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.

• Fri. 18th Sept. in Carslaw Building, Room 275 at 2:00pm; specialist lecture by Danny Calegari (Caltech): "Stable commutator length".

• 18th in Carslaw Building Room 275, Sydney Univ. at 4:00pm; specialist lecture by Mohammed Abouzaid (MIT): "A mirror construction for hypersurfaces in toric varieties".

Both events will be at the Sydney University.

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: (Abouzaid) A mirror construction for hypersurfaces in toric varieties

The Strominger–Yau–Zaslow conjecture gives an intrinsic explanation for Homological Mirror Symmetry in the case of Calabi–Yau manifolds. I will explain that by extending the SYZ conjecture beyond the Calabi–Yau case, one may associate a Landau–Ginzburg mirror to generic hypersurfaces in toric varieties. The key idea is to use tropical geometry to reduce the problem to understanding the mirror of hyperplanes.