Abstract: A measure μ on a complete metric space X is 1-regular if the 1-dimensional density μ(B(x,r))/r converges, as r converges to 0, to a positive and finite number μ-almost everywhere. A classical result of Besicovitch states that 1-regular measures on Euclidean space are 1-rectifiable. The proof of this result is valid in much greater generality and essentially relies on a type of strict convexity of balls in X. On the other hand, the result is unknown in R^2 equipped with the supremum norm. 

In this talk we characterise the rectifiable and purely unrectifiable parts of a 1-regular measure on a metric space in terms of tangent measures. Naturally, we first study 1-uniform metric measure spaces: those measures for which the measure of every ball is some fixed constant times the radius. Indeed, we will show that, up to scaling the metric and measure by a positive constant, there exist exactly three 1-uniform metric measure spaces. 

Abstract: For more than a century, people have been trying to understand the precise connection between the properties of solutions of a PDE and the geometric properties of the set where the equation is given. The scenarios are particularly interesting when the coefficients of the equation are non-smooth and the set is rough. In this talk, I will present a unified characterization of rectifiability of a set of any codimension in terms of the Green function. This result is built on a series of earlier works on the Green function and a regularized distance to the set, which I will also discuss. This is based on joint works with Joseph Feneuil and Svitlana Mayboroda.

Abstract: The topic of my talk are functional inequalities for systems of orthonormal functions. The game is to obtain an optimal dependence of the constant on the number of functions involved that is better than just combining the triangle inequality with the one-function  bound. In this talk we focus on so-called spectral cluster bounds, which are concerned with $L^p$ norms of (linear combinations of) eigenfunctions of the Laplace-Beltrami operator on a compact closed manifold. Since the seminal work of Sogge in the 1980’s these bounds have been generalized in various directions. Frank and Sabin recently established a version of Sogge’s bound for systems of orthonormal functions. The result is valid for smooth metrics. We will show that the same result holds for $C^{1,1}$ metrics. The analogue in the one-function case was proved by Smith. The talk is based on joint ongoing work with Ngoc Nhi Nguyen. 

Abstract: The extension operator E of harmonic analysis is the Fourier transform of functions defined on curved hypersurfaces in R^n. The restriction conjecture claims that, when these hypersurfaces have non-vanishing Gaussian curvature, E is L^{2n/(n-1)}-bounded (which would imply that the level sets of the extension operator are small). On the other hand, the Mizohata-Takeuchi conjecture aims to understand the shape of these level sets, by establishing L^2-weighted bounds for E. In this talk we will discuss a "mix" of the two conjectures, establishing the analogue of the Mizohata-Takeuchi conjecture but with L^{2n/(n-1)}-weighted bounds instead. This is work in progress,  involving Anthony Carbery and Bassam Shayya.

Abstract: Spaces of Lipschitz mappings may be equipped, in various ways, with a metric in a such a way that they become complete metric spaces. It then makes sense to ask about the behaviour of a typical Lipschitz mapping in such spaces, where the word typical should be understood in the sense of the Baire Category Theorem. We will discuss recent and ongoing investigations of the behaviour exhibited by typical Lipschitz mappings, in particular with respect to differentiability. The talk is based on joint work with Olga Maleva (Birmingham).