To celebrate the 60th birthday of Professor ShingTung Yau, a conference titled Geometric Analysis: Present and Future was held at Harvard University from August 27 to September 1, 2008.

The purpose of this conference is to summarize what has been achieved in the fields around geometric analysis in the past, to discuss the most recent results andt o map out directions for the future. The title Geometric Analysis was interpretedvery broadly and reflected the wide range of interests of Yau. It was also reflectedin the topics of many talks at the conference.

The launch of this Advanced Lectures in Mathematics series is aimed at keeping mathematicians informed of the latest developments in mathematics, as well as toaid in the learning of new mathematical topics by students all over the world. Each volume consists of either an expository monograph or a collection of signify cant introductions to important topics, This series emphasizes the history and sources of motivation for the topics under discussion, and also gives an overview of the current status of research in each particular field. These volumes are the first source to which people will turn in order to learn new subjects and to dis cover the latest results of many cutting-edge fields in mathematics.

This book contains many substantial papers from distinguished speakers of a conference Geometric Analysis: Present and Future" and an overview of the works of Professor Shing-Tung Yau. Contributors include E. Witten, Y.T. Siu, R. Hamilton, H. Hitchin, B. Lawson, A. Strominger, C. Vafa, W. Schmid, V. Guillemin, N. Mok, D. Christodoulou. This is a valuable reference that gives an up-to-dated summary of geometric analysis and its applications in many different areas of mathematics.

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

1.1 Outline

2 Mirror symmetry for toric manifolds

2.1 Hori-Vafa mirror

2.2 Categories in mirror symmetry

2.3 Results to date

3 T-duality

3.1 Moment polytope

3.2 Geometry of the open orbit

3.3 Statement of symplectic results

3.4 T-dual of an equivariant line bundle

4 Microlocalization

4.1 Algebraic preliminaries

4.2 The cast of categories

4.3 Fukaya-Oh theorem

4.4 Building the equivalence

4.5 Equivalence and the inverse functor

4.6 Singular support and characteristic cycles

4.7 Comments on technicalities

4.8 Statement of results

5 Coherent-constructible correspondence

6 Examples

6.1 Taking the mapping cone

6.2 Toric Fano surfaces

6.3 Hirzebruch surfaces

References

Superspace: a Comfortably Vast Algebraic Variety

T. Hiibsch

1 Introduction

1.1 Basic ideas and definitions

1.2 The traditional superspace

2 Off-shell worldline supermultiplets

2.1 Adinkraic supermultiplets

2.2 Various hangings

2.3 Projected supermultiplets

2.4 Supermultiplets vs. superfields

3 Superspace, by construction

3.1 Superpartners of time

3.2 A telescoping deformation structure

3.3 Nontrivial superspace geometry

3.4 Higher-dimensional spacetime

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

6 Conclusions

References

Hermitian-Yang-Mills Connections on Kahler Manifolds

Jun Li

1 Introduction

1.1 Hermitian-Yang-Mills connections

1.2 HYM connections lead to stable bundles

1.3 Stable bundles and their moduli spaces

1.4 Flat bundles and stable bundles on curves

2 Donaldson-Uhlenbeck-Yau theorem

2.1 Donaldson's proof for algebraic surfaces

2.2 Uhlenbeck-Yau's proof for Kahler manifolds

3 Hermitian-Yang-Mills connections on curves

4 Hermitian-Yang-Mills connections on surfaces

4.1 Extending DUY correspondence

4.2 Stable topology of the moduli spaces

4.3 Donaldson polynomial invariants

5 HYM connections on high dimensional varieties

5.1 Extending the DUY correspondence in high dimensions

5.2 Donaldson-Thomas invariants

6 Concluding remark

References ~

Additivity and Relative Kodaira Dimensions

Tian-Jun Li and Weiyi Zhang

1 Introduction

2 Kodaira Dimensions and fiber bundles

2.1 h for complex manifolds and up to dimension 3

2.2 Ks for symplectic 4~manifolds

2.3 Additivity for a fiber bundle

3 Embedded symplectic surfaces and relative Kod. dim. in dim. 4.

3.1 Embedded symplectic surfaces and maxinmlity

3.2 The adjoint class

3.3 Existence and Uniqueness of relatively minimal model

3.4 (M,w,F)

4 Relative Kod. dim. in dim. 2 and fibrations over a surface

4.1 (F,D), Riemann-Hurwitz formula and Seifert fibrations...

4.2 Lefschetz fibrations

References

Descendent Integrals and Tautological Rings of

Moduli Spaces of Curves

Kefen9 Liu and Hao Xu

1 Introduction

2 Intersection numbers and the Witten-Kontsevich theorem

2.1 Witten-Kontsevich theorem

2.2 Virasoro constraints

3 The n-point function

3.1 A recursive formula of n-point functions

3.2 An effective recursion formulae of descendent integrals

4 Hodge integrals

4.1 Faber's algorithm

4.2 Hodge integral formulae

5 Higher Weil-Petersson volumes

5.1 Generalization of Mirzakhani's recursion formula

5.2 Recursion formulae of higher Weil-Petersson volumes

6 Faber's conjecture on tautological rings

6.1 The Faber intersection number conjecture

6.2 Relations with n-point functions

7 Dimension of tautological rings

7.1 Ramanujan's mock theta functions

7.2 Asymptotics of tautological dimensions

8 Gromov-Witten invariants

8.1 Universal equations of Gromov-Witten invariants

8.2 Some vanishing identities

9 Witten's r-spin numbers

9.1 Generalized Witten's conjecture

9.2 An algorithm for computing Witten's r-spin numbers

References

A General Voronoi Summation Formula for GL(n, Z)

Stephen D. Miller and Wilfried Sehmid

1 Introduction

2 Automorphic Distributions

3 Vanishing to infinite 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

1.1 Examples of equivariant embeddings into the projective space

1.2 Non-standard holomorphic isometries of the Poincar disk into polydisks

1.3 A non-standard holomorphic isometry of the Poincar disk into a Siegel upper half-plane

1.4 Examples of holomorphic isometries with arbitrary normalizing constants A > 1

2 Analytic continuation of germs of holomorphic isometries

2.1 Analytic continuation of holomorphic isometries into the projective space equipped with the Fubini-Study metric

2.2 An extension and rigidity problem arising from commutators of modular correspondences

2.3 Analytic continuation of holomorphic isometries up to normalizing constants with respect to the Bergman metric - extension beyond the boundary

2.4 Canonically embeddable Bergman manifolds and Bergman meromorphic compactifications

3 Holomorphic isometries of the Poincar disk into bounded symmetric domains

3.1 Structural equations on the norm of the second fundamental form and asymptotic vanishing order

3.2 Holomorphic isometries of the Poincar disk into polydisks: structural results

3.3 Calculated examples on the norm of the second fundamental form

3.4 Holomorphic isometries of the Poincar5 disk into polydisks: uniqueness results

3.5 Asymptotic total geodesy and applications

4 Measure-preserving algebraic correspondences on irreducible

bounded symmetric domains

4.1 Statements of results

4.2 Extension results on strictly pseudoconvex algebraic hypersurfaces

4.3 Alexander-type extension results in the higher-rank case

4.4 Total geodesy of germs of measure-preserving holomorphie

……

Abundance Conjecture

Sasaki-Einstein Geometry

A Simple Proof of the Chiral Gravity Conjecture

Geometry of Grand Unification

Quantum Invariance Under Flop Transitions

The Problem of gauge Theory

Part 4 Appendices

Shing-Tung Yau,a Manifold Man of Mathematics

Perspectives on Geometric Analysis

A Survey of Calabi-Yau Manifolds