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An addendum to symmetry regularization . Finally, I can write a very short paper. The key takeaway is that while the Poincare group provides a crucial hint, imagination is still needed for the seemingly impossible task of embedding the zero-energy surface (odd dimensional) as a dense open set of a co-adjoint orbit (even dimensional). Here, the imagination involves replacing the zero-energy surface with its space of co-normal vectors, which appear identical when observed from a distance perspective.
On Fock Transform and Moser Regularization Map. The main contribution lies in the characterization of the Moser regularization map. The article focuses on clarification, simplification, and characterization. We firmly believe that, for a subject as elegant and significant as the Kepler problem, every aspect should be presented in the clearest and simplest manner possible.
Ligon-Schaaf Map Revisited, J. Math. Phys. Volume 64, 122902 (2023). This article deals with the regularization for Kepler motions, a topic that has been explored in the past, but I have taken it a step further by revealing some mathematical signatures of relativity theory in the Kepler problem, including time-like geodesics. A duality emerges out of this work and it might be named as de Sitter - Anti-de Sitter duality.

A. Algebra

Legendre Transform, Hessian Conjecture and Tree Formula, Appl. Math. Lett. 19 (2006), 503-510. The Jacobian conjecture is one of the most fundamental open problems in mathematics, and has attracted the attention of many researchers due to its deceptively simple statement. In my attempt to solve this conjecture, I introduced the concepts of Legendre transform and Hessian conjecture. The Hessian conjecture states that a polynomial whose Hessian is a nonzero constant must have a polynomial Legendre transform. Although my attempt to solve the Jacobian conjecture was unsuccessful, my work attracted the attention of experts in the field. In my paper, I showed the equivalence of the Hessian conjecture with the Jacobian conjecture and derived a tree formula for the Legendre transform using the technique of Feynman diagrams. This was the first time that these concepts were introduced and applied in this context.

B. Geometric Topology

The Stability Theorem for Smooth Concordance Imbeddings (1993, unpublished thesis). In the field of geometric topology, a natural sequence of homotopy functors arises, denoted by A, B, and so on, with the first one being the Waldhausen A-functor. My advisor T. Goodwillie developed calculus for homotopy functors and computed the derivative for the A-functor in Waldhausen's K-theory of spaces. This allowed him to determine the A-functor, up to a constant, in a much more explicit form. My research focused on computing the derivative of the next functor in the sequence. One surprising result, as noted by Goodwillie, was the appearance of the module group in my answer. This raises the question of whether elliptic cohomology is related to the B-functor in a similar way that topological K-theory is related to the A-functor.

C. Low Dimensional Topology

Bracket models for weight systems and the universal Vassiliev invariants, Topology Appl. 76 (1997), no. 1, 47--60. When we view the invariants of knots and links as functions, the weight systems can be seen as their derivatives. L. Kauffman made the groundbreaking discovery that the Jones polynomial invariant has a simple bracket model interpretation. Building on this, I realized that the weight system for HOMFLY and Kauffman polynomial invariants also have a simple bracket model interpretation. This discovery has a few applications, including the derivation of the first low bound for the dimension of the weight system.
(Joint with C.H. Taubes) "SW=Milnor torsion, Math. Res. Lett. 3 (1996), no. 5, 661--674. The Seiberg-Witten invariants are analytically defined smooth invariants of smooth, compact, oriented 4-manifolds. On the other hand, Milnor torsion is a topologically defined improved version of the multi-variable Alexandar polynomial invariant of compact, oriented 3-manifolds with a possible toroidal boundary. In our research, we discovered the formula:
SW(X x S^1) = Milnor Torsion (X).
This formula holds for any compact, oriented 3-manifold X with a positive 1st Betti number and possible toroidal boundaries. These findings have implications for the study of smooth, compact, oriented 4-manifolds and their relationship with compact, oriented 3-manifolds.

D. 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. This article presents a formulation of Tulczyjew's mechanics for particles in gauge fields, including both abelian and non-abelian fields. As a result, we demonstrate 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. While this was initially formulated by S. K. Wong in the Hamiltonian/Poisson setting (1970) and later by S. Sternberg in the Hamiltonian/symplectic setting (1978), our approach offers a fresh perspective on this important problem.

E. Mathematical Physics

On Quantum Hall Effects in High Dimensions: The quantum Hall effect is a fundamental macro quantum effect that occurs in two dimensions. In my research, I explored the mathematical aspects of this phenomenon. While some researchers have suggested that quantum Hall effects in high dimensions may have real-world implications, I believe that this claim is far-fetched. I was surprised to discover that quantum Hall effects can exist in all even dimensions, despite the conventional wisdom that they exist only in certain special even dimensions.
On Dirac and Yang Monopoles: The Dirac monopoles are a well-known concept in the theoretical physics community, and their generalization to dimension five by C. N. Yang is less well-known. In my research, I developed a conceptual framework for understanding the Dirac and Yang monopoles, which led to an automatic extension of Yang's work to all odd dimensions. This has the potential to deepen our understanding of these important concepts and their applications in theoretical physics.

F. Kepler Problem, Symplectic Geometry, and Representation Theory

0. Conference Papers and Lectures

Kepler Problem and Jordan Algebras. In: Kielanowski P., Odzijewicz A., Previato E. (eds) Geometric Methods in Physics XXXVI. Trends in Mathematics.
Tulczyjew Mechanics for Particles in Gauge Fields, Banach Center Publications, Volume 110, Warszawa 2016, 193-199.
The Classical Magnetized Kepler Problems in Higher Odd Dimensions, JGSP 32 (2013) 15-23
Generalized Kepler Problems and Euclidean Jordan Algebras, In: Geometry, Integrability and Quantization, I. Mladenov, Guowu Meng and Akira Yoshioka (Eds), Avangard Prima, Sofia 2016, pp 72 - 94

1. 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 Schrodinger 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. There emerges a correspondence between representations: for each half of an integer u, the irreducible reps. of Spin(2k) with highest weight (|u|, ..., |u|, u) corresponds to the irreducible unitary highest weight reps. of Spin(2, 2k+2) with highest weight (-k-|u|, |u|, ..., |u|, u). The correspondence is referred to as charge-model correspondence. The gauge group is Spin(2k).
Generalized MICZ-Kepler Problems and Unitary Highest Weight Modules -- II , J. London Math. Soc. 2010 81(3): 663-678. There emerges a correspondence between representations: for u = 0 or 1/2, the irreducible reps. of Spin(2k-1) with highest weight (u, ..., u) corresponds to the irreducible unitary highest weight reps. of Spin(2, 2k+1) with highest weight (-k-u+1/2, u, ..., u). In this charge-model correspondence, there is only one allowable magnetic charge (case u = 1/2). Case u = 0 means no magnetic charge.
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

In all cases here the charge-model correspondence is precisely the local Theta correspondence.

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

Here I developed a general theory of integrable models based on Euclidean Jordan algebra. First of all, there is a universal model for each Euclidean Jordan algebra. Secondly, for each suitable Poisson realisation of the conformal algebra of the Jordan algebra, we have a classical integrable model of Kepler type. Thirdly, for each suitable operator realisation of the conformal algebra of the Jordan algebra, we have a quantum integrable model of Kepler type. This general theory unites the Kepler problem with the oscillator problem under the same roof, just as abstract integration theory does to Riemann integral and Series. A surprising consequence from this theory is the discovery of the mathematical signatures (such as light cone and Lorentz transformations) of the relativity theory in the non-relativistic models.

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.
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
(With S. Bouarroudj) The classical dynamic symmetry for the U(1)-Kepler problems, J. Geom. Phys. 124 (2018), 1-15
(With S. Bouarroudj) The classical dynamic symmetry for the Sp(1)-Kepler problems, J. Math. Phys. 58, 091702 (2017)
Magnetized Kepler Manifolds and their Quantization, Journal of Geometry and Physics 159 (2021) 103891

6. Symmetry Regularizations and Bound/Unbound Duality

Ligon-Schaaf Map Revisited, J. Math. Phys. Volume 64, 122902 (2023). This article deals with the regularization for Kepler motions, a topic that has been explored in the past, but I have taken it a step further by revealing some mathematical signatures of relativity theory in the Kepler problem, including time-like geodesics. A duality emerges out of this work and it might be named as de Sitter - Anti-de Sitter duality.

G. Fundamental Physics

It is of course speculative because the basic assumption cannot be directly tested. My starting point is this: 1) replace the Lorentz structure of the space-time by the more fundamental Euclidean Jordan algebra structures, 2) assume that quarks and leptons are grouped together by an approximate symmetry group as a single particle. Here is a few logical consequences:

The absolute spacetime assumed by Minkowski also has an absolute time direction.
By matching with the real world data, we conclude that Euclidean Jordan algebra must be the Dirac type Jordan algebra Γ(10), so the space time is a Lorentz manifold of dimension 1 + 10.
There are infinite many fundamental matter generations, consequently there is a 5th fundamental force which violates CP symmetry or time reversal symmetry. The 5th fundamental force and the gravity are mixed together just as weak force and electric force are mixed together. Let us call the unified force the gravity-5th force.
Dark matter exists, but only participates in the gravity-5th force. If true, the current detection of dark matter based on weak force cannot find dark matter.