Oxford

Abstract regular polytopes are finite quotients of Coxeter complexes
with string diagram, satisfying a natural intersection property, see
e.g. [MMS2002]. They arise in a number of geometric and group-theoretic
contexts. The first class of such objects, beyond the
well-understood examples coming from finite and affine Coxeter groups,
are locally toroidal cases, e.g.  extensions of quotients of the affine
F_4 complex [3,3,4,3].  In 1996 P.McMullen & E.Schulte constructed a
number of examples of locally toroidal abstract regular polytopes of
type [3,3,4,3,3], and conjectured completeness of their list. We
construct counterexamples to the conjecture using a Y-shaped
presentation for a subgroup of the Monster, and discuss various
related questions.
 

The Beilinson-Drinfeld Grassmannian, which classifies a G-bundle trivialised away from a finite set of points on a curve, is one of the basic objects in the geometric Langlands programme. Similar construction in higher dimensions in the algebraic and analytic settings are not very interesting because of Hartogs' theorem. In this talk I will discuss a differentiable version. I will also explain a theory of D-modules on differentiable spaces and use it
to define differentiable chiral and factorisation algebras. By linearising the Grassmannian we get examples of differentiable chiral algebras. This is joint work with Dennis Borisov.

The International Congresses of Mathematicians (ICMs) have taken place at (reasonably) regular intervals since 1897, and although their participants may have wanted to confine these events purely to mathematics, they could not help but be affected by wider world events.  This is particularly true of the 1936 ICM, held in Oslo.  In this talk, I will give a whistle-stop tour of the early ICMs, before discussing the circumstances of the Oslo meeting, with a particular focus on the activities of the Nazi-led German delegation.

We construct a space of vector fields that are normal to differentiable curves in the plane. Its basis functions are defined via saddle point variational problems in reproducing kernel Hilbert spaces (RKHSs). First, we study the properties of these basis vector fields and show how to approximate them. Then, we employ this basis to discretise shape Newton methods and investigate the impact of this discretisation on convergence rates.

Structure from Motion (SfM) is a problem which asks: given photos of an object from different angles, can we reconstruct the object in 3D? This problem is important in computer vision, with applications including urban planning and autonomous navigation. A key part of SfM is bundle adjustment, where initial estimates of 3D points and camera locations are refined to match the images. This results in a high-dimensional nonlinear least-squares problem, which is typically solved using the Gauss-Newton method. In this talk, I will discuss how dimensionality reduction methods such as block coordinates and randomised sketching can be used to improve the scalability of Gauss-Newton for bundle adjustment problems.

In oil and gas reservoir simulation, standard preconditioners involve solving a restricted pressure system with AMG. Initially designed for isothermal models, this approach is often used in the thermal case. However, it does not incorporate heat diffusion or the effects of temperature changes on fluid flow through viscosity and density. We seek to develop preconditioners which consider this cross-coupling between pressure and temperature. In order to study the effects of both pressure and temperature on fluid and heat flow, we first consider a model of non-isothermal single phase flow through porous media. By focusing on single phase flow, we are able to isolate the properties of the pressure-temperature subsystem. We present a numerical comparison of different preconditioning approaches including block preconditioners.

I will overview my research for a general math audience.

 First I will present the biological questions and motivate why systems biology needs computational algebraic biology and topological data analysis. Then I will present the mathematical methods I've developed to study these biological systems. Throughout I will provide examples.