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My Ph.D work was in geometric topology, specifically in the group actions on topological manifolds. Since coming to Hong Kong, I turned my attention to other fields. This resulted in my work in combinatorics. Together with my colleague Dr. Bei-Fang Chen, we established the comprehensive theory of Eulerian stratification. After this, I started collaboration with Dr. Yong-Chang Zhu and Jiang-Hua Lu on Hopf algebra on one hand and with Dr. Ji-Shan Hu on Painleve analysis on the other hand. The former is a comprehensive set-theoretical theory of Hopf algebras and the associated solutions of the Yang-Baxter equation. The later is a discovery of a new analytical approach in the Painleve analysis that may lead to many fruitful consequences.

Geometric Topology

I am able to prove some cases of the equivariant periodicity for the homotopy classification of the group actions on topological manifolds. The work suggests that the equivariant periodicity should hold for twice of any complex representation. The technique involved may also be used to construct the functoriality and the induction theory for the homotopy classification, which may be very useful for computing the homotopy classifications. Technically, the work is interesting in using non-manifold constructions (the stratified technique) to prove certain facts about manifolds.

1. The Periodicity in Stable Equivariant Surgery. Comm. in Pure and App. Math. 46(1993) 1013-1040 dvi ps pdf

2. Equivariant Periodicity in Surgery for Actions of Some Nonabelian Groups. In: Proceedings of 1993 Georgia Topology Conference, ed. W. Kazez, 478-508, AMS and International Press 1997 dvi ps pdf

3. Equivariant Periodicity for Abelian Group Actions. Adv. in Geom. 1(2001) 49-70 (with S. Weinberger) dvi ps pdf

Combinatorics

It was shown by Victor Klee in 1964 that the Euler equation and the Dehn-Sommerville equations are the only linear conditions on simplicial complexes that are manifolds from the viewpoint of the Euler characteristic. The stratified technique in my topology work is well suited for extending this theory to general polyhedra. We showed the Eulerian condition is essentially topological (rather than combinatorial) and grdually set up a theory from the simple case of manifold with boundary to the most general case of Eulerian stratification. Our viewpoint also leads to a generalization of angle-sum formula.

1. Linear Conditions on the Number of Faces of Manifolds with Boundary. Adv. in App. Math. 19(1997) 144-168 dvi ps pdf

2. f-vectors of Polyhedra. Geom. Dedicata 68(1997) 21-28 dvi ps pdf

3. The Geometric Cone Relations for Simplicial and Cubical Complexes. Discrete Math 183(1998) 39-46 dvi ps pdf

4. Eulerian Stratified Spaces. Adv. in App. Math. 21(1998) 22-57 dvi ps pdf

5. Singularities from Eulerian Viewpoint. Trans. Math. Inst. Steklova, vol.221 (1998), 305-319 dvi ps pdf

6. Eulerian 2-strata Spaces. J. of Comb. Theory A 85(1999) 1-28 dvi ps pdf

(all except 2. are joint with B.F. Chen)

Hopf Algebra

The first paper is what we believe the right definition of the homogeneous spaces for Hopf algebra actions. The construction is motivated by Drinfel?d identification of homogeneous spaces for Lie bialgebras and transversal lagrangians in the Drinfel?d double. My collaborator further used the theory to obtain some interesting results in Hopf algebra. The subsequent papers are based on the simple idea that the linearization of set-theoretical analogues of linear algebraic structures will have bases so that all the structure constants relative to the bases are non-negative (in fact either 0 or 1). We completely classified the so-called Hopf algebras with positive bases and found them to be set-theoretical. Then we further classified the positive quasi-triangular structures on such Hopf algebras. Since such structures are again set-theoretical, they lead to a comprehensive construction of set-theoretical solutions of the Yang-Baxter equation, which includes all the earlier constructions as special cases.

1. Stabilizer for Hopf Algebra Actions. Comm. in Alg. 26(1998) 3885-3898 dvi ps pdf

2. On Hopf Algebras with Positive Bases. J. Algebra 237(2001) 421-445 dvi ps pdf

3. Quasi-triangular Structures on Hopf Algebra with a Positive Basis. In: New Trends in Hopf Algebra Theory, ed. N. Andruskiewitsch, W.R.F. Santos, A-J. Schneider, 339-356, Contemporary Math vol.267, AMS Press 2000 dvi ps pdf

4. On Set-theoretical Yang-Baxter Equation. Duke Math. J. 104(2000) 1-18 dvi ps pdf

(1. is joint with Y.C. Zhu, all others are joint with J.H. Lu and Y.C. Zhu)

Integrable System

A fundamental problem of the theory of differential equations is the relation between the integrability and the Painleve property (all movable singularities are single-valued) of solutions. The strong belief on such a relation is behind the Painleve test, the most effective method for detecting integrability since it was first used more than a century ago. However, the Painleve test is only a formal algebraic process and the reason for it to have important analytical consequences has been heuristic at best. We discovered an algorithm that is equivalent to the Painleve test but yields a regular system (which we call the mirror system) at the end. We believe this is the analytical ingredient missing from the Painleve test and could be used to rigorously establish a relation between the integrability and Painleve analysis.

1. The Mirror Systems of Integrable Equations. Studies in App. Math. 104(2000) 67-90 dvi ps pdf

2. Singularity Analysis for Integrable Systems by their Mirrors. Nonlinearity 12(1999) 1531-1543 dvi ps pdf

3. Local Analyticity of Solutions in the Painleve Test. In: Proceedings of the workshop on Nonlinearity, Integrability and all that: Twenty years after NEEDS '79, ed. M Boiti, et. al.,146-152, World Scientific 2000

4. Analytical Aspects of the Painleve Test. preprint

5. Symplectic Structure of the Painleve Test. J. Nonlinear Math. Phys. 8(2001) 145-148 dvi ps pdf

6. Mirror Transforms of Hamiltonian System. Physica D 152-153(2001) 110-123 dvi ps pdf

7. An Elementary and Direct Proof of the Painleve Property for P1, P2 and P4. Journal d'Analyse Mathematique 91(2003) 105-121 dvi ps pdf

8. A Link between two Fundamental Contributions of Kowalevski. In: The Kowalevski Property, CRM Proceedings & Lecture Notes, ed. V. B. Kuznetsov, 149-156, American Mathematical Society, Providence, RI, 2002

(6. is joint with J.S. Hu and T.L. Yee, all others are joint with J.S. Hu)

Other

The first is my Bachelor's thesis

1. The higher-order variation method in differential geometry and its applications. J. Fudan Univ. Natur. Sci. 24 (1985), no. 4, 453-458.

2. Extension of Convex Functions. preprint dvi ps pdf