Abstract:  
Computing the L2 cohomology of moduli spaces of monopoles and instantons is a challenging problem.  It is significant in physics having an interpretations as counting the BPS states in quantum gauge theories, as well as in mathematics, manifesting itself in the geometric Langlands correspondence for complex surfaces.  

We propose a rather unconventional compactification of these moduli spaces in terms as exploded geometry.  The corresponding De Rham theory was developed by Brett Parker and, as we expect, should produce the desired cohomology spaces.  In this talk I shall describe how exploded geometry naturally emerges in the study of monopole walls (doubly periodic monopoles).  Then I shall describe the exploded twistor description of monopoles in R^3, which automatically incorporates their asymptotic form and provides the desired compactification of their moduli space.