Hall algebras are a deformation of the K-group (Grothendieck group) of an abelian category, which encode some information about non-trivial extensions in the category.
A main feature of Hall algebras is that in addition to the product (which deforms the product in the K-group) there is a natural coproduct, which in certain cases makes the Hall algebra a (braided) bi-algebra. This is the content of Green's theorem and supplies the main ingredient in a construction of quantum groups.

We give a noncommutative analogue of Castelnuovo's classic theorem that (-1) lines on a smooth surface can be contracted, and show how this may be used to construct an explicit birational map between a noncommutative P^2 and a noncommutative quadric surface. This has applications to the classification of noncommutative projective surfaces, one of the major open problems in noncommutative algebraic geometry. We will not assume a background in noncommutative ring theory.  The talk is based on joint work with Rogalski and Staffor

Diffusion processes are commonly used in image processing. In particular, complex diffusion models have been successfully applied in medical imaging denoising. The interpretation of a complex diffusion equation as a cross-diffusion system motivates the introduction of more general models of this type and their study in the context of image processing. In this talk we will discuss the use of nonlinear cross-diffusion systems to perform image restoration. We will analyse the well-posedness, scale-space properties and
long time behaviour of the models along with their performance to treat image filtering problems. Examples of application will be highlighted.

Low-rank compression of matrices and tensors is a huge and growing business.  Closely related is low-rank compression of multivariate functions, a technique used in Chebfun2 and Chebfun3.  Not all functions can be compressed, so the question becomes, which ones?  Here we focus on two kinds of functions for which compression is effective: those with some alignment with the coordinate axes, and those dominated by small regions of localized complexity.

We purify and generalize some techniques which were successful in the limit theory of graphs and other discrete structures. We demonstrate how this technique can be used for solving different combinatorial problems, by defining the limit problems of the Manickam--Miklós--Singhi Conjecture, the Kikuta–Ruckle Conjecture and Alpern's Caching Game.