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.

• 31st August, Behrend Memorial Lecture in Copland Lecture Theatre, Melbourne University, 6:00 pm;
  public lecture by Terence Tao (UCLA): Mathematical research and the internet (flyer)

For more information: Jan Thomas, tel. 03 8344 1774 email: jan@amsi.org.au
or Robyn Trethowen, tel. 03 8344 4392 (Science Events)

• 1st September, in Casey Plaza Theatre (Building 10, Level 4, Room 27) RMIT, 3:30pm; colloquium talk by Terence Tao (UCLA): Compressed Sensing

• 2nd September in Room 345, Building 28, Monash University and via the Access Grid, 2:30 pm; specialist talk by Terence Tao (UCLA): Discrete Random Matrices

AGR Contact: Simon Clarke – Monash Univ. (simon.clarke@sci.monash.edu.au)

Abstract: Mathematical research and the internet

Prof. Tao will talk about some of his personal experiences on how the internet is transforming the way he and other mathematicians do research, from mundane tools as email, home pages and search engines, to blogs, pre–print servers, wikis, and more.

Abstract: Compressed Sensing

Suppose one wants to recover an unknown signal \boldsymbol{x} in \mathbb{R}^n from a given vector A\boldsymbol{x}=\boldsymbol{b} in \mathbb{R}^m of linear measurements of the signal \boldsymbol{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 \boldsymbol{x} is not unique.

However, if we also know that \boldsymbol{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 \boldsymbol{x} from the measurements \boldsymbol{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.

Abstract: Discrete Random Matrices

The spectral theory of continuous random matrix models (e.g., real or complex gaussian random matrices) has been well studied, and very precise information on the distribution of eigenvalues and singular values is now known. But many of the results rely quite heavily on the special algebraic properties of the matrix ensemble (e.g., the invariance properties with respect to the orthogonal or unitary group). As such, the results do not easily extend to discrete random matrix models, such as the Bernoulli model of matrices with random \pm1 signs as entries.

Recently, however, tools from additive combinatorics and elementary linear algebra have been applied to establish several results for such discrete ensembles, such as the circular law for the distribution of eigenvalues, and also explicit asymptotic distributions for the least singular values of such matrices. We survey some of these developments in this talk.