The research proposal concerns the further development of integration theory in superspace; i.e. integration of functions in commuting and anti-commuting variables. This proposal focuses on two main parts:
(1) integration of super-differential forms over super-chains;
(2) Radon decompositions and plane waves in superspace.
Although there exist methods for integration over general (super) domains and their boundaries, a theory of integration over k-surfaces, and more in general, over super-chains, has not yet been developed. In the first part of this project, we will establish an integration theory of superdifferential forms over super chains, which allows for integration over orientable and also nonorientable surfaces. To this end, we will further develop the theory of super-forms combined with Clifford analysis techniques. In the second part, we will determine several Radon decompositions for the Dirac delta distribution in flat superspace, which will allow us to reduce problems of arbitrary superdimension to 2D or 1D problems. This will be obtained by decomposing monogenic superfunctions (null solutions of the super Dirac operator) in terms of plane waves. This part will also lead to extensions of the Szego-Radon and Bargmann-Radon projections of some Hilbert modules of monogenic functions onto plane wave submodules. Summarizing, this project aims at further developing both integration theory and transform analysis in superspace following a Clifford approach.