Topology and Physics

The following sections are included:

• Foreword

• Preface

• Short Biography of Lin

• Mathematics of Lin

• Organizing Committees

• List of Participants

• Program

• Welcome Speech of Weiping Zhang

• Speech of Boju Jiang

• Contents

In this paper, we derive a partial result related to a question of Yau: “Does a simply-connected complete Kähler manifold M with negative sectional curvature admit a bounded non-constant holomorphic function?”

Main Theorem. Let M²ⁿ be a simply-connected complete Kähler manifold M with negative sectional curvature ≤−1 and S_∞(M) be the sphere at infinity of M. Then there is an explicit bounded contact form β defined on the entire manifold M²ⁿ.

Consequently, if M²ⁿ is a simply-connected Kähler manifold with negative sectional curvature −a² ≤ sec_M ≤ −1, then the sphere S_∞(M) at infinity of M admits a bounded contact structure and a bounded pseudo-Hermitian metric in the sense of Tanaka-Webster.

We also discuss several open modi_ed problems of Calabi and Yau for Alexandrov spaces and CR-manifolds.

The following sections are included:

• Introduction

• Jones representations
  • Braid statistics
  • Generic Jones representation of the braid groups
  • Unitary Jones representations
  • Uniqueness of Jones-Wenzl projectors

• Diagram TQFTs for closed manifolds
  • "d-isotopy", local relation, and skein relation
  • Picture classes
  • Skein classes
  • Recoupling theory
  • Handles and S-matrix
  • Diagram TQFTs for closed manifolds
  • Boundary conditions for picture TQFTs
  • Jones-Kauffman skein spaces

• Morita equivalence and cut-paste topology
  • Bimodules over picture category
  • Cutting and paste as Morita equivalence
  • Annualization and quantum double

• Temperley-Lieb-Jones categories
  • Annular Markov trace
  • Representation of Temperley-Lieb-Jones categories
  • Rectangular Tempeley-Lieb-Jones categories for low levels
  • Annular Temperley-Lieb-Jones theories for low levels
  • Temperley-Lieb-Jones categories for primitive 4rth roots of unity
  • Temperley-Lieb-Jones categories for primitive 2rth root or rth root of unity, r odd

• The definition of a TQFT
  • Refined labels for TQFTs
  • Anomaly of TQFTs and extended manifolds
  • Axioms for TQFTs
  • More consequences of the axioms
  • Framed link invariants and modular representation
  • Verlinde algebras and Verlinde formulas

• Diagram and Jones-Kauffman TQFTs
  • Diagram TQFTs
  • Jones-Kauffman TQFTs

• WRT and Turaev-Viro SU(2)-TQFTs
  • Flagged TLJ categories
  • Turaev-Viro Unitary TQFTs
  • WRT Unitary TQFTs

• Black-White TQFTs
  • Black-white TLJ categories
  • Labels for black-white theories
  • BW TQFTs

• Classification and Unitarity
  • Classification of diagram local relations
  • Unitary TQFTs
  • Classification 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

Some entangled states can be generated based on Temperley-Lieb algebra. The dynamics in the entangling degree (θ in this paper) space is set up through the Yang-Baxterization and the Berry phase is found in the Yang-Baxter approach. The Yang-Baxterization has been presented for A(u) and B(u), i.e. the 2-dimensional braid matrices related to 2-state anyon model. We also show the equivalence between the usual 4 × 4 Ř(u)-matrix and the set of A(u) and B(u).

We establish the second part of Milnor's conjecture on the volume of simplexes in hyperbolic and spherical spaces. A characterization of the closure of the space of the angle Gram matrices of simplexes is also obtained.

We use an action, of 2l-component string links on l-component string links, defined by Habegger and Lin, to lift the indeterminacy of finite type link invariants. The set of links up to this new indeterminacy is in bijection with the orbit space of the restriction of this action to the stabilizer of the identity.

Structure theorems for the sets of links up to C_n-equivalence and Self-C_n--equivalence are also given.

In this paper we investigate a kind of generalized Ricci flow which possesses a gradient form. We study the monotonicity of the given function under the generalized Ricci flow and prove that the related system of partial differential equations are strictly and uniformly parabolic. Based on this, we show that the generalized Ricci flow defined on an n-dimensional compact Riemannian manifold admits a unique short-time smooth solution. Moreover, we also derive the evolution equations for the curvatures, which play an important role in our future study.

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. The simplest case of these models is the Fibonacci model, itself universal for quantum computation. We here formulate these braid group representations in a form suitable for computation and algebraic work. In particular, we give quantum algorithms for computing colored Jones polynomials and the Witten-Reshetikhin-Turaev invariant of three-manifolds.

In this article, we obtain a universal method to compute the Gromov-Witten type invariants using the localization technique. This method can be applied to any natural cohomology class on the moduli space of curves . As applications, we illustrate a new proof of the Witten's conjecture as well as the proof of Mariño-Vafa formula.

The A – B slice problem is a reformulation of the topological 4–dimensional surgery conjecture in terms of decompositions of the 4–ball and link homotopy. We show that link groups, a recently developed invariant of 4–manifolds, provide an obstruction for the class of model decompositions, introduced by M. Freedman and X.-S. Lin. This unifies and extends the previously known partial obstructions in the A – B slice program. As a consequence, link groups satisfy Alexander duality when restricted to the class of model decompositions, but not for general submanifolds of the 4–ball.

In this paper, we construct a knot invariant with SL(2, ℂ) representations by using the fundamental L²–representation of the fundamental group of a knot complement, which may be thought of as an twisted L²-Alexander(–Conway) invariant of the knot in S³. Both L²–Alexander and L²-Alexander-Conway invariants define functions from the representation space of the knot group to nonnegative real numbers, and descend to functions from the SL(2, ℂ) character variety of the knot group. We also show that the twisted L²-Alexander invariant is a L²-Reidemeister torsion twisted by the SL(2, ℂ) representation. The L²-Alexander (and L²-Alexander-Conway) invariant twisted by GL(n, ℂ) representations of the knot group is given.

The Faddeev knots are the energy minimizers topologically stratified by the Hopf invariant in the Faddeev quantum field theory model governing the interaction of baryons and mesons. Recent progress made on the existence theory indicates that two growth laws expressed in terms of the Faddeev energy and the Hopf invariant are essential. The first one, called the Substantial Inequality, describes the energy splitting pattern in a minimization process and gives valuable information on compactness or convergence of a minimizing sequence. The second one, which will be shown to be universally valid for a broad range of energy functionals, ensures that knotted structures are preferred over multiple-soliton structures in high Hopf numbers.

We establish an isomorphism between the ring of translation invariant symmetric polynomials in n variables and the full polynomial ring in n − 1 variables, over any field of characteristic 0. In addition, we give a counterexample to a conjecture of Haldane³ regarding the structure of translation invariant symmetric polynomials. Our motivation is the fractional quantum Hall effect, where translation invariant (anti)symmetric complex polynomials in n variables characterize n-electron wavefunctions.

The main goal of the present paper is to construct new invariants of knots with additional structure by adding new gradings to the Khovanov complex. The ideas given below work in the case of virtual knots, closed braids and some other cases of knots with additional structure. The source of our additional grading may be topological or combinatorial; it is axiomatised for many partial cases. As a by product, this leads to a complex which in some cases coincides (up to grading renormalisation) with the usual Khovanov complex and in some other cases with the Lee-Rasmussen complex.

The grading we are going to construct behaves well with respect to some generalisations of the Khovanov homology, e.g., Frobenius extensions. These new homology theories give sharper estimates for some knot characteristics, such as minimal crossing number, atom genus, slice genus, etc.

Our gradings generate a natural filtration on the usual Khovanov complex. There exists a spectral sequence starting with our homology and converging to the (graded version associated with) usual Khovanov homology.

We survey some recent developments in the large N duality between the Chern-Simons gauge theory and the topological string theory. Some related problems and its applications to different areas of mathematics are also discussed. We dedicate this paper to the memory of Professor X.-S. Lin.

Let M be a compact orientable 3-manifold which contains a non-separating closed incompressible surface F. Let M′ = M − η(F) where η(F) is an open regular neighborhood of F in M. In the paper we show that if M′ has a Heegaard splitting V′ ∪_S′ W′ with d(S′) > 2g(M′), then g(M) ≥ g(M′) − g(F). Furthermore, if F is a torus, then g(M) ≥ g(M′)+1.

The purpose of this short note it to give an elementary description of a recent work¹⁸ of the author on the geometric categorification of tensor products of U_q(sl₂)-modules. That is, we show the main ideas without going into specific techniques.

The following sections are included:

• Editors' Notes

• Markov Theorems for Links in 3-Manifolds

• Vertex Models, Quantum Groups, and Vassiliev's Knot Invariants

• Knot Invariants and Iterated Integrals

• Invariants of Legendrian Knots

• The Motion Group of the Unlink and its Representations

The following sections are included:

• Lin Award Information

• Selected Speeches at the Funeral

• Freedman's Writings on Lin

• Jian-Pin He's Speech