Mathematics

The study of maps, that is of graphs embedded in surfaces, is a popular subject that has implications in many branches of mathematics, the most famous aspects being purely graph-theoretical, such as the four-color theorem. The study of random maps has met an increasing interest in the recent years. This is motivated in particular by problems in theoretical physics, in which random maps serve as discrete models of random continuum surfaces. The probabilistic interpretation of bijective counting methods for maps happen to be particularly fruitful, and relates random maps to other important combinatorial random structures like the continuum random tree and the Brownian snake. This course will survey these aspects and present recent developments in this area.

Michael Bennett (President, Canadian Mathematical Society; Professor of Mathematics, University of British Columbia)

Diophantine equations are one of the oldest, frequently celebrated and most abstract objects in mathematics. They crop up in areas ranging from recreational mathematics and puzzles, to cryptography, error correcting codes, and even in studying the structure of viruses. In this talk, Dr. Bennett will attempt to show some of the roles these equations play in modern mathematics and beyond.

Inverse problems arise in many imaging applications, such as image
reconstruction (e.g., computed tomography), image deblurring, and
digital super-resolution. These inverse problems are very difficult
to solve; in addition to being large scale, the underlying
mathematical model is often ill-posed, which means that noise and
other errors in the measured data can be highly magnified in computed
solutions. Regularization methods are often used to overcome this
difficulty. In this talk we describe hybrid Krylov subspace based
regularization approaches that combine matrix factorization methods
with iterative solvers. The methods are very efficient for large scale
imaging problems, and can also incorporate methods to automatically
estimate regularization parameters. We also show how the approaches
can be adapted to enforce sparsity and nonnegative constraints.

We will use many imaging examples that arise in medicine and astronomy
to illustrate the performance of the methods, and at the same time
demonstrate a new MATLAB software package that provides an easy to use
interface to their implementations.

This is joint work with Silvia Gazzola (University of Bath) and
Per Christian Hansen (Technical University of Denmark).

Phase retrieval is a problem that arises in a wide range of imaging
applications, including x-ray crystallography, x-ray diffraction imaging
and ptychography. The data in the phase retrieval problem are samples of
the modulus of the Fourier transform of an unknown function. To
reconstruct this function one must use auxiliary information to determine
the unmeasured Fourier transform phases. There are many algorithms to
accomplish task, but none work very well. In this talk we present an
analysis of the geometry that underlies these failures and points to new
approaches for solving this class of problems.