Some results on perverse sheaves and Bernstein-Sato polynomials

The first part of this thesis concerns intersection cohomology sheaves on a smooth projective variety with a torus action that has finitely many fixed points. Under some additional assumptions, we consider tensor products of intersection cohomology sheaves on a Bia\l{}ynicki-Birula stratification of the variety. We give a formula for the hypercohomology of the tensor product in terms of the tensor products of the individual sheaves, as well as the cohomology of the variety. We prove a similar result in the setting of equivariant cohomology. In the second part of this thesis, we study the Bernstein--Sato polynomial, or the $b$-function, which is an invariant of singularities of hypersurfaces. We are interested in the $b$-function of hyperplane arrangements of Weyl arrangements, which are the arrangements of root systems of semi-simple Lie algebras. It has been conjectured that the poles of the local topological zeta function, which is another invariant of hypersurface singularities, are all roots of the $b$-function. Using the work of Opdam and Budur--Musta\c{t}\u{a}--Teitler, we prove this conjecture for all Weyl arrangements. We also give an upper bound for the $b$-function of the Vandermonde determinant, which cuts out the Weyl arrangement in type $A$.