Topology and Physics - Proceedings of Th: Proceedings of the Nankai International Conference in Memory of Xiao-Song Lin

This unique volume, resulting from a conference at the Chern Institute of Mathematics dedicated to the memory of Xiao-Song Lin, presents a broad connection between topology and physics as exemplified by the relationship between low-dimensional topology and quantum field theory. The volume includes works on picture (2+1) - TQFTs and their applications to quantum computing, categorification and Khovanov homology, Gromov - Witten type invariants, twisted Alexander polynomials, Faddeev knots, generalized Ricci flow, Calabi - Yau problems for CR manifolds, Milnor's conjecture on volume of simplexes, Heegaard genera of 3-manifolds, and the (A,B) - slice problem. It also includes five unpublished papers of Xiao-Song Lin and various speeches related to the memorial conference. CONTENTS......Page 30 Foreword......Page 10 Preface......Page 12 Short Biography of Lin......Page 14 Mathematics of Lin......Page 16 Organizing Committees......Page 19 List of Participants......Page 21 Program......Page 23 Welcome Speech of Weiping Zhang......Page 26 Speech of Boju Jiang......Page 28 Part A Invited Contributions......Page 34 The Modified Calabi-Yau Problems for CR-manifolds J. Cao and S.-C. Chang......Page 36 0. Introduction......Page 37 1. Bounded solutions to d = on manifolds with negative curvature......Page 38 A. Sup-harmonic functions on Alexandrov spaces with nonnegative sectional curvature......Page 43 B. The generalized Calabi problems for K ahler domains with boundaries......Page 44 C. The Calabi-Escobar type problem for K ahler domains with boundaries......Page 46 References......Page 47 1. Introduction......Page 52 2.1. Braid statistics......Page 56 2.2. Generic Jones representation of the braid groups......Page 57 2.3. Unitary Jones representations......Page 59 2.4. Uniqueness of Jones-Wenzl projectors......Page 63 3.1. "d-isotopy", local relation, and skein relation......Page 69 3.2. Picture classes......Page 72 3.3. Skein classes......Page 73 3.4. Recoupling theory......Page 74 3.5. Handles and S-matrix......Page 77 3.6. Diagram TQFTs for closed manifolds......Page 79 3.7. Boundary conditions for picture TQFTs......Page 82 3.8. Jones-Kauffman skein spaces......Page 83 4. Morita equivalence and cut-paste topology......Page 88 4.1. Bimodules over picture category......Page 90 4.2. Cutting and paste as Morita equivalence......Page 96 5. Temperley-Lieb-Jones categories......Page 98 5.2. Representation of Temperley-Lieb-Jones categories......Page 100 5.3.1. Level=1, d2 = 1......Page 102 5.3.3. Level=3, d2 = 1 + d or d2 = 1......Page 103 5.4.2. Level=2, d2 = 2......Page 104 5.5. Temperley-Lieb-Jones categories for primitive 4rth roots of unity......Page 105 5.6. Temperley-Lieb-Jones categories for primitive 2rth root or rth root of unity, r odd......Page 111 6. The definition of a TQFT......Page 112 6.1. Redined labels for TQFTs......Page 113 6.2. Anomaly of TQFTs and extended manifolds......Page 114 6.3. Axioms for TQFTs......Page 115 6.4. More consequences of the axioms......Page 118 6.5. Framed link invariants and modular representation......Page 119 6.6. Verlinde algebras and Verlinde formulas......Page 120 7. Diagram and Jones-Kau man TQFTs......Page 121 7.1. Diagram TQFTs......Page 122 8. WRT and Turaev-Viro SU(2)-TQFTs......Page 123 8.1. Flagged TLJ categories......Page 124 8.3. WRT Unitary TQFTs......Page 125 9.1. Black-white TLJ categories......Page 126 9.2.2. Level=3......Page 127 10. Classification and Unitarity......Page 128 10.1. Classification of diagram local relations......Page 129 10.2. Unitary TQFTs......Page 130 10.3. Classi cation and unitarity......Page 131 Elementary excitations as particles......Page 132 Appendix B. Representation of linear category......Page 133 General representation theory......Page 134 Gluing......Page 137 References......Page 138 1. Introduction......Page 140 2. Berry phase in Yang-Baxter approach......Page 142 3. Berry phase for hamiltonian H2( ; (t))......Page 145 4. Yang-Baxterization of a simple model in 2-dimensional braid relation.......Page 147 5. Yang-Baxterization of Eq(4.3)......Page 151 References......Page 153 Milnor's conjecture......Page 155 A characterization of angle Gram matrices......Page 156 2. Normal vectors of Euclidean simplexes......Page 157 3. Degenerate hyperbolic simplexes......Page 162 4. Proof of theorem 1......Page 165 5. Proof of theorem 3......Page 166 References......Page 169 1. Introduction......Page 171 Acknowledgments......Page 172 2. Preliminaries......Page 173 3. The Habegger-Lin Classi cation Scheme......Page 174 4. Structure Theorems for Cn-equivalence and for Self-Cn-equivalence.......Page 177 5. The Indeterminacy of Finite Type Invariants.......Page 179 References......Page 182 1. Introduction......Page 184 2. Local Existences and Uniqueness......Page 186 3. The Monotonicity Formula......Page 192 4. Critical points......Page 196 5. Evolution of Curvatures......Page 197 References......Page 203 1. INTRODUCTION......Page 205 2. Spin Networks and Temperley { Lieb Recoupling Theory......Page 208 2.1. Evaluations......Page 213 2.2. Symmetry and Unitarity......Page 214 3. Explicit Form of the Braid Group Representations......Page 217 3.1. The Fibonacci Model......Page 220 4. Quantum Computation of Colored Jones Polynomials and the Witten-Reshetikhin-Turaev Invariant......Page 221 4.1. The Hadamard Test......Page 224 References......Page 226 1. Introduction......Page 228 2.1. The Gromov-Witten invariants......Page 229 2.2.2. Atiyah-Bott Localization Formula......Page 230 2.2.4. Virtual Functorial Localization Formula......Page 231 2.3. The moduli space of relative stable morphisms......Page 232 3. A new approach to the Gromov-Witten theory......Page 234 4. Localization proof of the Witten conjecture......Page 237 5. Proof of Marino-Vafa formula......Page 242 6. Future Research problems......Page 245 6.1. Generalized Witten conjecture......Page 246 6.2. Faber's conjecture on Hodge integrals......Page 247 6.3. General Virasoro conjecture......Page 248 References......Page 250 1. Introduction......Page 253 2. Surgery and the A......Page 255 3. Link groups and Bing cells......Page 257 3.1. The associated tree......Page 258 4. An obstruction for model decompositions.......Page 261 Acknowledgement......Page 267 References......Page 268 1. Introduction......Page 269 2. Group von Neumann algebra and Fuglede-Kadison determinant......Page 271 3. Twisted SL(2; C) L2-Alexander invariant of a knot......Page 273 4. Twisted L2-Reidemeister torsion......Page 281 5. Twisted L2–Alexander(-Conway) invariant with parameters in character variety......Page 287 6. Twisted L2-Alexander invariant with GL(n; C) representations......Page 289 References......Page 291 1. Introduction......Page 293 2. Faddeev Knots at Unit Charge......Page 297 3. Growth Law Perspectives......Page 298 4. Knot Energy in General Hopf Dimensions......Page 302 5. The Universal Growth Law......Page 303 6. Overlook......Page 305 References......Page 307 1. Introduction......Page 312 2. The ring of translation invariant symmetric polynomials......Page 313 3. Haldane's conjecture......Page 314 4. Proof of Theorem 2.1......Page 317 References......Page 320 1. Introduction......Page 321 1.1. Acknowledgments......Page 323 2.1. Atoms and Knots......Page 324 2.2. Usual Khovanov homology......Page 332 3. Bourgoin's twisted knots. Additional gradings......Page 333 4. Additional grading: the general case......Page 336 4.1. Explanation for the second and the third moves......Page 339 5. More gradings; more examples......Page 340 5.1. Examples......Page 341 5.2. Braids......Page 343 5.3. Further gradings......Page 344 6. Khovanov's Frobenius theory......Page 345 6.2. Khovanov Frobenius theory modulo Z2 in the general case......Page 349 6.2.1. The Z2-case......Page 352 7. Gradings or ltrations? The spectral sequence......Page 353 8. Applications......Page 354 9. The relation to other papers......Page 355 References......Page 356 1. Introduction......Page 359 2.1. Partition and symmetric function......Page 360 2.2. Quantum group invariants of links......Page 362 3. Large N CS/TS duality......Page 364 4.1. Marino-Vafa formula......Page 365 4.2. Topological vertex theory......Page 366 4.3. Labastida-Marino-Ooguri-Vafa Conjecture......Page 368 4.4. U(N) Chern-Simons gauge theory......Page 369 4.5. Uniqueness of the cut-and-join system......Page 370 References......Page 371 1. Introducion......Page 374 2. An extension of Schultens's Lemma......Page 376 3. The proof of Theorem 1......Page 377 References......Page 380 1. Introduction......Page 381 2. Quantum sl......Page 382 3. A toy model: simple Uq(sl2)-modules......Page 383 4. Intersection complex......Page 386 References......Page 389 Part B Xiao-Song Lin's Unpublished Papers......Page 390 Editors' Notes......Page 392 1. Coxeter groups and their corresponding braid groups......Page 393 2. Study links via open book decompositions......Page 397 3. The case of S2 S1......Page 400 References......Page 402 1. Introduction......Page 403 2. The quantum group U and its finite dimensional representations......Page 405 3. Vertex models and quantum groups......Page 407 4. VBL-functionals and their states models......Page 412 References......Page 416 Knot Invariants and Iterated Integrals......Page 417 1. The associator KZ......Page 418 2. Non-associative tangles......Page 422 3. Some calculations......Page 425 References......Page 427 1. Introduction......Page 428 2. Legendrian knots and Legendrian braids......Page 429 3. Extended configuration spaces and a flat formal connection......Page 434 4. An invariant of Legendrian knots......Page 437 6. A sketch of the proof of Theorem 3.1......Page 439 7. An irreducible weight system on dotted chord diagrams......Page 442 References......Page 443 1. Introduction......Page 444 2. Configuration space of disjoint small loops......Page 445 3. Elementary loop braids and relations among them......Page 447 4. Presentation of loop braid group......Page 449 References......Page 450 Part C Lin Award, Speeches and Writings......Page 452 3. Xiao-Song Lin fund members......Page 454 4. Timeline and cash prize......Page 455 Michael Freedman's Speech:......Page 456 Gang Tian's Speech:......Page 458 Zhenghan Wang's Speech:......Page 461 2. For the LinBook......Page 463 Jian-Pin He's Speech......Page 465 This unique volume, resulting from a conference at the Chern Institute of Mathematics dedicated to the memory of Xiao-Song Lin, presents a broad connection between topology and physics as exemplified by the relationship between low-dimensional topology and quantum field theory.The volume includes works on picture (2+1)-TQFTs and their applications to quantum computing, Berry phase and Yang-Baxterization of the braid relation, finite type invariant of knots, categorification and Khovanov homology, Gromov-Witten type invariants, twisted Alexander polynomials, Faddeev knots, generalized Ricci flow, Calabi-Yau problems for CR manifolds, Milnor's conjecture on volume of simplexes, Heegaard genera of 3-manifolds, and the (A,B)-slice problem. It also includes five unpublished papers of Xiao-Song Lin and various speeches related to the memorial conference. This unique volume, resulting from a conference at the Chern Institute of Mathematics dedicated to the memory of Xiao-Song Lin, presents a broad connection between topology and physics as exemplified by the relationship between low-dimensional topology and quantum field theory. The volume includes works on picture (2+1)-TQFTs and their applications to quantum computing, Berry phase and Yang-Baxterization of the braid relation, finite type invariant of knots, categorification and Khovanov homology, Gromov-Witten type invariants, twisted Alexander polynomials, Faddeev knots, generalized Ricci flo