Senior Honors Thesis: Symplectic Geometry: Reduction, Convexity, and Unimodularity

Abstract:

In this undergraduate thesis, I will discuss the structural and combinatorial aspects of symplectic geometry, focusing on the Marsden-Weinstein-Meyer Theorem, the Atiyah-Guillemin-Sternberg Theorem, and Delzant's classification of symplectic toric manifolds. I begin with preliminaries covering symplectic manifolds, compatible triples (ω, g, J), Morse-Bott theory, and Hamiltonian actions, including illustrative examples like the action of Lie subgroups and the canonical action on C^d with the Fubini-Study form. Applications to spectral theory and geometric representation theory will also be discussed. Finally, I will conclude with Weinstein’s generalization of the convexity theorem for semisimple Lie group actions.