# Manchester Geometry Seminar 2009/2010

**Thursday 11 March 2010. ** *The Frank Adams Room (Room 1.212), the Alan Turing Building. 4.15pm*
##
Witten-Hodge Theory for Manifolds with Boundary and a Torus Action

Qusay Al-Zamil (University of Manchester)

`Qusay.Abdul-Aziz@postgrad.manchester.ac.uk`

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 X_{M} be the associated vector field of X on M where X is in the Lie algebra.
One defines Witten 's inhomogeneous operator d_{XM}=d+ι_{XM}: Ω_{G}^{ev/odd} →Ω_{G}^{odd/ev}, where Ω_{G}^{ev/odd} is the space of invariant forms of even or odd degree. We define the operator δ_{XM} = (-1)^{n(k+1)+1}∗d_{XM}∗.

Then we present the following:

- The relevant version of the invariant Hodge decomposition theory in terms of the operators d
_{XM} and δ_{XM} when there is no boundary of M;
- The relevant version of the invariant Hodge-Morrey-Friedrichs decomposition theory in terms of the operators d
_{XM} and δ_{XM} when there is a boundary of M;
- In case there is a boundary of M, we present the relative and absolute X
_{M}-cohomology and X_{M}-Poincaré-Lefschetz duality.

If I have time I will explain how the X_{M}-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.