CONTENTS -- Foreword -- Preface -- Short Biography of Lin -- Mathematics of Lin -- Organizing Committees -- List of Participants -- Program -- Welcome Speech of Weiping Zhang -- Speech of Boju Jiang -- Part A Invited Contributions -- The Modified Calabi-Yau Problems for CR-manifolds J. Cao and S.-C. Chang -- 0. Introduction -- 1. Bounded solutions to d = on manifolds with negative curvature -- 2. The modified Calabi-Yau problems for singular spaces and CR-manifolds -- A. Sup-harmonic functions on Alexandrov spaces with nonnegative sectional curvature -- B. The generalized Calabi problems for K ahler domains with boundaries -- C. The Calabi-Escobar type problem for K ahler domains with boundaries -- Acknowledgments -- References -- On Picture (2+1)-TQFTs M. Freedman, C. Nayak, K. Walker and Z. Wang -- 1. Introduction -- 2. Jones representations -- 2.1. Braid statistics -- 2.2. Generic Jones representation of the braid groups -- 2.3. Unitary Jones representations -- 2.4. Uniqueness of Jones-Wenzl projectors -- 3. Diagram TQFTs for closed manifolds -- 3.1. "d-isotopy", local relation, and skein relation -- 3.2. Picture classes -- 3.3. Skein classes -- 3.4. Recoupling theory -- 3.5. Handles and S-matrix -- 3.6. Diagram TQFTs for closed manifolds -- 3.7. Boundary conditions for picture TQFTs -- 3.8. Jones-Kauffman skein spaces -- 4. Morita equivalence and cut-paste topology -- 4.1. Bimodules over picture category -- 4.2. Cutting and paste as Morita equivalence -- 4.3. Annualization and quantum double -- 5. Temperley-Lieb-Jones categories -- 5.1. Annular Markov trace -- 5.2. Representation of Temperley-Lieb-Jones categories -- 5.3. Rectangular Tempeley-Lieb-Jones categories for low levels -- 5.3.1. Level=1, d2 = 1 -- 5.3.2. Level=2, d2 = 2 -- 5.3.3. Level=3, d2 = 1 + d or d2 = 1 -- 5.4. Annular Temperley-Lieb-Jones theories for low levels.

5.4.1. Level=1, d2 = 1 -- 5.4.2. Level=2, d2 = 2 -- 5.4.3. Level=3, d2 = 1 + d or d2 = 1 -- 5.5. Temperley-Lieb-Jones categories for primitive 4rth roots of unity -- 5.6. Temperley-Lieb-Jones categories for primitive 2rth root or rth root of unity, r odd -- 6. The definition of a TQFT -- 6.1. Redined labels for TQFTs -- 6.2. Anomaly of TQFTs and extended manifolds -- 6.3. Axioms for TQFTs -- 6.4. More consequences of the axioms -- 6.5. Framed link invariants and modular representation -- 6.6. Verlinde algebras and Verlinde formulas -- 7. Diagram and Jones-Kau man TQFTs -- 7.1. Diagram TQFTs -- 7.2. Jones-Kau man TQFTs -- 8. WRT and Turaev-Viro SU(2)-TQFTs -- 8.1. Flagged TLJ categories -- 8.2. Turaev-Viro Unitary TQFTs -- 8.3. WRT Unitary TQFTs -- 9. Black-White TQFTs -- 9.1. Black-white TLJ categories -- 9.2. Labels for black-white theories -- 9.2.1. Level=2, d2 = 2 -- 9.2.2. Level=3 -- 9.3. BW TQFTs -- 10. Classification and Unitarity -- 10.1. Classification of diagram local relations -- 10.2. Unitary TQFTs -- 10.3. Classi cation and unitarity -- Appendix A. Topological phases of matter -- Ground states manifolds as modular functors -- Elementary excitations as particles -- Braid statistics -- Appendix B. Representation of linear category -- General representation theory -- Appendix C. Gluing and Maslov index -- Gluing -- Maslov index -- References -- Berry Phase in Yang-Baxter Approach Related to Entangled States and the Yang-Baxterization of Braid Relation for Twostate Anyon Model M.-L. Ge, L. Kau man and K. Xue -- 1. Introduction -- 2. Berry phase in Yang-Baxter approach -- 3. Berry phase for hamiltonian H2( -- (t)) -- 4. Yang-Baxterization of a simple model in 2-dimensional braid relation. -- 5. Yang-Baxterization of Eq(4.3) -- References -- On a Conjecture of Milnor about Volume of Simplexes R. Guo and F. Luo -- 1. Introduction.

Milnor's conjecture -- A characterization of angle Gram matrices -- Acknowledgment -- 2. Normal vectors of Euclidean simplexes -- 3. Degenerate hyperbolic simplexes -- 4. Proof of theorem 1 -- 5. Proof of theorem 3 -- References -- On the Classification of Links up to Finite Type N. Habegger and J.B. Meilhan -- 1. Introduction -- Acknowledgments -- 2. Preliminaries -- 3. The Habegger-Lin Classi cation Scheme -- 4. Structure Theorems for Cn-equivalence and for Self-Cn-equivalence. -- 5. The Indeterminacy of Finite Type Invariants. -- References -- Generalized Ricci Flow I: Local Existence and Uniqueness C.-L. He, S. Hu, D.-X. Kong and K. Liu -- 1. Introduction -- 2. Local Existences and Uniqueness -- 3. The Monotonicity Formula -- 4. Critical points -- 5. Evolution of Curvatures -- Acknowledgements: -- References -- Unitary Representations of the Artin Braid Groups and Quantum Algorithms for Colored Jones Polynomials and the Witten-Reshetikhin Invariant L. H. Kau man and S. J. Lomonaco Jr. -- 1. INTRODUCTION -- 2. Spin Networks and Temperley { Lieb Recoupling Theory -- 2.1. Evaluations -- 2.2. Symmetry and Unitarity -- 3. Explicit Form of the Braid Group Representations -- 3.1. The Fibonacci Model -- 4. Quantum Computation of Colored Jones Polynomials and the Witten-Reshetikhin-Turaev Invariant -- 4.1. The Hadamard Test -- References -- A New Approach to Deriving Recursion Relations for the Gromov-Witten Theory Y.-S. Kim and K. Liu -- 1. Introduction -- 2. Preliminaries -- 2.1. The Gromov-Witten invariants -- 2.2. Localization technique -- 2.2.1. Equivariant Cohomology -- 2.2.2. Atiyah-Bott Localization Formula -- 2.2.3. Functorial Localization Formula -- 2.2.4. Virtual Functorial Localization Formula -- 2.3. The moduli space of relative stable morphisms -- 3. A new approach to the Gromov-Witten theory -- 4. Localization proof of the Witten conjecture.

5. Proof of Marino-Vafa formula -- 6. Future Research problems -- 6.1. Generalized Witten conjecture -- 6.2. Faber's conjecture on Hodge integrals -- 6.3. General Virasoro conjecture -- References -- Link Groups and the A-B Slice Problem V.-S. Krushkal -- 1. Introduction -- 2. Surgery and the A -- 3. Link groups and Bing cells -- 3.1. The associated tree -- 4. An obstruction for model decompositions. -- Acknowledgement -- References -- Twisted L2-Alexander-Conway Invariants for Knots W. Li and W. Zhang -- 1. Introduction -- 2. Group von Neumann algebra and Fuglede-Kadison determinant -- 3. Twisted SL(2 -- C) L2-Alexander invariant of a knot -- 4. Twisted L2-Reidemeister torsion -- 5. Twisted L2-Alexander(-Conway) invariant with parameters in character variety -- 6. Twisted L2-Alexander invariant with GL(n -- C) representations -- References -- Existence of Knots of Minimum Energy and Topological Growth Laws in the Faddeev Model F. Lin and Y. Yang -- 1. Introduction -- 2. Faddeev Knots at Unit Charge -- 3. Growth Law Perspectives -- 4. Knot Energy in General Hopf Dimensions -- 5. The Universal Growth Law -- 6. Overlook -- Acknowledgments. -- References -- On Translation Invariant Symmetric Polynomials and Haldane's Conjecture J. Liptrap -- 1. Introduction -- 2. The ring of translation invariant symmetric polynomials -- 3. Haldane's conjecture -- 4. Proof of Theorem 2.1 -- Acknowledgements -- References -- Additional Gradings in Khovanov Homology V.O. Manturov -- 1. Introduction -- 1.1. Acknowledgments -- 2. Preliminaries: Virtual knots, Kau man bracket, atoms, and Khovanov homology -- 2.1. Atoms and Knots -- 2.2. Usual Khovanov homology -- 3. Bourgoin's twisted knots. Additional gradings -- 4. Additional grading: the general case -- 4.1. Explanation for the second and the third moves -- 5. More gradings -- more examples -- 5.1. Examples -- 5.2. Braids.

5.3. Further gradings -- 6. Khovanov's Frobenius theory -- 6.1. Yet another de nition of the Khovanov homology -- 6.2. Khovanov Frobenius theory modulo Z2 in the general case -- 6.2.1. The Z2-case -- 7. Gradings or ltrations? The spectral sequence -- 8. Applications -- 9. The relation to other papers -- References -- Towards the Large N Duality between the Chern-Simons Gauge Theory and the Topological String Theory P. Peng -- 1. Introduction -- 2. Quantum Group Invariants -- 2.1. Partition and symmetric function -- 2.2. Quantum group invariants of links -- 3. Large N CS/TS duality -- 4. Known results -- 4.1. Marino-Vafa formula -- 4.2. Topological vertex theory -- 4.3. Labastida-Marino-Ooguri-Vafa Conjecture -- 4.4. U(N) Chern-Simons gauge theory -- 4.5. Uniqueness of the cut-and-join system -- 5. Concluding remarks -- References -- On the Heegaard Genera of 3-manifolds Containing Nonseparating Surfaces R. Qiu and F. Lei -- 1. Introducion -- 2. An extension of Schultens's Lemma -- 3. The proof of Theorem 1 -- References -- A Geometric Categori cation of Representations of Uq(sl2) H. Zheng -- 1. Introduction -- 2. Quantum sl -- 3. A toy model: simple Uq(sl2)-modules -- 4. Intersection complex -- Acknowledgement -- References -- Part B Xiao-Song Lin's Unpublished Papers -- Editors' Notes -- Markov Theorems for Links in 3-Manifolds -- 1. Coxeter groups and their corresponding braid groups -- 2. Study links via open book decompositions -- 3. The case of S2 S1 -- References -- Vertex Models, Quantum Groups, and Vassiliev's Knot Invariants -- 1. Introduction -- 2. The quantum group U and its finite dimensional representations -- 3. Vertex models and quantum groups -- 4. VBL-functionals and their states models -- References -- Knot Invariants and Iterated Integrals -- 1. The associator KZ -- 2. Non-associative tangles -- 3. Some calculations.

References.