Differential equations and neural networks are two of the most widespread modelling paradigms. I will talk about how to combine the best of both worlds through neural differential equations. These treat differential equations as a learnt component of a differentiable computation graph, and as such integrates tightly with current machine learning practice. Applications are widespread. I will begin with an introduction to the theory of neural ordinary differential equations, which may for example be used to model unknown physics. I will then move on to discussing recent work on neural controlled differential equations, which are state-of-the-art models for (arbitrarily irregular) time series. Next will be some discussion of neural stochastic differential equations: we will see that the mathematics of SDEs is precisely aligned with the machine learning of GANs, and thus NSDEs may be used as generative models. If time allows I will then discuss other recent work, such as how the training of neural differential equations may be sped up by ~40% by tweaking standard numerical solvers to respect the particular nature of the differential equations. This is joint work with Ricky T. Q. Chen, Xuechen Li, James Foster, and James Morrill.

Let $p$ be a prime. Minkowski (1887) gave a bound for the order of a finite $p$-subgroup of the linear group $\mathsf{GL}(n,\mathbf Z)$ as a function of $n$, and this necessarily holds for $p$-subgroups of $\mathsf{GL}(n,\mathbf Q)$ also. A few years ago, Serre asked me whether some analogous result might be obtained for subgroups of $\mathsf{GL}(n,\mathbf C)$ using the methods I employed to obtain optimal bounds for Jordan's theorem.

Bounds can be so obtained and I will explain how but, while Minkowski's bound is achieved, no linear bound (as Serre initially suggested) can be achieved. I will discuss progress on this problem and the issues that arise in seeking an ideal form for the solution.

The Zilber-Pink conjecture predicts how large the intersection of a d-dimensional subvariety of an abelian variety/algebraic torus/Shimura variety/... with the union of special subvarieties of codimension > d can be (where the definition of "special" depends on the setting). In joint work with Fabrizio Barroero, we have reduced this conjecture for complex abelian varieties to the same conjecture for abelian varieties defined over the algebraic numbers. In work in progress, we introduce the notion of a distinguished category, which contains both connected commutative algebraic groups and connected mixed Shimura varieties. In any distinguished category, special subvarieties can be defined and a Zilber-Pink statement can be formulated. We show that any distinguished category satisfies the defect condition, introduced as a useful technical tool by Habegger and Pila. Under an additional assumption, which makes the category "very distinguished", we show furthermore that the Zilber-Pink statement in general follows from the case where the subvariety is defined over the algebraic closure of the field of definition of the distinguished variety. The proof closely follows our proof in the case of abelian varieties and leads also to unconditional results in the moduli space of principally polarized abelian surfaces as well as in fibered powers of the Legendre family of elliptic curves.

Motivated by string dualities we propose topological gravity as the early phase of our universe.  The topological nature of this phase naturally leads to the explanation of many of the puzzles of early universe cosmology.  A concrete realization of this scenario using Witten's four dimensional topological gravity is considered.  This model leads to the power spectrum of CMB fluctuations which is controlled by the conformal anomaly coefficients $a,c$.  In particular the strength of the fluctuation is controlled by $1/a$ and its tilt by $c g^2$ where $g$ is the coupling constant of topological gravity.  The positivity of $c$, a consequence of unitarity, leads automatically to an IR tilt for the power spectrum.   In contrast with standard inflationary models, this scenario predicts $\mathcal{O}(1)$ non-Gaussianities for four- and higher-point correlators and the absence of tensor modes in the CMB fluctuations.

Coadmissible modules over Frechet-Stein algebras arise naturally in p-adic representation theory, e.g. in the study of locally analytic representations of p-adic Lie groups or the function spaces of rigid analytic Stein spaces. We show that in many cases, the category of coadmissible modules admits an exact and fully faithful embedding into the category of complete bornological modules, also preserving tensor products. This allows us to introduce derived methods to the study of coadmissible modules without forsaking the analytic flavour of the theory. As an application, we introduce six functors for Ardakov-Wadsley's D-cap-modules and discuss some instances where coadmissibility (in a derived sense) is preserved.

Multivariate point processes are used to model event-type data in a wide range of domains. One interesting application is to model the emission of electric impulses of biological neurons. In this context, the point process model needs to capture the time-dependencies and interactions between neurons, which can be of two kinds: exciting or inhibiting. Estimating these interactions, and in particular the functional connectivity of the neurons are problems that have gained a lot of attention recently. The general nonlinear Hawkes process is a powerful model for events occurring at multiple locations in interaction. Although there is an extensive literature on the analysis of the linear model, the probabilistic and statistical properties of the nonlinear model are still mainly unknown. In this paper, we consider nonlinear Hawkes models and, in a Bayesian nonparametric inference framework, derive concentration rates for the posterior distribution.  We also infer the graph of interactions between the dimensions of the process and prove that the posterior distribution is consistent on the graph adjacency matrix.

In this talk I will present an efficient algorithm to produce a provably dense sample of a smooth compact algebraic variety. The procedure is partly based on computing bottlenecks of the variety. Using geometric information such as the bottlenecks and the local reach we also provide bounds on the density of the sample needed in order to guarantee that the homology of the variety can be recovered from the sample.