Quivers, representation theory, non-commutative symplectic geometry, stratifications and singular symplectic quotients

In this dissertation we use ideas from the representation theory of quivers and noncommutative symplectic geometry to study problems in singular reduction theory: Symmetry, Singularity, Stratification of Singular symplectic spaces and their embeddings. V. Ginzburg proved in [21] that: Smooth affine quiver varieties can be embedded as coadjoint orbits in the dual of an appropriate infinite dimensional Lie algebra and in addition showed the existence of an infinitesimal transitive action of the Lie algebra on the smooth affine quiver varieties. Our main Theorems give a generalization of this result to singular quiver varieties, using stratifications, orbital convexity, saturation and techniques from R. Sjamaar's Holomorphic Slice Theorem [72]. We also prove Singular Reduction Theorem in this context. Furthermore, we generalize to the singular case a result of W. L. Gan and V. Ginzburg [20]: Showing that solving the Maurer-Cartan equation is equivalent to constructing Hamiltonian reduction and that any differential graded Lie algebra equipped with an even non-degenerate invariant bilinear form gives rise to modular stacks with symplectic structures.