On Restricted Tangent Bundles of Grassmannian, and Betti Numbers of The moduli of Stable Sheaves on P^2

This thesis is based on work done on two different problems. The first problem is regarding restricted tangent bundles of the Grassmannian to rational curves. Let n >= 4, 2 <= r <= n-2 and e >= 1. We show that the intersection of the locus of degree e morphisms from P^1 to G(r, n) with the restricted universal sub-bundles having a given splitting type and the locus of degree e morphisms with the restricted universal quotient-bundle having a given splitting type is non-empty and generically transverse. As a consequence, we get that the locus of degree e morphisms from P^1 to G(r, n) with the restricted tangent bundle having a given splitting type need not always be irreducible. The second problem is regarding the Betti numbers of the moduli space of sheaves on the projective plane. Let r >= 2 be an integer, and let a be an integer coprime to r. We show that if c_2 >= n+ (r-1)a^2 /2r + (r^2 + 1)/2 , then the 2nth Betti number of the moduli space M_{P^2,O_{P^2} (1)}(r,O_{P^2}(a), c_2) stabilizes.