In this talk, we discuss Sobolev embeddings into rearrangement-invariant function spaces on (regular) domains in $\mathbb{R}^n$ endowed with measures whose decay on balls is dominated by a power $d$ of their radius, called $d$-Frostman measures. We show that these embeddings can be deduced from one-dimensional inequalities for an operator depending on $n$, $d$ and the order $m$ of the Sobolev space. We also point out an interesting feature of this theory - namely that the results take a substantially different form depending on whether the measure is decaying fast ($d\geq n-m$) or slowly ($d<n-m$). This is a
joint work with Andrea Cianchi and Lubos Pick.

The Kruzkhov's semi-group of a scalar conservation law extends as a semi-group over $L^1$, thanks to its contraction property. M. Crandall raised in 1972 the question of whether its trajectories can be distributional, entropy solutions, or if they are only "abstract" solutions. We solve this question in the case of the multi-dimensional Burgers equation, which is a paradigm for non-degenerate conservation laws. Our answer is the consequence of dispersive estimates. We first establish $L^p$-decay rate by applying the recently discovered phenomenon of Compensated Integrability. The $L^\infty$-decay follows from a De Giorgi-style argument. This is a collaboration with Luis Sivestre (University of Chicago).

Nonlocal minimal surfaces are hypersurfaces of Euclidean space that minimize the fractional perimeter, a geometric functional introduced in 2010 by Caffarelli, Roquejoffre, and Savin in connection with phase transition problems displaying long-range interactions.

In this talk, I will introduce these objects, describe the most important progresses made so far in their analysis, and discuss the most challenging open questions.

I will then focus on the particular case of nonlocal minimal graphs and present some recent results obtained on their regularity and classification in collaboration with X. Cabre, A. Farina, and L. Lombardini.

Witten-Reshetikhin-Turaev quantum invariants of links and 3 dimensional manifolds are obtained from quantum sl(2). There exist different versions of quantum sl(2) leading to other families of invariants. We will briefly overview the original construction and then discuss two variants. First one, so called unrolled quantum sl(2), allows construction of invariants of 3-manifolds involving C* flat connections. In simplest case it recovers Reidemeister torsion. The second one is the non restricted version at a root of unity. It enables construction of invariants of links equipped with a gauge class of SL(2,C) flat connection. This is based respectively on joined work with Costantino, Geer, Patureau and Geer, Patureau, Reshetikhin.

I will describe new solutions to the stationary Ginzburg-Landau equation in 3 dimensions with vortex lines given by interacting helices, with degree one around each filament and total degree an arbitrary positive integer. I will also present results on the asymptotic behavior of vortices in the entire plane for a dissipative Ginzburg-Landau equation. This is work in collaboration with Manuel del Pino, Remy Rodiac, Maria Medina, Monica Musso and Juncheng Wei.

String sigma-models relevant in the AdS/CFT correspondence are highly non-trivial two-dimensional field theories for which predictions at finite coupling exist, assuming integrability and/or the duality itself.  I will discuss general features of the perturbative approach to these models, and present progress on how to go extract finite coupling information in the most possibly general way, namely via the use of lattice field theory techniques. I will also present new results on certain ``defect-CFT’' correlators  at strong coupling. 

Given two metric spaces $X$ and $Y$, it is natural to ask how faithfully, from the point of view of the metric, one can embed $X$ into $Y$. One way of making this precise is asking whether there exists a coarse embedding of $X$ into $Y$. Positive results are plentiful and diverse, from Assouad's embedding theorem for doubling metric spaces to the elementary fact that any finitely generated subgroup of a finitely generated group is coarsely embedded with respect to word metrics. Moreover, the consequences of admitting a coarse embedding into a sufficiently nice space can be very strong. By contrast, there are few invariants which provide obstructions to coarse embeddings, leaving many seemingly elementary geometric questions open.
I will present new families of invariants which resolve some of these questions. Highlights of the talk include a new algebraic dichotomy for connected unimodular Lie groups, and a method of calculating a lower bound on the conformal dimension of a compact Ahlfors-regular metric space.