UCL

We develop a new construction of complete non-compact 8-manifolds with holonomy equal to $\Spin(7)$. As a consequence of the holonomy reduction, these manifolds are Ricci-flat. These metrics are built on the total spaces of principal $T^2$-bundles over asymptotically conical Calabi Yau manifolds. The resulting metrics have a new geometry at infinity that we call asymptotically $T^2$-fibred conical ($AT^2C$) and which generalizes to higher dimensions the ALG metrics of 4-dimensional hyperkähler geometry. We use the construction to produce infinite diffeomorphism types of $AT^2C$ $\Spin(7)$-manifolds and to produce the first known example of complete toric $\Spin(7)$-manifold.

Determining stationary compact configurations of vorticity described by the 2D Euler equations is a classic problem dating back to the late 19th century. The aim is to find equilibrium distributions of vorticity, in the form  of point vortices, vortex sheets, vortex patches, and hollow vortices. This endeavour has driven the development of mathematical and numerical techniques such as Hamiltonian vortex dynamics and contour dynamics.

In the case of vortex sheets, methods and results are presented for finding rotating equilibria, some in the presence of point vortices. To begin, a numerical approach based on that recently developed by Trefethen, Costa, Baddoo, and others for solving Laplace's equation in the complex plane by series and rational approximation is described. The method successfully reproduces the exact vortex sheet solutions found by O'Neil (2018) and Protas & Sakajo (2020). Some new solutions are found.

The numerical approach suggests an analytical method based on conformal mapping for finding exact closed-form vortex sheet equilibria. Examples are presented.

Finally, new numerical solutions are computed for steady, doubly-connected vortex layers of uniform vorticity surrounding a solid object such that the fluid velocity vanishes on the outer free boundary. While dynamically unrelated, these solutions have mathematical analogy and application to the industrial free boundary problem arising in the dip-coating of objects by a viscous fluid.

This talk is rescheduled and will take place on 21 January 2025

Let X --> S be a family of algebraic varieties parametrized by an infinitesimal disk S, possibly of mixed characteristic. The Bloch conductor conjecture expresses the difference of the Euler characteristics of the special and generic fibers in algebraic and arithmetic terms. I'll describe a proof of some new cases of this conjecture, including the case of isolated singularities. The latter was a conjecture of Deligne generalizing Milnor's formula on vanishing cycles. 

This is joint work with Massimo Pippi; our methods use derived and non-commutative algebraic geometry. 

The PDE formulation of Mean Field Games (MFG) is described by nonlinear systems in which a Hamilton—Jacobi—Bellman (HJB) equation and a Kolmogorov—Fokker—Planck (KFP) equation are coupled. The advective term of the KFP equation involves a partial derivative of the Hamiltonian that is often assumed to be continuous. However, in many cases of practical interest, the underlying optimal control problem of the MFG may give rise to bang-bang controls, which typically lead to nondifferentiable Hamiltonians. In this talk we present results on the analysis and numerical approximation of second-order MFG systems for the general case of convex, Lipschitz, but possibly nondifferentiable Hamiltonians.
In particular, we propose a generalization of the MFG system as a Partial Differential Inclusion (PDI) based on interpreting the partial derivative of the Hamiltonian in terms of subdifferentials of convex functions.

We present theorems that guarantee the existence of unique weak solutions to MFG PDIs under a monotonicity condition similar to one that has been considered previously by Lasry & Lions. Moreover, we introduce a monotone finite element discretization of the weak formulation of MFG PDIs and prove the strong convergence of the approximations to the value function in the H¹-norm and the strong convergence of the approximations to the density function in L^q-norms. We conclude the talk with some numerical experiments involving non-smooth solutions. 

This is joint work with my supervisor Iain Smears. 

Abstract: We give a generalized Kuznetsov formula that allows one to impose additional conditions at finitely many primes.  The formula arises from the relative trace formula. I will discuss applications to spectral large sieve inequalities and subconvexity. This is work in progress with M.P. Young and Y. Hu.

In March 2020, as COVID-19 cases started to surge for the first time in the UK, a team spanning UCL engineers, University College London Hospital (UCLH) intensivists and Mercedes Formula 1 came together to design, manufacture and deploy non-invasive breathing aids for COVID-19 patients. We reverse engineered and an off-patent CPAP (continuous positive airways pressure) device, the Philips WhisperFlow, and changed its design to minimise its oxygen utilisation (given that hospital oxygen supplies are under extreme demand). The UCL-Ventura received regulatory approvals from the MHRA within 10 days, and Mercedes HPP manufactured 10,000 devices by mid-April. UCL-Ventura CPAPs are now in use in over 120 NHS hospitals.

In response to international need, the team released all blueprints open source to enable local manufacture in other countries, alongside a support package spanning technical, manufacturing, clinical and regulatory components. The designs have been downloaded 1900 times across 105 countries, and around 20 teams are now manufacturing at scale and deploying in local hospitals. We have also worked closely with NGOs, on a non-profit basis, to deliver devices directly to countries with urgent need, including Palestine, Uganda and South Africa.

A connected matching is a matching contained in a connected component. A well-known method due to Łuczak reduces problems about monochromatic paths and cycles in complete graphs to problems about monochromatic matchings in almost complete graphs. We show that these can be further reduced to problems about monochromatic connected matchings in complete graphs.
    
I will describe Łuczak's reduction, introduce the new reduction, and mention potential applications of the improved method.

An 8-dimensional Riemannian manifold with holonomy group contained in Spin(7) is Ricci-flat, but not Kahler. The condition that the holonomy reduces to Spin(7) is equivalent to a complicated system of non-linear PDEs. In the non-compact setting, symmetries can be used to reduce this complexity. In the case of cohomogeneity one manifolds, i.e. where a generic orbit has codimension one, the non-linear PDE system
reduces to a nonlinear ODE system. I will discuss recent progress in the construction of 1-parameter families of complete cohomogeneity one Spin(7) holonomy metrics. All examples are asymptotically conical (AC) or asymptotically locally conical (ALC).