Diffeological Symplectic Frobenius Reciprocity and the Co-Moment Map Portrait of Quantum Hydrodynamics

This thesis addresses two problems in symplectic geometry and mathematical physics, connected by the common framework of Hamiltonian spaces. The first concerns a symplectic analogue of Frobenius reciprocity, which is realized as a bijection between certain symplectically reduced spaces that are not necessarily manifolds. To address the singularities arising in such quotients, we employ diffeology, a generalization of classical differential geometry that provides a natural setting for treating non-smooth spaces. Motivated by a conjecture in a past work, we show that the symplectic Frobenius reciprocity holds as a diffeomorphism between diffeological spaces, preserving the reduced forms they may carry. This raises the foundational question of when such forms exist. We provide new sufficient conditions, showing that local freeness, or strictness, or properness of the group action guarantees their existence. Parallel results are obtained for prequantum spaces. The second problem originates in quantum hydrodynamics. Recent work has shown that the Madelung transform defines a moment map correspondence between wave functions and the cotangent bundle of densities. This correspondence, however, breaks down on nodal lines, where the probability density vanishes and the phase is undefined. We seek analogous relations that overcome this limitation by building on the Clebsch geometry of the probability current, resulting in a network of symplectic manifolds naturally associated with both quantum-mechanical and hydrodynamical structures. These manifolds are connected by moment maps, and their relations remain valid when restricted to the nodal line, thereby extending the geometric interplay between quantum mechanics and fluid dynamics into this singular regime.