An Ky Duy Nguyen

Decomposability of Quadratic Killing Tensors on Symmetric Spaces

My current PhD research studies the decomposability of quadratic Killing tensors on Riemannian symmetric spaces: a problem about whether higher-order conserved quantities of the geodesic flow are genuinely new, or already forced by the classical symmetries of the space. In this setting, decomposability means that every quadratic Killing tensor can be generated from Killing vector fields, so that apparent hidden symmetries reduce to the visible geometry of the isometry group.

My work develops a structural and computational approach to this problem across major families of symmetric spaces. Building on the top-slot model of Vladimir Matveev and Yuri Nikolayevsky, I construct novel Lie-theoretic reductions, induction arguments through totally geodesic submanifolds, and exact algebraic computations. In this new framework, the overdetermined Killing tensor equation is transformed into a finite-dimensional decomposability problem, where geometric structures and rigorous SageMath computer algebra come together to give a complete and affirmative resolution of the problem across ALL compact classical symmetric spaces.

Wolf’s Homogeneity Conjecture and Clifford-Wolf Quotients

Wolf’s Homogeneity Conjecture concerns Riemannian quotients of homogeneous spaces by discrete groups of isometries of constant displacement. It proposed that if every deck transformation is a Clifford-Wolf translation, then the quotient should itself remain homogeneous. Xu and Deng recently disproved the conjecture through a counterexample on the compact Lie group Sp(2).

My current work investigates the broader geometric mechanism behind this phenomenon on compact Lie groups and develops infinite families on Sp(n) and SU(2k). For every prescribed cyclic order N ≥ 3, the construction uses suitable left-invariant metrics close to a bi-invariant metric for which every element of the cyclic deck group has constant displacement, while the resulting quotient is non-homogeneous.

The proof brings together constant-length Killing fields, a uniform persistence theorem for globally minimising geodesics under metric deformation, the structure of connected isometry groups of left-invariant metrics, covering-space arguments, and exact centraliser-stabiliser dimension calculations. The project is currently a working manuscript undergoing expert mathematical review.