Oxford

The stack of Higgs bundles of a given rank and degree over a non-singular projective curve can be stratified in two ways: according to its Higgs Harder-Narasimhan type (its instability type) and according to the Harder-Narasimhan type of the underlying vector bundle (instability type of the underlying bundle). The semistable stratum is an open stratum of the former and admits a coarse moduli space, namely the moduli space of semistable Higgs bundles. It can be constructed using Geometric Invariant Theory (GIT) and is a widely studied moduli space due to its rich geometric structure.

In this talk I will explain how recent advances in Non-Reductive GIT can be used to refine the Higgs Harder-Narasimhan and Harder-Narasimhan stratifications in such a way that each refined stratum admits a coarse moduli space. I will explicitly describe these refined stratifications and their intersection in the case of rank 2 Higgs bundles, and discuss the topology and geometry of the corresponding moduli spaces

Finding Riemannian metrics with holonomy G_2 is a challenging problem with links in mathematics to Einstein metrics and area-minimizing submanifolds, and to M-theory in theoretical physics.  I will provide a brief survey on recent progress towards studying this problem using a geometric flow approach, including connections to calibrated fibrations.

Snapshot mosaic multispectral imagery acquires an undersampled data cube by acquiring a single spectral measurement per spatial pixel. Sensors which acquire p frequencies, therefore, suffer from severe 1/p undersampling of the full data cube.  We show that the missing entries can be accurately imputed using non-convex techniques from sparse approximation and matrix completion initialised with traditional demosaicing algorithms.

When solving partial differential equations driven by additive spatial white noise, the efficient sampling of white noise realizations can be challenging. In this talk we focus on the efficient sampling of white noise using quasi-random points in a finite element method and multilevel Quasi Monte Carlo (MLQMC) setting. This work is an extension of previous research on white noise sampling for MLMC.

We express white noise as a wavelet series expansion that we divide in two parts. The first part is sampled using quasi-random points and contains a finite number of terms in order of decaying importance to ensure good QMC convergence. The second part is a correction term which is sampled using standard pseudo-random numbers.

We show how the sampling of both terms can be performed in linear time and memory complexity in the number of mesh cells via a supermesh construction. Furthermore, our technique can be used to enforce the MLQMC coupling even in the case of non-nested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments.

In this talk we will show how the floating point errors in the simulation of SDEs (stochastic differential equations) can be modelled as stochastic. Furthermore, we will show how these errors can be corrected within a multilevel Monte Carlo approach which performs most calculations with low precision, but a few calculations with higher precision. The same procedure can also be used to correct for errors in converting from uniform random numbers to approximate Normal random numbers. Numerical results will be generated on both CPUs (using single/double precision) and GPUs (using half/single precision).

A conjecture from the '80s claims that the isometry type of a closed, negatively curved Riemannian manifold should be uniquely determined by the lengths of its closed geodesics. By work of Otal, this is essentially equivalent to the problem of extending cross-ratio preserving maps between Gromov boundaries of simply connected, negatively curved manifolds. Progress on the conjecture has been remarkably slow, with only the 2-dimensional and locally symmetric cases having been solved so far (Otal '90 and Hamenstädt '99).
Still, it is natural to try leaving the world of manifolds and address the conjecture in the general context of non-positively curved metric spaces. We restrict to the class of CAT(0) cube complexes, as their geometry is both rich and well-understood. We introduce a new notion of cross ratio on their horoboundary and use it to provide a full answer to the conjecture in this setting. More precisely, we show that essential, hyperplane-essential cubulations of Gromov-hyperbolic groups are completely determined by their combinatorial length functions. One can also consider non-proper non-cocompact actions of non-hyperbolic groups, as long as the cube complexes are irreducible and have no free faces.
Joint work with J. Beyrer and M. Incerti-Medici.

Stable commutator length (scl) is a well established invariant of elements g in the commutator subgroup (write scl(g)) and has both geometric and algebraic meaning.  A group has a \emph{gap} in stable commutator length if for every non-trivial element g, scl(g) > C for some C > 0.
SCL may be interpreted as an 'algebraic translation length' and such a gap may be thus interpreted an 'algebraic injectivity radius'.
Many classes of groups have such a gap, like hyperbolic groups, mapping class groups, Baumslag-Solitar groups and graph of groups.
In this talk I will show that Right-Angled Artin Groups have the optimal scl-gap of 1/2. This yields a new invariant for the vast class of subgroups of Right-Angled Artin Groups.

We begin by reviewing numerical methods for problems in one variable and find that univariate polynomials are the starting point for most of them.  A similar review in several variables, however, reveals that multivariate polynomials are not so important.  Why?  On the other hand in pure mathematics, the field of algebraic geometry is precisely the study of multivariate polynomials.  Why?