“Legendre Transform, Hessian Conjecture and Tree Formula”, Appl. Math. Lett. 19 (2006), 503-510. Jacobian conjecture is one of fundamental open problems in mathematics; its deceivingly simplicity has attracted many people to give it a try. This piece of work is a result of my failed attempt; though unsuccessful, it has attracted the attention of experts in the field. Legendre transform and Hessian conjecture (a polynomial whose Hessian is a nonzero constant must have a polynomial Legendre transform) are introduced here first time. Here, we also showed the equivalence of the Hessian conjecture with the Jacobian conjecture and derived a tree formula for the Legendre transform by using the technique of Feynman diagrams.
"The Stability Theorem for Smooth Concordance Imbeddings" (1993, unpublished thesis). From geometric topology, there arises a natural sequence of homotopy functors : A, B, .... with the first one being the Waldhausen A-functor. Goodwillie developed calculus for homotopy functors and computed the derivative for the A-functor in Waldhausen's K-theory of spaces, so that he can determine (up to a constant) the A-functor in a homotopically much more explicit form. My job was to compute the derivative of the next functor. One surprise as pointed out by Goodwillie is that the module group appears in my answer. Does that mean elliptic cohomology is related to the B-functor in a similar way that the topological K-theory is related to the A-functor?
"Bracket models for weight systems and the universal Vassiliev invariants", Topology Appl. 76 (1997), no. 1, 47--60. If we view the invariants of knots and links as functions, then the weight systems will be viewed as the derivatives of the invariants. It is a discovery of L. Kauffman that the Jones polynomial invariant has a simple bracket model interpretation. I realized that the the weight system for HOMFLY and Kauffman polynomial invariants has a simple bracket model interpretation, too. One application mentioned here is the first low bound for the dimension of weight system.
"(Joint with C.H. Taubes) "SW=Milnor torsion", Math. Res. Lett. 3 (1996), no. 5, 661--674. The Seiberg-Witten invariants of smooth compact oriented 4-manifolds are analytically defined smooth invariants. Milnor torsion is the topologically defined improved version of multi-variable Alexandar polynomial invariant of compact oriented 3-manifolds with a possible toroidal boundary. The formula we discovered is
SW(X x S^1) = Milnor Torsion (X)
for any compact oriented 3-manifold X with positive 1st Betti number and possible toroidal boundaries.
"Geometric construction of the Quantum Hall Effect in all even dimensions", J. Phys. A 36 (2003) 9415-9424. The quantum Hall effect is an interesting two-dimensional fundamental macro quantum effect, but what I did is purely mathematical, and I am not convinced that the quantum Hall effects in high dimensions is physics. It was believed that quantum Hall effects can exist only in certain special even dimensions, but I saw that it actually exists in all even dimensions.
A path integral derivation of $\chi\sb y$-genus, J. Phys. A 36 (2003), no. 4, 1083--1086. A direct path integral derivation of the Hirzebruch chi-y-genus formula is given here in details. This paper is rigorous only by the current standard of physics.
“Dirac and Yang monopoles revisited”, CEJP. 5 (2007), pp. 570-575. The Dirac monopoles are well-known in the theoretical physics community, what is less-known is their generalization by C. N. Yang to dimension five. I saw a conceptual way of looking into the Dirac and Yang monopoles, and this leads to an automatic extension of Yang's work to all odd dimensions.
The research here is an interplay between the super integrable system on the one hand and the unitary highest weight theory for real non-compact simple Lie groups of hermitian type on the other hand. Here are four surprises we have revealed: 1) The family of super integrable systems of Kepler type is much bigger than previously believed, 2) There is a simple geometric realizations for certain special family of unitary highest weight modules, 3) The correspondence between the magnetic charges and the Hilbert spaces of bound states resembles Howe's local theta-correspondence over real numbers, 4) The family of super integrable systems of Kepler type is intimately related to symmetric domains.
Initial Breakthrough
“MICZ-Kepler problems in all dimensions”, J. Math. Phys. 48, 032105 (2007). The Kepler problem is a fundamental physics problem, and the MICZ-Kelper problems are its magnetized cousins where the nucleus carries both electric and magnetic charge. The MICZ-Kepler problems were generalized to dimension five by Iwai in 1990, and were generalized to any dimension here.
"A generalization of the Kepler problem", Physics of Atomic Nuclei, Vol. 71, No. 5 (2008), 946-950. The Kepler problem admits various natural mathematical generalizations. Historically it is Schrödinger who started the 1st generalization. Since then, various further generalizations have appeared, which are all essentially done along three directions: dimension, curvature and magnetic charge. However, the full generalization along the afore-mentioned three directions is missing mainly because people thought magnetized Kepler Problems can only exist in dimensions 3 and 5. Here, we give the desired full generalization.
2. An "intrusion" into the representation theory
(With R. B. Zhang) "Generalized MICZ-Kepler Problems and Unitary Highest Weight Modules", J. Math. Phys. 52, 042106 (2011)
"Generalized MICZ-Kepler Problems and Unitary Highest Weight Modules -- II", J. London Math. Soc. 2010 81(3): 663-678.
"The Representation Aspect of the Generalized Hydrogen Atoms", Journal of Lie Theory 18 (2008), No. 3, 697-715. Here a simple algebraic characterization is found for the unitary highest weight modules that can be realized by the Hilbert space of bound states of a generalized MICZ-Kepler problem.
3. Local Theta correspondences over R
The O(1)-Kepler Problems, J. Math. Phys. 49, 102111 (2008).
The U(1)-Kepler Problems, J. Math. Phys. 51, 122105 (2010).
The Sp(1)-Kepler Problems, J. Math. Phys. 50, 072107 (2009).
4. Connection with Jordan Algebras
Euclidean Jordan Algebras, Hidden Actions, and J-Kepler Problems, J. Math. Phys. 52, 112104 (2011)
The Universal Kepler Problem, J. Geom. Symm. Phys. {\bf 36} (2014) 47-57. DOI:10.7546/jgsp-36-2014-47-57
Generalized Kepler Problems I: Without Magnetic Charges, J. Math. Phys. 54, 012109(2013)
Lorentz Group and Oriented MICZ-Kepler Orbits, J. Math. Phys. 53, 052901 (2012)
5. Connection with Symplectic Geometry
The Poisson Realization of $\mathfrak{so}(2, 2k+2)$ on Magnetic Leaves. J. Math. Phys. 54, 052902 (2013).
(With Z. Q. Bai and E. X. Wang) On the orbits of the magnetized Kepler problems in dimension $2k+1$. Journal of Geometry and Physics Volume 73, November 2013, Pages 260–269.
On the trajectories of O(1)-Kepler Problems. J. Math. Phys. 56, 052901(2015).
On the trajectories of U(1)-Kepler Problems. In: Geometry, Integrability and Quantization, I. Mladenov, A. Ludu and A. Yoshioka (Eds), Avangard Prima, Sofia 2015, pp 219 - 230.
On the trajectories of Sp(1)-Kepler Problems. Journal of Geometry and Physics, Volume 96, October 2015, Pages 123-132
F. Fundamental Physics
G. Geometric Foundation of Physics
Tulczyjew's Approach for Particles in Gauge Fields. J. Phys. A: Math. Theor. 48 (2015) 145201 doi:10.1088/1751-8113/48/14/145201. In this article, Tulczyjew's formulation of mechanics is worked out for particles in gauge fields (both abelian and non-abelian). As a consequence, it is demonstrated for the first time how the dynamics of a charged particle in the presence of a background non-abelian gauge field can be formulated in the Lagrangian setting. (This was formulated initially by S. K. Wong in the Hamiltonian/Poisson setting (1970) and later by S. Sternberg in the Hamiltonian/symplectic setting (1978). )