This thesis investigates various rigidity and flexibility phenomena of convex projective structures on hyperbolic manifolds, particularly in dimension 3. Let M[superscipt n] be a finite volume hyperbolic n-manifold where [mathematical equation] and [mathematical symbol] be its fundamental group. Mostow rigidity tells us that there is a unique conjugacy class of discrete faithful representation of [mathematical symbol] into PSO(subscript n, 1). In light of this fact we examine when this representations can be non-trivially deformed into the larger Lie group of PGL[subscript n+1](R) as well as the relationship between these deformations and convex projective structures on M. Specifically, we show that various two-bridge knots do not admit such deformations into PGL[subscript 4](R) satisfying certain boundary conditions. We subsequently use this result to show that certain orbifold surgeries on amphicheiral knot complements do admit deformations.

(2013-08) Haque, Mohammad Moinul; Helm, David, doctor of mathematics

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In this thesis we construct an analogue in tropical geometry for a class of Schubert varieties from classical geometry. In particular, we look at the collection of tropical lines contained in the fan tropical plane. We call these tropical spaces "tropical Schubert prevarieties", and develop them after creating a tropical analogue for flag varieties that we call the "flag Dressian". Having constructed this tropical analogue of Schubert varieties we then determine that the 2-skeleton of these tropical Schubert prevarieties is realizable. In fact, as long as the lift of the fan tropical plane is in general position, only the 2-skeleton of the tropical Schubert prevariety is realizable.