Heat Kernels on Metric Measure Spaces with Regular Volume Growth&Alexander Grigor'yan 1
1 Introduction 1
1.1 Heat kernel in Rn 2
1.2 Heat kernels on Riemannian manifolds 3
1.3 Heat kernels of fractional powers of Laplacian 4
1.4 Heat kernels on fractal spaces 5
1.5 Summary of examples 7
2 Abstract heat kernels 8
2.1 Basic definitions 8
2.2 The Dirichlet form 11
2.3 Identifying Ф in the non-local case 13
2.4 Volume of balls 17
3 Besov spaces 21
3.1 Besov spaces in Rn 21
3.2 Besov spaces in a metric measure space 23
3.3 Embedding of Besov spaces into H？lder spaces 24
4 The energy domain 26
4.1 A local case 26
4.2 Non-local case 31
4.3 Subordinated heat kernel 32
4.4 Bessel potential spaces 35
5 The walk dimension 36
5.1 Intrinsic characterization of the walk dimension 36
5.2 Inequalities for the walk dimension 39
6 Two-sided estimates in the local case 46
6.1 The Dirichlet form in subsets 46
6.2 Maximum principles 47
6.3 A tail estimate 47
6.4 Identifying Ф in the local case 55
References 57
A Convexity Theorem and Reduced Delzant Spaces&Bong H.Lian,Bailin Song 61
1 Introduction 61
2 Convexity of image of moment map 64
3 Rationality of moment polytope 69
4 Realizing reduced Delzant spaces 74
5 Classification of reduced Delzant spaces 82
References 94
Localization and some Recent Applications&Bong H.Lian,Kefeng Liu 97
1 Introduction 97
2 Localization 100
3 Mirror principle 102
4 Hori-Vafa formula 112
5 The Mari？o-Vafa Conjecture 115
6 Two partition formula 123
7 Theory of topological vertex 125
8 Gopakumar-Vafa conjecture and indices of elliptic operators 128
9 Two proofs of the ELSV formula 129
10 A localization proof of the Witten conjecture 132
11 Final remarks 134
References 134
Gromov-Witten Invariants of Toric Calabi-Yau Threefolds&Chiu-Chu Melissa Liu 139
1 Gromov-Witten invariants of Calabi-Yau 3-folds 139
1.1 Symplectic and algebraic Gromov-Witten invariants 139
1.2 Moduli space of stable maps 139
1.3 Gromov-Witten invariants of compact Calabi-Yau 3-folds 140
1.4 Gromov-Witten invariants of noncompact Calabi-Yau 3-folds 141
2 Traditional algorithm in the toric case 142
2.1 Localization 142
2.2 Hodge integrals 143
3 Physical theory of the topological vertex 144
4 Mathematical theory of the topological vertex 146
4.1 Locally planar trivalent graph 146
4.2 Formal toric Calabi-Yau(FTCY)graphs 148
4.3 Degeneration formula 150
4.4 Topological vertex 152
4.5 Localization 153
4.6 Framing dependence 154
4.7 Combinatorial expression 154
4.8 Applications 155
4.9 Comparison 155
5 GW/DT correspondences and the topological vertex 156
Acknowledgments 156
References 156
Survey on Affine Spheres&John Loftin 161
1 Introduction 161
2 Affine structure equations 163
3 Examples 164
4 Two-dimensional affine spheres and Titeica's equation 165
5 Monge-Ampère equations and duality 168
6 Global classification of affine spheres 172
7 Hyperbolic affine spheres and invariants of convex cones 173
8 Projective manifolds 176
9 Affinc manifolds 181
10 Affine maximal hypersurfaces 185
11 Affine normal flow 186
References 187
Convergence and Collapsing Theorems in Riemannian Geometry&Xiaochun Rong 193
Introduction 193
1 Gromov-Hausdorff distance in space of metric spaces 194
1.1 The Gronov-Hausdorff distance 194
1.2 Examples 199
1.3 An alternative formulation of GH-distance 202
1.4 Compact subsets of(Met,dGH) 204
1.5 Equivariant GH-convergence 206
1.6 Pointed GH-convergence 209
2 Smooth limits-fibrations 217
2.1 The fibration theorem 217
2.2 Sectional curvature comparison 219
2.3 Embedding via distance functions 223
2.4 Fibrations 226
2.5 Proof of theorem 2.1.1 231
2.6 Center of mass 234
2.7 Equivariant fibrations 235
2.8 Applieations of the fibration theorem 240
3 Convergence theorems 245
3.1 Cheeger-Gromov's convergence theorem 245
3.2 Injectivity radius estimate 248
3.3 Some elliptic estimates 253
3.4 Harmonic radius estimate 255
3.5 Smoothing metrics 259
4 Singular limits-singular fibrations 260
4.1 Singular fibrations 261
4.2 Controlled homotopy structure by geometry 265
4.3 The π2-finiteness theorem 269
4.4 Collapsed manifolds with pinched positive sectional curvature 271
5 Almost flat manifolds 273
5.1 Gromov's theorem on almost flat manifolds 273
5.2 The Margulis lemma 275
5.3 Flat connections with small torsion 277
5.4 Flat connection with a parallel torsion 281
5.5 Proofs—part Ⅰ 285
5.6 Proofs—part Ⅱ 290
5.7 Refined fibration theorem 294
References 297
Geometric Transformations and Soliton Equations&Chuu-Lian Terng 301
1 Introduction 301
2 The moving frame method for submanifolds 306
3 Line congruences and B？cklund transforms 309
4 Sphere congruences and Ribaucour transforms 315
5 Combescure transforms,O-surfaces,and k-tuples 317
6 From moving frame to Lax pair 320
7 Soliton hierarchies constructed from symmetric spaces 329
8 The U/K-system and the Gauss-Codazzi equations 336
9 Loop group actions 343
10 Action of simple elements and geometric transforms 347
References 355
Affine Integral Geometry from a Differentiable Viewpoint&Deane Yang 359
1 Introduction 359
2 Basic definitions and notation 361
2.1 Linear group actions 361
3 Objects of study 362
3.1 Geometric setting 362
3.2 Convex body 362
3.3 The space of all convex bodies 362
3.4 Valuations 362
4 Overall strategy 363
5 Fundamental constructions 363
5.1 The support function 363
5.2 The Minkowski sum 364
5.3 The polar body 365
5.4 The inverse Gauss map 366
5.5 The second fundamental form 366
5.6 The Legendre transform 366
5.7 The curvature function 367
6 The homogeneous contour integral 368
6.1 Homogeneous functions and differential forms 368
6.2 The homogeneous contour integral for a differential form 369
6.3 The homogeneous contour integral for a measure 369
6.4 Homogeneous integral calculus 373
7 An explicit construction of valuations 374
7.1 Duality 375
7.2 Volume 375
8 Classification of valuations 376
9 Scalar valuations 376
9.1 SL(n)-invariant valuations 376
9.2 Hug's theorem 378
10 Continuous GL(n)-homogeneous valuations 378
10.1 Scalar valuations 378
10.2 Vector-valued valuations 379
11 Matrix-valued valuations 380
11.1 The Cramer-Rao inequality 381
12 Homogeneous function-and convex body-valued valuations 383
13 Questions 384
References 385
Classification of Fake Projective Planes&Sai-Kee Yeung 391
1 Introduction 391
2 Uniformization of fake projective planes 393
3 Geometric estimates on the number of fake projective planes 396
4 Arithmeticity of lattices associated to fake projective planes 398
5 Covolume formula of Prasad 410
6 Formulation of proof 411
7 Statements of the results 419
8 Further studies 423
References 427