Oxford

The aim of this talk is to describe joint work with Gergely Berczi using a recent extension to non-reductive actions of geometric invariant theory, and its links with moment maps in symplectic geometry, to study hyperbolicity of generic hypersurfaces in a projective space. Using intersection theory for non-reductive GIT quotients applied to  compactifications of bundles of invariant jet differentials over complex manifolds leads to a proof of the Green-Griffiths-Lang conjecture for a generic projective hypersurface of dimension n whose degree is greater than n^6. A recent result of Riedl and Yang then implies the Kobayashi conjecture for generic hypersurfaces of degree greater than (2n-1)^6.

Given a smooth variety X with a normal crossings divisor D (or more generally a smooth log variety) we consider the ring of logarithmic differential operators: the subring of differential operators on X generated by vector fields tangent to D. Modules over this ring are called logarithmic D-modules and generalize the classical theory of regular meromorphic connections. They arise naturally when considering compactifications.

We will discuss which parts of the theory of D-modules generalize to the logarithmic setting and how to overcome new challenges arising from the logarithmic structure. In particular, we will define holonomicity for log D-modules and state a conjectural extension of the famous Riemann-Hilbert correspondence. This talk will be very example-focused and will not require any previous knowledge of D-modules or logarithmic geometry. This is joint work with Mattia Talpo.
 

Stability conditions on triangulated categories were introduced by Bridgeland, based on ideas from string theory. Conjecturally, they control existence of solutions to the deformed Hermitian Yang-Mills equation and the special Lagrangian equation (on the A-side and B-side of mirror symmetry, respectively). I will focus on the symplectic side and sketch a program which replaces special Lagrangians by "spectral networks", certain graphs enhanced with algebraic data. Based on joint work in progress with Katzarkov, Konstevich, Pandit, and Simpson.

The two courses I took from Wilkinson as a graduate student at Stanford influenced me greatly.  Along with some reminiscences of those days, this talk will touch upon backward error analysis, Gaussian elimination, and Evariste Galois.  It was originally presented at the Wilkinson 100th Birthday conference in Manchester earlier this year.

Subspace methods have the potential to outperform conventional methods, as the derivatives only need to be computed in a smaller dimensional subspace. The sub-problem that needs to be solved at each iteration is also smaller in size, and thus the Linear Algebra cost is also lower. However, if the subspace is not selected "properly", the progress per iteration can be significantly much lower than the progress of the equivalent full-space method. Thus, "improper" selection of the subspace results in subspace methods which are actually more computationally expensive per unit of progress than their full-space alternatives. The popular subspace selection methods (such as randomized) fall into this category when the objective function does not have a known (exploitable) structure. We provide a simple and effective rule to choose the subspace in the "right way" when the objective function does not have a structure. We focus on Gauss-Newton and Least-Squares, but the idea can be generalised to any other solvers and/or objective functions. We show theoretically that the cost of this strategy per unit progress lies in between (approximately) 50% and 100% of the cost of Gauss-Newton, and give an intuition why in practice, it should be closer to the favorable end of the spectrum. We confirm these expectations by running numerical experiments on the CUTEst32 test set. We also compare the proposed selection method with randomized subspace selection. We briefly show that the method is globally convergent and has a 2-step quadratic asymptotic rate of convergence for zero-residual problems.
 

The Approximate Message Passing (AMP) algorithm is a powerful iterative method for reconstructing undersampled sparse signals. Unfortunately, AMP is sensitive to the type of sensing matrix employed and frequently encounters convergence problems. One case where AMP tends to fail is compressed sensing MRI, where Fourier coefficients of a natural image are sampled with variable density. An AMP-inspired algorithm constructed specifically for MRI is presented that exhibits a 'state evolution', where at every iteration the image estimate before thresholding behaves as the ground truth corrupted by Gaussian noise with known covariance. Numerical experiments explore the practical benefits of such effective noise behaviour.
 

Boundary integral equations (BIEs) are well established for solving scattering at bounded infinitely thin objects, so-called screens, which are modelled as “open surfaces” in 3D and as “open curves” in 2D. Moreover, the unknowns of these BIEs are the jumps of traces across $\Gamma$. Things change considerably when considering scattering at multi-screens, which are arbitrary arrangements of thin panels that may not be even locally orientable because of junction points (2D) or junction lines (3D). Indeed, the notion of jumps of traces is no longer meaningful at these junctions. This issue can be solved by switching to a quotient space perspective of traces, as done in recent work by Claeys and Hiptmair. In this talk, we present the extension of the quotient-space approach to the Galerkin boundary element (BE) discretization of first-kind BIEs. Unlike previous approaches, the new quotient-space BEM relies on minimal geometry information and does not require any special treatment at junctions. Moreover, it allows for a rigorous numerical analysis.
 

Topology optimisation finds the optimal material distribution of a fluid or solid in a domain, subject to PDE and volume constraints. There are many formulations and we opt for the density approach which results in a PDE, volume and inequality constrained, non-convex, infinite-dimensional optimisation problem without a priori knowledge of a good initial guess. Such problems can exhibit many local minima or even no minima. In practice, heuristics are used to obtain the global minimum, but these can fail even in the simplest of cases. In this talk, we will present an algorithm that solves such problems and systematically discovers as many of these local minima as possible along the way.  

Ocean biogeochemical models used in climate change predictions are very computationally expensive and heavily parameterised. With derivatives too costly to compute, we optimise the parameters within one such model using derivative-free algorithms with the aim of finding a good optimum in the fewest possible function evaluations. We compare the performance of the evolutionary algorithm CMA-ES which is a stochastic global optimization method requiring more function evaluations, to the Py-BOBYQA and DFO-LS algorithms which are local derivative-free solvers requiring fewer evaluations. We also use initial Latin Hypercube sampling to then provide DFO-LS with a good starting point, in an attempt to find the global optimum with a local solver. This is joint work with Coralia Cartis and Samar Khatiwala.
 

Complex spatial patterns such as superlattice patterns and quasipatterns occur in a variety of physical systems ranging from vibrated fluid layers to crystallising soft matter. Reduced order models that describe such systems are usually PDEs. Close to a phase transition, modal expansion along with perturbation methods can be applied to convert the PDEs to normal form equations in the form of coupled ODEs. I use equivariant bifurcation theory along with homotopy methods (developed in computational algebraic geometry) to obtain all solutions of the normal form equations in a non-iterative method. I want to talk about how this approach allows us to ask new questions about the physical systems of interest and what extensions to this method might be possible. This forms a step in my long-term interest to explore how to better ‘complete’ a bifurcation diagram!