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.

• 3rd September, 6:00 pm in Social Sciences Lecture Theatre; public lecture talk by Terence Tao (UCLA): Cosmic Distance Ladder.

• 4th September, 1:00 pm in Engineering Lecture Theatre 1 and via the Access Grid; specialist talk by Terence Tao (UCLA): Compressed Sensing

Both events will be at the University of Western Australia.

Contact: Professor Cheryl Praeger, School of Mathematics and Statistics (praeger@maths.uwa.edu.au)

Abstract: Cosmic Distance Ladder

How do we know the distances from the earth to the sun and moon, from the sun to the other planets, and from the sun to other stars and distant galaxies? Clearly we cannot measure these directly. Nevertheless there are many indirect methods of measurement, combined with basic high-school mathematics, which can allow one to get quite convincing and accurate results without the need for advanced technology (for instance, even the ancient Greeks could compute the distances from the earth to the sun and moon to moderate accuracy). These methods rely on climbing a cosmic distance ladder, using measurements of nearby distances to then deduce estimates on distances slightly further away; we shall discuss several of the rungs in this ladder in this talk.

Abstract: Compressed sensing

Suppose one wants to recover an unknown signal \vec{x} in \mathbb{R}^n from a given vector A \vec{x}=\vec{b} in \mathbb{R}^m of linear measurements of the signal \vec{x}. If the number of measurements m is less than the degrees of freedom n of the signal, then the problem is underdetermined and the solution \vec{x} is not unique.

However, if we also know that \vec{x} is sparse or compressible with respect to some basis, then it is a remarkable fact that (given some assumptions on the measurement matrix A) we can reconstruct \vec{x} from the measurements \vec{b} with high accuracy, and in some cases with perfect accuracy. Furthermore, the algorithm for performing the reconstruction is computationally feasible. This observation underlies the newly developing field of compressed sensing.

In this talk we will discuss some of the mathematical foundations of this field.