**Ben Sibley ** will speak on

** A complex analytic structure on the compactification of the Hermitian-Yang-Mills moduli space
**

**Abstract:** A key aspect of gauge theory is finding a suitable compactification for the moduli space of instantons. For higher dimensional manifolds posessing certain additional geometric structures, Tian has defined a notion of instanton and made progress towards a compactification analogous to Uhlenbeck's compactification of the moduli space of anti-self-dual connections on a four-manifold. In the case when the base manifold is Kähler, and the bundle in question is hermitian, instantons which are unitary and give rise to a holomorphic structures are Hermitian-Yang-Mills connections. A sequence of such connections is known to bubble at most along a codimension 2 analytic subvariety, and so one might hope that there is a gauge theoretic compactification which has the structure of a complex analytic space. I will attempt to explain why this true in the case when the base is projective. This gives a higher dimensional analogue of a theorem of Jun Li for algebraic surfaces. This is joint work in progress with Daniel Greb, Matei Toma, and Richard Wentworth.