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.

• Thurs. 17 Sep. in E6A Level 2, AG room and via the Access Grid, 2:00–3:00pm; specialist talk by Terence Tao (UCLA): Recent progress on the Kakeya problem.

• Thurs. 17 Sep. in E6A Level 2, AG room and via the Access Grid, 4:00–5:00pm; seminar lecture, by Mohammed Abouzaid (MIT): Understanding hypersurfaces through tropical geometry.

Both events will be at Macquarie University.

AGR Contact: John Porte (jporte@science.mq.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: (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.