The launch of this Advanced Lectures in Mathematics series is aimed at keeping mathematicians informed of the latest developments in mathematics, as well as to aid 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 signifi-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.

Geometric Analysis combines differential equations and differential geometry. An important aspect is to solve geometric problems by studying differential equations.Besides some known linear differential operators such as the Laplace operator,many differential equations arising from differential geometry are nonlinear. A particularly important example is the IVlonge-Ampere equation; Applications to geometric problems have also motivated new methods and techniques in differen-rial equations. The field of geometric analysis is broad and has had many striking applications. This handbook of geometric analysis provides introductions to and surveys of important topics in geometric analysis and their applications to related fields which is intend to be referred by graduate students and researchers in related areas.

Heat Kernels on Metric Measure Spaces with Regular Volume Growth

 1 Introduction

1.1 Heat kernel in Rn

1.2 Heat kernels on Riemannian manifolds

1.3 Heat kernels of fractional powers of Laplacian

1.4 Heat kernels on fractal spaces

1.5 Summary of examples

 2 Abstract heat kernels

2.1 Basic definitions

2.2 The Dirichlet form

2.3 Identifying φ in the non-local case

2.4 Volume of balls

 3 Besov spaces

3.1 Besov spaces in IRn

3.2 Besov spaces in a metric measure space

3.3 Embedding of Besov spaces into HSlder spaces

 4 The energy domain

4.1 A local case

4.2 Non-local case

4.3 Subordinated heat kernel

4.4 Bessel potential spaces

 5 The walk dimension

5.1 Intrinsic characterization of the walk dimension

5.2 Inequalities for the walk dimension

 6 Two-sided estimates in the local case

6.1 The Dirichlet form in subsets

6.2 Maximum principles

6.3 A tail estimate

6.4 Identifying φ in the local case

 References

A Convexity Theorem and Reduced Delzant Spaces

 1 Introduction

 2 Convexity of image of moment map

 3 Rationality of moment polytope

 4 Realizing reduced Delzant spaces

 5 Classification of reduced Delzant spaces

 References

Localization and some Recent Applications

 1 Introduction

 2 Localization

 3 Mirror principle

 4 Hori-Vafa formula

 5 The Marifio-Vafa Conjecture

 6 Two partition formula

 7 Theory of topological vertex.

 8 Gopakumar-Vafa conjecture and indices of elliptic operators

 9 Two proofs of the ELSV formula

 10 A localization proof of the Wittcn conjecture

 11 Final remarks

 References

Gromov-Witten Invariants of Toric Calabi-Yau Threefolds

 1 Gromov-Witten invariants of Calabi-Yau 3-folds

1.1 Symplectic and algebraic Gromov-Wittcn invariants

1.2 Moduli space of stable maps

1.3 Gromov-Witten invariants of compact Calabi-Yau 3-folds

1.4 Gromov-Witten invariants of noncompact Calabi-Yau 3-folds

 2 Traditional algorithm in the toric case

2.1 Localization

2.2 Hodge integrals

 3 Physical theory of the topological vertex

 4 Mathematical theory of the topological vertex

4.1 Locally planar trivalent graph

4.2 Formal toric Calabi-Yau (FTCY) graphs

4.3 Degeneration formula

4.4 Topological vertex

4.5 Localization

4.6 Framing dependence

4.7 Combinatorial expression

4.8 Applications

4.9 Comparison

 5 GW/DT correspondences and the topological vertex

 Acknowledgments

 References

Survey on Affine Spheres

 1 Introduction

 2 Affine structure equations

 3 Examples

 4 Two-dimensional affine spheres and Titeica's equation

 5 Monge-Ampere equations and duality

 6 Global classification of affine spheres

 7 Hyperbolic affine spheres and invariants of convex cones

 8 Projective manifolds

 9 Affine manifolds

 10 Affine maximal hypersurfaces

 11 Affine normal flow

 References

Convergence and Collapsing Theorems in Riemannian Geometry

 Introduction

 1 Gromov-tlausdorff distance in space of metric spaces

1.1 The Gromov-Hausdorff distance

1.2 Examples

1.3 An alternative formulation of GH-distance

1.4 Compact subsets of (Met, dGH)

1.5 Equivariant GH-convergcnce

1.6 Pointed GH-convergence

 2 Smooth limits-fibrations

2.1 The fibration theorem

2.2 Sectional curvature comparison

2.3 Embedding via distance functions

2.4 Fibrations

2.5 Proof of theorem 2.1.1

2.6 Center of mass

2.7 Equivariant fibrations

2.8 Applications of the fibration theorem

 3 Convergence theorems

3.1 Cheeger-Gromov's convergence theorem

3.2 Injectivity radius estimate

3.3 Some elliptic estimates

3.4 Harmonic radius estimate

3.5 Smoothing metrics

 4 Singular limits-singular fibrations

4.1 Singular fibrations

4.2 Controlled homotopy structure by geometry

4.3 The π2-finiteness theorem

4.4 Collapsed manifolds with pinched positive sectional curvature

 5 Almost fiat manifolds

5.1 Gromov's theorem on almost flat manifolds

5.2 The Margulis lemma

5.3 Flat connections with small torsion

5.4 Flat connection with a parallel torsion

5.5 Proofs--part I

5.6 Proofs--part II

5.7 Refined fibration theorem

 References

Geometric Transformations and Soliton Equations

 1 Introduction

 2 The moving frame method for submanifolds

 3 Line congruences and Biicklund transforms

 4 Sphere congruences and Ribaucour transforms

 5 Combescure transforms, O-surfaces, and k-tuples

 6 From moving frame to Lax pair

 7 Soliton hierarchies constructed from symmetric spaces

 8 The U/K-system and the Gauss-Codazzi equations

 9 Loop group actions

 10 Action of simple elements and geometric transforms

 References

Affine Integral Geometry from a Differentiable Viewpoint

 1 Introduction

 2 Basic definitions and notation

2.1 Linear group actions

 3 Objects of study

3.1 Geometric setting

3.2 Convex body

3.3 The space of all convex bodies

3.4 Valuations

 4 Overall strategy

 5 Fundamental constructions

5.1 The support function

5.2 The Minkowski sum

5.3 The polar body

5.4 The inverse Gauss map

5.5 The second fundamental form

5.6 The Legendre transform

5.7 The curvature function

 6 The homogeneous contour integral

6.1 Homogeneous functions and differential forms

6.2 The homogeneous contour integral for a differential form

6.3 The homogeneous contour integral for a measure

6.4 Homogeneous integral calculus

 7 An explicit construction of valuations

7.1 Duality

7.2 Volume

 8 Classification of valuations

 9 Scalar valuations

9.1 SL(n)-invariant valuations

9.2 Hug's theorem

 10 Continuous GL(n)-homogeneous valuations

10.1 Scalar valuations

10.2 Vector-valued valuations

 11 Matrix-valued valuations

11.1 The Cramer-Rao inequality

 12 Homogeneous function- and convex body-valued valuations

 13 Questions

 References

Classification of Fake Projective Planes

 1 Introduction

 2 Uniformization of fake projective planes

 3 Geometric estimates on the number of fake projective planes

 4 Arithmeticity of lattices associated to fake projective planes

 5 Covolume formula of Prasad

 6 Formulation of proof

 7 Statements of the results

 8 Further studies

 References