Abstract: L\'evy processes are increasingly popular for modelling stochastic process data with jump behaviour. In practice statisticians only observe discretely sampled increments of the process, leading to a statistical inverse problem. To understand the jump behaviour of the process one needs to make inference on the infinite-dimensional parameter given by the L\'evy measure. We discuss recent developments in the analysis of this problem, including in particular functional limit theorems for commonly used estimators of the generalised distribution function of the L\'evy measure, and their application to statistical uncertainty quantification methodology (confidence bands and tests). 

We consider the problem of minimising the commute or shuttle time for a diffusion between the endpoints of an interval. The control is the scale function for the diffusion. We show that the dynamic version of the problem has the same solution as the static version if we start at an end point and consider the much harder case where the starting point is in the interior.

We present some recent work - joint with Arno Fehm - in which we give an `existential Ax-Kochen-Ershov principle' for equicharacteristic henselian valued fields. More precisely, we show that the existential theory of such a valued field depends only on the existential theory of the residue field. In residue characteristic zero, this result is well-known and follows from the classical Ax-Kochen-Ershov Theorems. In arbitrary (but equal) characteristic, our proof uses F-V Kuhlmann's theory of tame fields. One corollary is an unconditional proof that the existential theory of F_q((t)) is decidable. We will explain how this relates to the earlier conditional proof of this result, due to Denef and Schoutens.
 

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The field of transseries was introduced by Ecalle to give a solution to Dulac's problem, a weakening of Hilbert's 16th problem. They form an elementary extension of the real exponential field and have received the attention of model theorists. Another such elementary extension is given by Conway's surreal numbers, and various connections with the transseries have been conjectured, among which the possibility of introducing a Hardy type derivation on the surreal numbers. I will present a complete solution to these conjectures obtained in collaboration with Vincenzo Mantova.
 

 We describe how cross-kernel matrices, that is, kernel matrices between the data and a custom chosen set of `feature spanning points' can be used for learning. The main potential of cross-kernel matrices is that (a) they provide Nyström-type speed-ups for kernel learning without relying on subsampling, thus avoiding potential problems with sampling degeneracy, while preserving the usual approximation guarantees and the attractive linear scaling of standard Nyström methods and (b) the use of non-square matrices for kernel learning provides a non-linear generalization of the singular value decomposition and singular features. We present a novel algorithm, Ideal PCA (IPCA), which is a cross-kernel matrix variant of PCA, showcasing both advantages: we demonstrate on real and synthetic data that IPCA allows to (a) obtain kernel PCA-like features faster and (b) to extract novel features of empirical advantage in non-linear manifold learning and classification.

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The Renormalization Group (RG) was pioneered by the physicist Kenneth Wilson in the early 70's and since then it has become a fundamental tool in physics. RG remains the most general philosophy for understanding how many models in statistical mechanics behave near their critical point but implementing RG analysis in a mathematically rigorous way remains quite challenging.

I will describe how analysis of RG flows translate into statements about continuum limits, universality, and cross-over phenomena - as a concrete example I will speak about some joint work with Abdelmalek Abdesselam and Gianluca Guadagni.

Let $K\subset {\mathbb R}$ be a compact definable set in an o-minimal structure over $\mathbb R$, e.g. a semi-algebraic or a real analytic set. A definable family $\{S_\delta\ |  0<\delta\in{\mathbb R}\}$ of compact subsets of $K$, is called a monotone family if $S_\delta\subset S_\eta$ for all sufficiently small $\delta>\eta>0$. The main result in the talk is that when $\dim K=2$ or $\dim K=n=3$ there exists a definable triangulation of $K$ such that for each (open) simplex $\Lambda$ of the triangulation and each small enough $\delta>0$, the intersections $S_\delta\cap\Lambda$ is equivalent to one of five (respectively, nine) standard families in the standard simplex (the equivalence relation and a standard family will be formally defined). As a consequence, we prove the two-dimensional case of the topological conjecture on approximation of definable sets by compact families.

This is joint work with Andrei Gabrielov (Purdue).