# Hodge theory on singular spaces

This talk concerns recent work on the Hodge theorem, a fundamental theorem in differential geometry which identifies a topological feature of a closed manifold (namely its de Rham cohomology) with a special class of differential forms called harmonic forms. I will give a gentle introduction to this theorem and then explain how it has been extended to certain singular spaces or non-compact spaces, in particular I will explain the deep significance of the Hodge theorem of Hausel-Hunsicker-Mazzeo. I will then discuss work toward extending the Hodge theorem to certain singular spaces, namely those for which the singularity arises as one approaches a given subspace, near which the geometry brakes up into components which have uniform rates of collapse. This includes the moduli space of punctured Riemann surfaces with the Weil-Petersson metric, and we will discuss this important example if time permits. This is based on joint works with R. Melrose and with J. Swoboda.