University College London

Joint work Marta Betcke and Bolin Pan

In photoacoustic tomography (PAT) with flat sensor, we routinely encounter two types of limited data. The first is due to using a finite sensor and is especially perceptible if the region of interest is large relatively to the sensor or located farther away from the sensor. In this talk we focus on the second type caused by a varying sensitivity of the sensor to the incoming wavefront direction which can be modelled as binary i.e. by a cone of sensitivity. Such visibility conditions result, in Fourier domain, in a restriction of the data to a bowtie, akin to the one corresponding to the range of the forward operator but further narrowed according to the angle of sensitivity. 

We show how we can separate the visible and invisible wavefront directions in PAT image and data using a directional frame like Curvelets, and how such decomposition allows for decoupling of the reconstruction involving application of expensive forward/adjoint solvers from the training problem. We present fast and stable approximate Fourier domain forward and adjoint operators for reconstruction of the visible coefficients for such limited angle problem and a tailored UNet matching both the multi-scale Curvelet decomposition and the partition into the visible/invisible directions for learning the invisible coefficients from a training set of similar data.

For many spaces of interest to number theorists one can construct cycles which in some ways behave like the coefficients of modular forms. The aim of this talk is to give an introduction to this idea by focusing on examples coming from modular curves and Heegner points and the relevant work of Zagier, Gross-Kohnen-Zagier and Borcherds. If time permits I will discuss generalizations to other spaces.

Harmonic maps from surfaces to other manifolds is a fundamental object of geometric analysis with many applications, for example to minimal surfaces. In particular, there are many available methods of constructing them such, such as using complex geometry, min-max methods or flow techniques. By contrast, much less is known for harmonic maps from higher dimensional manifolds. In the present talk I will explain the role of dimension in this problem and outline the recent joint work with D. Stern, where we provide a min-max construction for higher-dimensional harmonic maps. If time permits, an application to eigenvalue optimisation problems will be discussed. Based on joint work with D. Stern.

The elliptic Gamma function — a generalization of the q-Gamma function, which is itself the q-analog of the ordinary Gamma function — is a meromorphic special function in several variables that mathematical physicists have shown to satisfy modular functional equations under SL(3,Z). In this talk I will present evidence (numerical and theoretical) that this function often takes algebraic values that satisfy explicit reciprocity laws and that are related to derivatives of Hecke L-functions at s=0. Thus this function conjecturally allows to extend the theory of complex multiplication to complex cubic fields as envisioned by Hilbert's 12^th problem. This is joint work with Nicolas Bergeron and Pierre Charollois.

The formulation of Mean Field Games (MFG) via partial differential equations typically requires continuous differentiability of the Hamiltonian in order to determine the advective term in the Kolmogorov--Fokker--Planck equation for the density of players. However, in many cases of practical interest, the underlying optimal control problem may exhibit bang-bang controls, which typically lead to nondifferentiable Hamiltonians. In this talk we will present results on the analysis and numerical approximation of stationary second-order MFG systems for the general case of convex, Lipschitz, but possibly nondifferentiable Hamiltonians. In particular, we will propose a generalization of the MFG system as a Partial Differential Inclusion (PDI) based on interpreting the derivative of the Hamiltonian in terms of subdifferentials of convex functions. We present results that guarantee the existence of unique weak solutions to the stationary MFG PDI under a monotonicity condition similar to one that has been considered previously by Lasry and Lions. Moreover, we will propose a monotone finite element discretization of the weak formulation of the MFG PDI, and present results that confirm the strong H^1-norm convergence of the approximations to the value function and strong L^q-norm convergence of the approximations to the density function. The performance of the numerical method will be illustrated in experiments featuring nonsmooth solutions. This talk is based on joint work with my supervisor Iain Smears.

The design and analysis of finite element methods for high Reynolds flow remains a challenging task, not least because of the difficulties associated with turbulence. In this talk we will first revisit some theoretical results on interior penalty methods using equal order interpolation for smooth solutions of the Navier-Stokes’ equations at high Reynolds number and show some recent computational results for turbulent flows.

Then we will focus on so called pressure robust methods, i.e. methods where the smoothness of the pressure does not affect the upper bound of error estimates for the velocity of the Stokes’ system. We will discuss how convection can be stabilized for such methods in the high Reynolds regime and, for the lowest order case, show an interesting connection to turbulence modelling.

In this talk we will look at the different ways to define city boundaries, and the relevance to consider socio-demographic and spatial connectivity in urban systems, in particular if interventions are to be considered.

How can one integrate singular functions over fractals? And why would one want to do this? In this talk I will present a general approach to numerical quadrature on the compact attractor of an iterated function system of contracting similarities, where integration is with respect to the relevant Hausdorff measure. For certain singular integrands of logarithmic or algebraic type the self-similarity of the integration domain can be exploited to express the singular integral exactly in terms of regular integrals that can be approximated using standard techniques. As an application we show how this approach, combined with a singularity-subtraction technique, can be used to accurately evaluate the singular double integrals that arise in Hausdorff-measure Galerkin boundary element methods for acoustic wave scattering by fractal screens. This is joint work with Andrew Gibbs (UCL) and Andrea Moiola (Pavia).

In this talk, we describe some recent work on applying tools from category theory in finite model theory, descriptive complexity, constraint satisfaction, and combinatorics.

The motivations for this work come from Computer Science, but there may be something of interest for model theorists and other logicians.

The basic setting involves studying the category of relational structures via a resource-indexed family of adjunctions with some process category - which unfolds relational structures into treelike forms, allowing natural resource parameters to be assigned to these unfoldings.

One basic instance of this scheme allows us to recover, in a purely structural, syntax-free way:

- the Ehrenfeucht-Fraisse game

- the quantifier rank fragments of first-order logic

- the equivalences on structures induced by (i) the quantifier rank fragments, (ii) the restriction to the existential-positive part, and (iii) the extension with counting quantifiers

- the combinatorial parameter of tree-depth (Nesetril and Ossona de Mendez).

Another instance recovers the k-pebble game, the finite-variable fragments, the corresponding equivalences, and the combinatorial parameter of treewidth.

Other instances cover modal, guarded and hybrid fragments, generalized quantifiers, and a wide range of combinatorial parameters.

This whole scheme has been axiomatized in a very general setting, of arboreal categories and arboreal covers.

Beyond this basic level, a landscape is beginning to emerge, in which structural features of the resource categories, adjunctions and comonads are reflected in degrees of logical and computational tractability of the corresponding languages.

Examples include semantic characterisation and preservation theorems, Lovasz-type results on  isomorphisms, and classification of constraint satisfaction problems.