In this thesis, we introduce a notion of asymptotic stability for a holomorphic vector bundle with a global holomorphic section on a projective manifold. We prove that the special metric on the bundle studied by Bradlow ...

In this thesis, we investigate the birational geometry of the space of rational curves in various homogeneous spaces, with a focus on the quasi-map compactification induced by the Quot and Hyperquot functors. We first study ...

In this thesis we apply techniques from the bordered and sutured variants of Floer homology to study Legendrian knots. First, given a front diagram for a Legendrian knot K in S₃ which has been split into several pieces, ...

Abstract: A fundamental problem about irreducible representations of a reductive Lie group G is understanding their restriction to a maximal compact subgroup K. In certain important cases, known as the discrete series, we ...

The classical Brill-Noether theorems count the dimension of the family of maps from a general curve of genus g to non-degenerate curves of degree d in projective space Pr. These theorems can be extended to include ramification ...

The Greene-Kleitman theorem says that the lengths of chains and antichains in any poset are intimately related via an integer partition, but very little is known about the partition [lambda](P) for most posets P. Our first ...