Affine Stratifications and equivariant vector bundles on

of the Dissertation Affine Stratifications and equivariant vector bundles on the moduli of principally polarized abelian varieties by Anant Atyam Doctor of Philosophy in Mathematics Stony Brook University 2014 We explicitly construct a locally closed affine stratification of the coarse moduli space of principally polarized complex abelian four folds A4 and using [32], produce an upper bound for the coherent cohomological dimension and the constructible cohomological dimension ofA4, and make a conjecture on the cohomological dimension of Ag for any g. In the second part of the thesis we study a subring of K0(Ag) and study the Chern character map from this ring to the tautological ring of Ag which leads us to give a conjectural representation theoretic interpretation of the tautological ring of Ag.