Algebra

Prolog is a very unusual programming language, developed by Alain Colmerauer in one of the buildings on the way to the CIRM in Luminy. It is not fundamentally iterative in the way that, for example, GAP and Magma are. Instead it operates by taking a list of axioms as input, and responds at the command line to queries asking the language to achieve particular goals. It gained some notoriety by beating contestants on the game show Jeopardy in 2011. It is also the worlds fastest sudoku solver. I will describe some recent Prolog investigations to search for new simple Lie algebras over the field GF(2). We were able to discover some new examples in dimensions 15 and 31 and extrapolate from these to construct two new infinite families of simple Lie algebras. This is a work with David Cushing and George Stagg.

We will discuss recent results in modular Lie theory obtained in the setting of "diagrammatic Hecke categories”.  We will discuss the Kazhdan—Lusztig positivity conjecture, the counter examples to Lusztig’s conjecture, and how p-Kazhdan—Lusztig polynomials can be seen to control the representation theory of braid groups, cyclotomic Hecke algebras, and algebraic groups.  We also discuss when these polynomials can and cannot be explicitly calculated. 

A finite reductive group is a matrix group defined over a finite field, which we assume has characteristic p.  The prototypical example is the general linear group of all invertible n by n matrices with entries in the field of p elements. These groups take a central role in finite group theory because, after the classification of finite simple groups, almost all finite groups can be constructed from finite reductive groups.

There are wonderful tools available to us, coming from geometry, when studying the representations of finite reductive groups. However, in this talk we will survey the impact that p-groups have had, in particular their role in classifying irreducible modular representations and computing character fields and Schur indices.

In this talk, I will discuss techniques to study irreducible representations of various quantum algebras at roots of unity. Our main example will be quantum Schubert varieties in the quantum grassmannian. This is based on joint work with Jason Bell, Samuel Lopes and Alexandra Rogers. 

I will discuss a new subject "Lie theory in tensor categories" (also to be discussed in my plenary lecture) and formulate a number of open problems in it.

The Virasoro Lie algebra Vir is one of the most important infinite-dimensional Lie algebras, ubiquitous in representation theory and mathematical physics.  In this talk we consider the dual space Vir*.  Although Vir famously has no adjoint group, Poisson geometry on Vir* still gives a meaningful notion of a coadjoint orbit, or more accurately pseudo-orbit.   We describe the geometry of these pseudo-orbits:  each central character corresponds to a single infinite-dimensional pseudo-orbit, and in addition for the trivial character there is a family of finite-dimensional pseudo-orbits parameterised by partitions and points in a finite-dimensional affine space.  Each finite-dimensional pseudo-orbit is actually an orbit:  a principal homogeneous space for a finite-dimensional solvable algebraic group.  We further show how to use the parameterisation of pseudo-orbits to give a complete description of Poisson subvarieties of Vir*. This is joint work with Alexey Petukhov.

Generalised double affine Hecke algebras were introduced by Etingof, Oblomkov and Rains as flat deformations of the group algebra of 2-dimensional crystallographic groups associated to simply laced star Dynkin diagrams. In this talk we focus on the D_4 and E_6 cases. We give an explicit matrix representation in terms of Fock Goncharov coordinates and describe the embedding of the D_4 GDAHA into the E_6 one.