In this talk we give a brief introduction to Equivariant intersection theory and explain the localization formula. We will derive a version of Bott's residue formula using the localization theorem and compute an example to illustrate its use in enumerative geometry.

In noncommutative geometry, a framework in classical geometry need not have a trivial generalization. In my defence, I will introduce pseudo-Riemannian calculus of modules over noncommutative algebras in order to investigate as to what extent the differential geometry of classical Riemannian manifolds can be extended to a noncommutative setting. In this framework, it is possible to prove an analogue of the Levi-Civita theorem, which states that there exists at most one connection, which satisfies torsion-free condition and metric compatible condition on a given smooth manifold. More significantly, the corresponding curvature operator has the same symmetry properties as in the classical curvature tensors. We consider a pseudo-Riemannian calculus over the noncommutative 3-sphere and the noncommutative 4-sphere to explicitly determine the torsion-free and metric compatible connection, and compute its scalar curvature. Lastly, I will also prove a Gauss-Bonnet-Chern type theorem for the noncommutative 4-sphere, which is the main result of my main thesis.