(jt work with David Cushing and George Stagg)

Prolog is a rather unusual programming language that was developed by Alain Colmerauer 50 years ago in one of the buildings on the way to the CIRM in Luminy. It is a declarative language that operates on a paradigm of first-order logic -- as distinct from imperative languages like C, GAP and Magma. Prolog operates by loading in a list of axioms as input, and then responds at the command line to queries that ask the language to achieve particular goals, given those axioms. It gained some notoriety through IBM’s implementation of ‘Watson’, which was a system designed to play the game show Jeopardy. Through a very efficiently implemented constraint logic programming module, it is also the worlds fastest sudoku solver. However, it has had barely any serious employment by pure mathematicians. So the aim of this talk is to advertise Prolog through an extended example: my co-authors and I used it to search for new simple Lie algebras over the field GF(2) and were able to classify a certain flavour of absolutely simple Lie algebra in dimensions 15 and 31, discovering a dozen or so new examples. With some further examples in dimension 63, we then extrapolated two previously undocumented infinite families of simple Lie algebras.

Kazhdan-Lusztig polynomials are remarkable polynomials associated to pairs of elements in a Coxeter group W. They describe the base change between the standard and Kazhdan-Lusztig bases for the corresponding Hecke algebra. They were discovered by Kazhdan and Lusztig in 1979 and have found applications throughout representation theory and geometry. In 1987, Deodhar introduced the parabolic Kazhdan-Lusztig polynomials associated to a Coxeter group W and a standard parabolic subgroup P. These describe the base change between the standard and Kazhdan-Lusztig bases for the anti-spherical module for the Hecke algebra. (We recover the original definition of Kazhdan and Lusztig by taking the trivial parabolic subgroup).

(Anti-spherical) Hecke categories first rose to mathematical celebrity as the centrepiece of the proof of the (parabolic) Kazhdan-Lusztig positivity conjecture. The Hecke category categorifies the Hecke algebra and the anti-spherical Hecke category categorifies the anti-spherical module. More precisely, it was shown by Elias-Williamson (and Libedinsky-Williamson) that the (parabolic) Kazhdan-Lusztig polynomials are precisely the graded decomposition numbers for the (anti-spherical) Hecke categories over fields of characteristic zero, hence proving positivity of their coefficients.
The (anti-spherical) Hecke categories can be defined over any field. Their graded decomposition numbers over fields of positive characteristic p, the so-called (parabolic) p-Kazhdan-Lusztig polynomials, have been shown to have deep connections with the modular representation theory of reductive groups and symmetric groups. However, these polynomials are notoriously difficult to compute.
Unlike in the case of the ordinary (parabolic) Kazhdan-Lusztig polynomials, there is not even a recursive algorithm to compute them in general.
In this talk, I will discuss the representation of the anti-spherical Hecke categories for (W,P) a Hermitian symmetric pair, over an arbitrary field. In particular, I will explain why the decomposition numbers are characteristic free in this case.
This is joint work with C. Bowman, A. Hazi and E. Norton.

I will explain how bornological and condensed structures can both be described as algebraic theories. I will also show how this permits the construction of functors between bornological and condensed structures. If time permits I will also briefly describe how to compare condensed derived geometry and bornological derived geometry and sketch how they relate to analytic geometry and Arakelov geometry

The local Langlands conjectures connect representations of p-adic groups to certain representations of Galois groups of local fields called Langlands parameters.  Recently, there has been a shift towards studying representations over more general coefficient rings and towards certain categorical enhancements of the original conjectures.  In this talk, we will focus on representations over coefficient rings with p invertible and how the corresponding category of representations of the p-adic group decomposes.  

In the last few years, major advances have been made in our understanding of the representation theory of reductive algebraic groups over algebraically closed fields of positive characteristic. Four key tools which are central to this progress are highest weight theory, reduction to the principal block, wall-crossing functors, and tilting modules. When considering instead the representation theory of the Lie algebras of these algebraic groups, more subtleties arise. If we look at those modules whose p-character is in so-called standard Levi form we are able to recover the four tools mentioned above, but they have been less well-studied in this setting. In this talk, we will explore the similarities and differences which arise when employing these tools for the Lie algebras rather than the algebraic groups. This research is funded by a research fellowship from the Royal Commission for the Exhibition of 1851.

We discuss the Local Langlands correspondence in connection with the Bernstein center and the Stable Bernstein center. We also give an example of stable Bernstein center as a stable essentially compact invariant distribution.

Geometric (even combinatorial) group theory suffers from the unfortunate situation that many obvious questions about group presentations (ex: is this a presentation of the trivial group? is this word the identity in that group?) cannot be answered. Not only "we don't know how to tell" but "we know that we cannot know how to tell" - this is called undecidability. This talk will serve as an introduction (for non-experts, since I am also such) to the area of group theoretic decision problems: I'll aim to cover some problems, some solutions (or half-solutions) and some of the general sources of undecidability, as well as featuring some of my (least?) favourite pathological groups.

Given a mathematical object, what can you compute about it? In some settings, you cannot say very much. Given an arbitrary group presentation, for example, there is no procedure to decide whether the group is trivial. In 3-manifolds, however, algorithms are a fruitful and active area of study (and some of them are even implementable!). One of the main tools in this area is normal surface theory, which allows us to describe interesting surfaces in a 3-manifold with respect to an arbitrary triangulation. I will discuss some results in this area, particularly around Seifert fibered spaces.

Stable commutator length (scl) is a measure of homological complexity in groups that has attracted attention for its various connections with geometric topology and group theory. In this talk, I will introduce scl and discuss the (hard) problem of computing scl in surface groups. I will present some results concerning isometric embeddings of free groups for scl, and how they generalise to surface groups for the relative Gromov seminorm.