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.

• Mon. 21st in Bernhard Neumann seminar room JD G35, ANU at 3:00pm; specialist lecture by Terry Tao (UCLA): Recent progress on the Kakeya problem.

• Tues. 22nd in Bernhard Neumann seminar room JD G35, ANU at 3:30pm; colloquium talk by Danny Calegari (Caltech):
  Faces of the stable commutator length ball.

• Tues. 22nd in Theatre 1, Manning Clarke Building, ANU at 5:30pm; public lecture by Terry Tao (UCLA): Structure and randomness in the prime numbers.

• Wed. 23rd in Bernhard Neumann seminar room JD G35, ANU at 2:00pm; colloquium talk by Mohammed Abouzaid (MIT):
  Understanding hypersurfaces through tropical geometry.

• Wed. 23rd in Baume AGR, Peter Baume Building, ANU and via the Access Grid at 3.30pm; colloquium talk by Terry Tao (UCLA):
  Recent progress in additive prime number theory.

• Thurs. 24th in Bernhard Neumann seminar room JD G35, ANU at 2.30pm; specialist lecture by Danny Calegari (Caltech):
  Stable commutator length.

• Thurs. 24th in Bernhard Neumann seminar room JD G35, ANU at 4.00pm; specialist lecture by Mohammed Abouzaid (MIT):
  A mirror construction for hypersurfaces in toric varieties.

All events will be at the Australian National University.

Enquiries: (02) 6125 2957 or email Katie.Lau@anu.edu.au,
or Andrew Hassell (Andrew.Hassell@anu.edu.au)

AGR Contact: Robert Sbragi (videoconference@anu.edu.au)
Other Clay–Mahler Access Grid events on the AMSI website

Abstract: (Tao) Recent progress on the Kakeya problem

The Kakeya needle problem asks: is it possible to rotate a unit needle in the plane using an arbitrarily small amount of area? The answer is known to be yes, but analogous problems in higher dimensions (where one now seeks to find sets of small dimension that contain line segments in each direction) remain open, and are related to many other important conjectures in harmonic analysis, PDE, and even number theory and computer science.

There have been many partial results on this problem, using such diverse techniques as geometric measure theory, incidence combinatorics, additive combinatorics, and PDE; more recently, algebraic geometry, and even algebraic topology have been used to obtain new breakthroughs in this subject. We will discuss many of these new developments in this talk.

Abstract: (Calegari) Faces of the stable commutator length norm ball

It often happens that a solution of an extremal problem in geometry has more regularity and nicer features than one has an a priori right to expect. I will show how a simple topological problem—when does an immersed curve on a surface bound an immersed subsurface?—is unexpectedly related to linear programming in nonseparable Banach spaces, and gives rise to geometric and dynamical rigidity and discreteness of symplectic representations.

Abstract: (Tao) Structure and randomness in the prime numbers

God may not play dice with the universe, but something strange is going on with the prime numbers — Paul Erdős.

The prime numbers are a fascinating blend of both structure (for instance, almost all primes are odd) and randomness. It is widely believed that beyond the obvious structures in the primes, the primes otherwise behave as if they were distributed randomly; this pseudorandomness then underlies our belief in many unsolved conjectures about the primes, from the twin prime conjecture to the Riemann hypothesis. This pseudorandomness has been frustratingly elusive to actually prove rigorously, but recently there has been progress in capturing enough of this pseudorandomness to establish new results about the primes, such as the fact that they contain arbitrarily long progressions. We survey some of these developments in this talk.

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: (Tao) Recent progress in additive prime number theory

Additive prime number theory is the study of additive patterns in the primes. We survey some recent advances in this subject, including the results of Goldston, Pintz and Yildirim on small gaps between primes, the results of Green and myself on arithmetic progressions in the primes, and the results of Bourgain, Gamburd, and Sarnak for detecting almost primes in orbits.

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.