**Presenter:** Jesse Gell-Redman, University of Melbourne

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.

### About Maths Colloquium

The Mathematics Colloquium is directed at students and academics working in the fields of pure and applied mathematics, and statistics.

We aim to present expository lectures that appeal to our wide audience.

Information for speakers

#### Information for speakers

Maths colloquia are usually held on Mondays, from 2pm to 3pm, in various locations at St Lucia.

Presentations are 50 minutes, plus five minutes for questions and discussion.

Available facilities include:

- computer
- data projector
- chalkboard or whiteboard

To avoid technical difficulties on the day, please contact us in advance of your presentation to discuss your requirements.