Solutions to elliptic boundary value problems on polygonal domains exhibit singularities at edges and corners. On the other hand, there has been much recent interest in nonlocal equations, e.g. involving the fractional Laplacian. We discuss a unified approach to their analysis, based on pseudodifferential techniques developed for elliptic differential equations with mixed boundary conditions. For the fractional Dirichlet problem in a domain with smooth boundary, our approach recovers recent results of G. Grubb, using independent methods. Applications of the results will be briefly discussed. (Joint with Heiko Gimperlein, Rafe Mazzeo, and Jakub Stocek.)

Let $X_n$ be a random vector uniformly distributed on the unit ball of $\ell_p^n$ and let $E_n$ be a random $k_n$-dimensional subspace with $\frac{k_n}{n}\to\lambda\in[0,1]$. We will study the behavior of the random variable $\Vert P_{E_n}X_n\Vert_2$ showing that it verifies a Central Limit Theorem as well as some Large Deviation Principles.

We consider a nonlinear Schrödinger equation with third-order dispersion and derivative nonlinearity, which arises as a mathematical model for the photonic crystal fiber oscillator. Although in the non-periodic case the associated Cauchy problem is known to be locally well-posed in Sobolev spaces, for the periodic problem we see that the "resonant" part of derivative nonlinearity causes ill-posedness (more precisely, non-existence of local-in-time solutions) of the Cauchy problem in Sobolev spaces and Gevrey classes. This talk is based on a joint work with Yoshio Tsutsumi (Kyoto University).

We show that the Cauchy problem associated to a class of dispersive perturbations of Burgers' equations containing the low dispersion Benjamin-Ono equation $$ \partial_tu-D_x^{\alpha}\partial_xu+u\partial_xu=0 \, ,$$ with $0<\alpha \le 1$, is locally well-posed in $H^s(\mathbb R)$ for $s>s_\alpha: = \frac 32-\frac {5\alpha} 4$. Then, as a consequence of this result and of the hamiltonian structure of the equation, we obtain global well-posedness in the energy space $H^{\frac{\alpha}2}(\mathbb R)$ as soon as $\frac\alpha 2> s_\alpha$, i.e. $\alpha>\frac67$. The proof combines an energy method for strongly nonresonant dispersive equations introduced by Molinet and Vento with modified energies. It also uses a generalized Coifmann-Meyer estimate proved by Muscalu, Pipher, Tao and Thiele.

In 2012 Hochman and Shmerkin proved that, given Borel probability measures on [0,1] invariant under multiplication by 2 and 3 respectively, the Hausdorff dimension of the orthogonal projection of the product of these measures is equal to the maximum possible value in every direction except the horizontal and vertical directions. Their result holds beyond multiplication by 2,3 to natural numbers m,n which are multiplicatively independent. We discuss a generalisation of this theorem to include random cascade measures on subsets of [0,1] invariant under multiplication by multiplicatively independent m,n. We will define random cascade measures in a heuristic way, as a natural randomisation of invariant measures on symbolic space. The theorem of Hochman and Shmerkin fully resolved a conjecture of Furstenberg originating in the late 1960s concerning sumsets of these invariant sets. We apply our main result to present a random version of this conjecture which holds for products of percolations on $\times m, \times n$-invariant sets.

Affine invariant harmonic analysis refers to the reformulation of estimates relevant to harmonic analysis and involving submanifolds of $\mathbf{R}^d$ (such as Fourier restriction or $L^p$ smoothing for Radon averages) in an affine invariant fashion - with the goal of obtaining results that are uniform across large classes of submanifolds. The first step in this direction is introducing an affine invariant weight on the submanifold, but until 2018 the correct weight was known essentially only for the dimension 1 and codimension 1 cases (curves and hypersurfaces). P. Gressman then introduced a general procedure by which one can construct an affine invariant weight for a submanifold of arbitrary codimension; he also proved the weight to be essentially unique, up to normalisation. This beautiful construction and its many aspects will be presented in this talk and new avenues for research will be discussed. Time permitting we will also discuss more recent work of Gressman regarding $L^p$ smoothing estimates and its connection to the construction of the affine invariant weight.

All talks are held in Bayes Centre Lecture Theatre 5.10.

This meeting complements the EMS lecture by Felix Otto (see below).

Pierre Raphael (University of Cambridge)

Time: 14:00 - 15:00

Title: On singularity formation for non linear PDEs

Abstract: The question of the formation of singularities in non linear PDEs in connection with energy concentration mechanisms has produced a huge amount of litterature, both in mathematics and physics. Yet, despite some substantial progress in the last twenty years and the development of numerical methods, basic questions remain completely open like the singularity formation problem in incompressible fluid dynamics. In all known singularity formation problems, self similar solutions or stationary soliton like solutions are fundamental structures underlying the concentration of energy mechanisms. In a recent joint work with Merle (IHES), Rodnianski (Princeton) and Szeftel (Sorbonne Universite), we show how singularity formation can still occur despite the absence of such special solutions with applications to the first description of implosion for a viscous three dimensional compressible fluid, and the discovery of highly oscillatory blow up solutions to the energy super critical non linear Schrodinger equation which answer negatively a conjecture formulated by Bourgain in 2000.

Nicolas Burq (Université Paris-Sud)

Time: 15:00 - 16:00

Title: Geometry and concentration properties of solutions to PDEs

Abstract: In this talk I will present recent results on concentration properties of solutions to wave or Schrödinger (time dependent or not) equations. For the waves I will show how some precise analysis of these concentration properties can give an answer to a longstanding question of controllability for waves on the disc, while for Schrödinger, I will show how one can take benefit from the global properties of the geodesic flow to exhibit some particular behaviours.

Title: Effective behavior of random media

Abstract: In engineering applications, heterogeneous media are often described in statistical terms. This partial knowledge is sufficient to determine the effective, i.~e.~large-scale behavior. This effective behavior may be inferred from the Representative Volume Element (RVE) method. I report on last years' progress on the quantitative understanding of what is called stochastic homogenization of elliptic partial differential equations: optimal error estimates of the RVE method and the homogenization error, and the leading-order characterization of fluctuations. Methods connect to elliptic regularity theory, and in fact lead to a fresh look upon this classical area, and to concentration of measure arguments.

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Let $n < d$ be positive integers. Given a $(d-n)$-plane $V\in G(d,d-n)$, a point $x\in \mathbf{R}^d$, and $\alpha\in (0,1)$, we define $K(x,V,\alpha)$ to be the open cone centered at $x$, with direction $V$, and aperture $\alpha$. It is easy to see that if a set $E\subset \mathbf{R}^d$ satisfies for some $V\in G(d,d-n),\ \alpha\in (0,1),$ the condition \begin{equation} x\in E\quad\quad\Rightarrow\quad\quad E\cap K(x,V,\alpha)=\varnothing, \end{equation} then $E$ is contained in some $n$-dimensional Lipschitz graph $\Gamma$, and $\mathrm{Lip}(\Gamma)\le \frac{\sqrt{1-\alpha^2}}{\alpha}$. To what extent can we weaken the condition above and still get meaningful information about the geometry of $E$? It depends on what we mean by ``meaningful information'', of course. For example, one could ask for rectifiability of $E$, or if $E$ contains big pieces of Lipschitz graphs, or if nice singular integral operators are bounded in $L^2(E)$. In the talk I will discuss these three closely related questions.

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A fundamental problem in the context of Einstein's equations of general relativity is to understand the dynamical evolution of small initial data perturbations of known Einstein spacetimes. In this talk I will discuss the stability of Kaluza-Klein type spacetimes, which are given by the product of Minkowski spacetime with a Ricci-flat Riemannian manifold. The PDE methods required lie at the intersection of methods for quasilinear wave and Klein-Gordon equations. This stability result is related to a conjecture of Penrose concerning the validity of string theory. This talk is based on joint work with Lars Andersson, Pieter Blue and Shing-Tung Yau.

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