I will describe joint work with Bruno Premoselli which gives a new existence theorem for negatively curved Einstein 4-manifolds, which are obtained by smoothing the singularities of hyperbolic cone metrics. Let (M_k) be a sequence of compact 4-manifolds and let g_k be a hyperbolic cone metric on M_k with cone angle \alpha (independent of k) along a smooth surface S_k. We make the following assumptions:

1. The injectivity radius i(k) of M_k tends to infinity (where in defining injectivity radius we ignore those geodesics which hit the cone singularity)

2. The normal injectivity radius of S_k is at least i(k)/2.

3. The area of the singular locii satisfy A(S_k)\leq C \exp(5 i(k)/2) for some C independent of k.

When these assumptions hold, we prove that for all large k, M_k carries a smooth Einstein metric of negative curvature. The proof involves a gluing theorem and a parameter dependent implicit function theorem (where k is the parameter). As I will explain, negative curvature plays an essential role in the proof. (For those who may be aware of our arxiv preprint, https://arxiv.org/abs/1802.00608 [arxiv.org], the work
I will describe has a new feature, namely we now treat all cone angles, and not just those which are greater than 2\pi. This gives lots more examples of Einstein 4-manifolds.)

String compactifications are essential for connecting string theory to low energy particle physics and cosmology. Moduli stabilisation gives rise to effective Lagrangians that capture the low-energy degrees of freedom. Much recent interest has been on swampland consistency conditions on such effective
field theories - which low energy Lagrangians can arise from quantum gravity? Furthermore, given that moduli stabilisation scenarios often exist in AdS space, we can also ask: what do swampland conditions mean in the context of AdS/CFT? I describe work on developing a holographic understanding of moduli stabilisation and swampland consistency conditions. I focus in particular on the Large Volume Scenario, which is especially appealing from a holographic perspective as in the large volume limit all its interactions can be expressed solely in terms of the AdS radius, with no free dimensionless parameters.

Integrable models provide simplified examples of quantum field theories with self-interaction. As often in relativistic quantum theory, their local observables are difficult to control mathematically. One either tries to construct pointlike local quantum fields, leading to possibly divergent series expansions, or one defines the local observables indirectly via wedge-local quantities, losing control over their explicit form.

We propose a new, hybrid approach: We aim to describe local quantum fields; but rather than exhibiting their n-point functions and verifying the Wightman axioms, we establish them as closed operators affiliated with a net of von Neumann algebras. This is shown to work at least in the Ising model.

The renormalized expectation value of the stress energy tensor (RSET) is an object of central importance in quantum field theory in curved space-time, but calculating this on black hole space-times is far from trivial.  The vacuum polarization (VP) of a quantum scalar field is computationally simpler and shares some features with the RSET.  In this talk we consider the properties of the VP for a massless, conformally coupled scalar field on asymptotically anti-de Sitter black holes with spherical, flat and hyperbolic horizons.  We focus on the effect of the different horizon curvature on the VP, and the role played by the boundary conditions far from the black hole.     

I will review the idea that entanglement must ultimately be understood in terms of modes, rather than in terms of particles. The most striking instance of mode entanglement is a single particle entangled state, which I will discuss both in the case of bosons as well as in the case of fermions. I then proceed to show that the Aharonov-Bohm effect can be understood by using a single electron entangled state. Finally, I will argue that this demonstrates beyond doubt that the Aharonov-Bohm effect is non non-local, contrary to what is frequently claimed in the literature.

After reviewing our recent description of generalized Kahler structures in terms of holomorphic symplectic Morita equivalence, I will describe how this can be used for explicit constructions of toric generalized Kahler metrics.  Then I will describe how these ideas, combined with concepts from geometric quantization, provide a new approach to noncommutative algebraic geometry.

Mitral regurgitation is one of the most common valve diseases in the UK and contributes to 50% of the transcatheter mitral valve replacement (TMVR) procedures with bioprosthetic valves. TMVR is generally performed in frailer, older patients unlikely to tolerate open-heart surgery or further interventions. One of the side effects of implanting a bioprosthetic valve is a condition known as left ventricular outflow obstruction, whereby the implanted device can partially obstruct the outflow of blood from the left ventricle causing high flow resistance. The ventricle has then to pump more vigorously to provide adequate blood supply to the circulatory system and becomes hypertrophic. This ultimately results in poor contractility and heart failure.
We developed personalised image-based models to characterise the complex relationship between anatomy, blood flow, and ventricular function both before and after TMVR. The model prediction provides key information to match individual patient and device size, such as postoperative changes in intraventricular pressure gradients and blood residence time. Our pilot data from a cohort of 7 TMVR patients identified a correlation between the degree of outflow obstruction and the deterioration of ventricular function: when approximately one third of the outflow was obstructed as a result of the device implantation, significant increases in the flow resistance and the average time spent by the blood inside the ventricle were observed, which are in turn associated with hypertrophic ventricular remodelling and blood stagnation, respectively. Currently, preprocedural planning for TMVR relies largely on anecdotal experience and standard anatomical evaluations. The haemodynamic knowledge derived from the models has the potential to enhance significantly pre procedural planning and, in the long term, help develop a personalised risk scoring system specifically designed for TMVR patients.