A Delzant-Type Correspondence For Hamiltonian SO(3) actions by Theo Mollano '25

A Delzant-Type Correspondence For Hamiltonian SO(3) actions by Theo Mollano ’25, Wednesday April 23, 1:00 – 1:50pm, North Science Building 113, Wachenheim, Mathematics Thesis Defense

Physics and mathematics today lean into the power of symmetry principles. According to Nöther’s Theorem, every continuous symmetry of the action in a physical system corresponds to a conserved quantity, known as its Nöther charge. For example, in physics, the Nöther charge associated with rotational invariance is angular momentum. In the framework of symplectic geometry, we characterize a symmetry on a manifold $M$ by a Lie group action $G circlearrowright M$. Much like Nöther’s Theorem, under certain conditions, a Lie group action $G$ gives rise to a moment map $J$, which generalizes the notion of angular momentum.

Given a Lie group action $G$, to what precise extent does its moment image determine the original space? When $G = T^{n}$ is a toric action, Delzant showed that there is a bijective correspondence between symplectic manifolds and Delzant polytopes, which are themselves realized as the images of moment maps.

The simplest compact and connected non-Abelian group is the special orthogonal group $mathrm{SO}(3)$. The moment images for (compact and connected) Hamiltonian $mathrm{SO}(3)$-spaces—necessarily $mathrm{SO}(3)$-invariant subsets of $ mathfrak{so}(3)^* cong mathbb{R}^3$—can fall into one of four types: the origin, a sphere, a closed ball, or a closed hollow ball. It is easy to construct examples showing that all four possibilities do occur. But again: to what extent do these moment images determine the original space? What is a Delzant correspondence for Hamiltonian $mathrm{SO}(3)$-spaces, if there is even one at all? We show that, under certain conditions, a Delzant-type correspondence can be recovered for Hamiltonian $mathrm{SO}(3)$ actions.