Qusay Al-Zamil (University of Manchester)
Let M be a compact oriented smooth Riemannian manifold of dimension n without boundary or with boundary and we suppose G is a torus acting by isometries on M. Let XM be the associated vector field of X on M where X is in the Lie algebra. One defines Witten 's inhomogeneous operator dXM=d+ιXM: ΩGev/odd →ΩGodd/ev, where ΩGev/odd is the space of invariant forms of even or odd degree. We define the operator δXM = (-1)n(k+1)+1∗dXM∗.
Then we present the following:
If I have time I will explain how the XM-cohomology can help to solve special kinds of differential equations and our decomposition can be used to solve special kinds of the boundary-value problems for invariant differential forms.