Presenter: Dr Paul Bryan, UQ

Harnack inequalities are an important tool in the analysis of elliptic and parabolic PDE. For geometric flows such at the Ricci flow and the Mean Curvature Flow, the so-called Li-Yau-Hamilton-Harnack inequality plays a central role in a number of places, in particular in the study of singularity formation. Ancient solutions of a parabolic PDE are those solutions existing on an interval $(-\infty,T)$, and are closely related to the Harnack inequality in that solitons (a self-similar class of ancient solutions) achieve equality. Solitons are themselves of importance since they model general singularity formation. To date, most work on hypersurface flows has focused on Euclidean space, where a Harnack inequality for general hypersurface flows was obtained by Ben Andrews via a simple computation using the support function and Gauss map parametrisation. I will a describe a variation of this approach that works in arbitrary Riemannian backgrounds, tying together several different aspects of the Harnack inequality. I will also describe how this relates to ancient solutions, and time permitting will give a brief survey of several classification techniques.

We present regular seminars on a range of pure mathematics interests. Students, staff and visitors to UQ are welcome to attend, and to suggest speakers and topics.

Seminars are usually held on Tuesdays from 3pm to 3.50pm.

Talks comprise 45 minutes of speaking time plus five minutes for questions and discussion.

#### Information for speakers

Researchers in all pure mathematics fields attend our seminars, so please aim your presentation at a general mathematical audience.