The Mahler lectures are a biennial activity organised by the Australian Mathematical Society. In 2009 we have partnered with the Clay Mathematical Institute to combine the Mahler Lectures and the Clay Lectures into the 2009 Clay–Mahler Lecture Tour, with funding also from the Australian Mathematical Sciences Institute.

• Friday 25th September in Napier Lecture Theatre 102, Napier Building, North Terrace Campus, at 12:10–1:00pm (SA time — 1:40–2:30pm AEST);
  colloquium talk by Mohammed Abouzaid (MIT):
  Understanding hypersurfaces through tropical geometry.
  (This talk is part of the Differential Geometry Seminar series.)

• Friday 25th September in Napier Lecture Theatre 102, Napier Building, North Terrace Campus, at 1:40–2:30pm (SA time — 3:10–4:00pm AEST);
  colloquium talk by Danny Calegari (Caltech): Stable commutator length.
  (This talk is part of the Differential Geometry Seminar series.)

• Friday 25th September in Napier Lecture Theatre 102, Napier Building, North Terrace Campus, at 3:10–4:00pm (SA time — 4:40–5:30pm AEST);
  colloquium talk by Terry Tao (UCLA): The proof of the Poincaré conjecture.
  (This talk is part of the School of Mathematical Sciences Colloquium series.)

All events will be at the University of Adelaide.

Enquiries: Maths Admin, Tel. +61 8 8303 5407
or email: admin.maths@list.adelaide.edu.au

Abstract: (Abouzaid) Understanding hypersurfaces through tropical geometry

Given a polynomial in two or more variables, one may study the zero locus from the point of view of different mathematical subjects (number theory, algebraic geometry, ...). I will explain how tropical geometry allows to encode all topological aspects by elementary combinatorial objects called tropical varieties.

Abstract: (Calegari) Stable commutator length

The scl (stable commutator length) answers the question: what is the simplest surface in a given space with prescribed boundary? where simplest is interpreted in topological terms. This topological definition is complemented by several equivalent definitions:

• in group theory, as a measure of non-commutativity of a group; and
• in linear programming, as the solution of a certain linear optimization problem.

On the topological side, scl is concerned with questions such as computing the genus of a knot, or finding the simplest 4-manifold that bounds a given 3-manifold. On the linear programming side, scl is measured in terms of certain functions called quasimorphisms, which arise from hyperbolic geometry (negative curvature) and symplectic geometry (causal structures).

We will discuss how scl in free and surface groups is connected to such diverse phenomena as the existence of closed surface subgroups in graphs of groups, rigidity and discreteness of symplectic representations, bounding immersed curves on a surface by immersed subsurfaces, and the theory of multi-dimensional continued fractions and Klein polyhedra.

Abstract: (Tao) The proof of the Poincaré conjecture

In a series of three papers from 2002–2003, Grigori Perelman gave a spectacular proof of the Poincaré conjecture (every smooth compact simply connected three-dimensional manifold is topologically isomorphic to a sphere), one of the most famous open problems in mathematics, by developing several new ground-breaking advances in Hamilton's theory of Ricci flow on manifolds. In this talk I describe in broad detail how the proof proceeds, and briefly discuss some of the key turning points in the argument.