Abstract: I shall first derive the semi-classical approximation of Feynman path integrals and give several examples. This involves regularized determinants of differential operators in terms of spectral zeta functions. This idea will then be applied to a supersymmetric quantum system defined by the Dirac operator of a closed spin manifold and after some non-trivial algebraic manipulations will lead to a proof the Atiyah-Singer index theorem.

Abstract: Failure of some (important) properties of a holomorphic mapping manifest themselves as degeneracies in the family of fibres of the mapping. Among these properties, openness and flatness are our main object of interest. The first goal in my thesis is to develop criteria such that first, they effectively (i.e., computationally) detect such degeneracies in the family of fibres, and second, they are applicable to the case of mappings with singular targets. For flatness, no such algorithms that work in the general setting of a singular target were known before. We prove that a mapping (with locally irreducible target) is flat (open) if and only if the fibre above the origin of the pullback by the blowing-up is not an (isolated) irreducible component. Algebraically, this flatness criterion reduces to the following straightforward prescription: compute the the local ring of the pullback, and check to see if the exceptional divisor is not a zero divisor. The second goal is to characterize different modes of such degeneracies. We take an invariant, called verticality index, as a gauge which measures the level of non-openness for mappings. We obtain some results about verticality index, specially on its behaviour and computation over singular targets.