(Oxford University)

The Hull--Strominger system is a system of non-linear PDEs on heterotic string theory involving a pair of Hermitian metrics $(g,h)$ on a six dimensional manifold $M$. One of these equations dictates the metric $g$ on $M$ to be conformally balanced. We will begin the talk by giving a description of the geometry of cohomogeneity one manifolds and SU(3)-structures. Then, we will look for solutions to the Hull--Strominger system in the cohomogeneity one setting. We show that a six-dimensional simply connected cohomogeneity one manifold under the almost effective action of a connected Lie group $G$ admits no $G$-invariant balanced non-Kähler SU(3)-structures. This is a joint work with F. Salvatore.

Cohomological Hall algebras and vertex algebras are two structures whose origins are (at least in part) from physics. I will explain what these objects are, how the latter was related to moduli stacks by Joyce, and a theorem relating these two structures. The main tool is torus localisation, a method for "turning geometry into combinatorics", or rather a new formulation of it which works in the singular setting.

A field extension $F/K$ in characteristic $p$ is purely inseparable if for each $x$ in $F$, some power $x^{p^n}$ belongs to $K$. Using methods from homotopy theory, we construct a Galois correspondence for finite purely inseparable field extensions $F/K$, generalising a classical result of Jacobson for extensions of exponent one (where $x^p$ belongs to $K$ for all $x$ in $F$). This is joint work with Waldron.

It is expected that complete noncompact Calabi-Yau manifolds are in some sense governed by their asymptotics at infinity. In the maximal volume growth case, the asymptotics at infinity are given by Calabi-Yau cones. We are interested in deformations of such metrics that fix the asymptotic cones at infinity. In the asymptotically conical case, Conlon-Hein proved uniqueness under such deformations. Their method is based on the corresponding linearized problem, namely the study of subquadratic harmonic functions. We generalize their work to the maximal volume growth case, allowing the tangent cones at infinity to have non-isolated singularities. Part of the talk is based on work in progress joint with Gabor Szekelyhidi.

In the mean curvature flow, translating solutions are an important model for singularity formation. In this talk, I will describe the asymptotic structure of 2D mean curvature flow translators embedded in R^3 which have finite total curvature, which turns out to be highly rigid. I will outline the proof of this asymptotic description, in particular focusing on some novel and unexpected features of this proof.

I will discuss the multi-sorted structure of analytic covers H -> Y(N), where H is the upper half-plane and Y(N) are the N-level modular curves, all N, in a certain language, weaker than the language applied by Adam Harris and Chris Daw.  We define a certain locally modular reduct of the structure which is called "pure" structure - an extension of the structure of special subvarieties.  
The problem of non-elementary categorical axiomatisation for this structure is closely related to the theory of "canonical models for Shimura curves", in particular, the description of Gal_Q action on the CM-points of the Y(N). This problem for the case of curves is basically solved (J.Milne) and allows the beautiful interpretation in our setting:  the abstract automorphisms of the pure structure on CM-points are exactly the automorphisms induced by Gal_Q.  Using this fact and earlier theorem of Daw and Harris we prove categoricity of a natural axiomatisation of the pseudo-analytic structure.
If time permits I will also discuss a problem which naturally extends the above:  a categoricity statement for the structure of unramified analytic covers H -> X, where X runs over all smooth curves over a given number field.  

Interacting particle systems have found diverse applications in mathematics and several related fields, including statistical physics, population dynamics, and machine learning.  We will focus, in particular, on the zero range process and the symmetric simple exclusion process.  The large-scale behavior of these systems is essentially deterministic, and is described in terms of a hydrodynamic limit.  However, the particle process does exhibit large fluctuations away from its mean.  Such deviations, though rare, can have significant consequences---such as a concentration of energy or the appearance of a vacuum---which make them important to understand and simulate.

In this talk, which is based on joint work with Benjamin Gess, I will introduce a continuum model for simulating rare events in the zero range and symmetric simple exclusion process.  The model is based on an approximating sequence of stochastic partial differential equations with nonlinear, conservative noise.  The solutions capture to first-order the central limit fluctuations of the particle system, and they correctly simulate rare events in terms of a large deviations principle.

Recent advances in computing power and the potential to make more realistic assumptions due to increased flexibility have led to the increased prevalence of simulation models in economics. While models of this class, and particularly agent-based models, are able to replicate a number of empirically-observed stylised facts not easily recovered by more traditional alternatives, such models remain notoriously difficult to estimate due to their lack of tractable likelihood functions. While the estimation literature continues to grow, existing attempts have approached the problem primarily from a frequentist perspective, with the Bayesian estimation literature remaining comparatively less developed. For this reason, we introduce a widely-applicable Bayesian estimation protocol that makes use of deep neural networks to construct an approximation to the likelihood, which we then benchmark against a prominent alternative from the existing literature.
 

TBC
 

We explore reinforcement learning methods for finding the optimal policy in the linear quadratic regulator (LQR) problem. In particular, we consider the convergence of policy gradient methods in the setting of known and unknown parameters. We are able to produce a global linear convergence guarantee for this approach in the setting of finite time horizon and stochastic state dynamics under weak assumptions. The convergence of a projected policy gradient method is also established in order to handle problems with constraints. We illustrate the performance of the algorithm with two examples. The first example is the optimal liquidation of a holding in an asset. We show results for the case where we assume a model for the underlying dynamics and where we apply the method to the data directly. The empirical evidence suggests that the policy gradient method can learn the global optimal solution for a larger class of stochastic systems containing the LQR framework and that it is more robust with respect to model mis-specification when compared to a model-based approach. The second example is an LQR system in a higher-dimensional setting with synthetic data.