construction of geometric structures on bundles and manifolds
  References
 A Survey of Calabi-Yau Manifolds
  Shing-Tung Yau
  1 Introduction
  2 General Constructions of Complete Ricci-Flat Metrics
in Kahler Geometry
  3 Moduli and Arithmetic of Calabi-Yau Manifolds
  4 Calabi-Yau Manifolds in Physics
  5 Invariants of Calabi-Yau Manifolds
  6 Homological Mirror Symmetry
  7 SYZ geometric interpretation of mirror symmetry
  8 Geometries Related to Calabi-Yau Manifolds
  References
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