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.
 

 I will give an overview of my recent research on the definition of asymptotic charges in asymptotically flat spacetimes, including the definition of subleading and dual BMS charges and the relation to the conserved Newman-Penrose charges at null infinity.

For a bundle E over a manifold M, the associated "configuration-section spaces" are spaces of configurations of points in M together with a section of E over the complement of the configuration. One often considers subspaces where the behaviour of the section near a configuration point -- a kind of "monodromy" -- is restricted or prescribed. These are examples of "non-local configuration spaces", and may be interpreted physically as moduli spaces of "fields with prescribed singularities" in an ambient manifold.

An important class of examples is given by Hurwitz spaces, which are moduli spaces of branched G-coverings of the 2-disc, and which are homotopy equivalent to certain configuration-section spaces on the 2-disc. Ellenberg, Venkatesh and Westerland proved that, under certain conditions, Hurwitz spaces are (rationally) homologically stable; from this they then deduced an asymptotic version of the Cohen-Lenstra conjecture for function fields, a purely number-theoretical result.

We will discuss another homological stability result for configuration-section spaces, which holds (with integral coefficients) whenever the base manifold M is connected and open. We will also show that the "stabilisation maps" are split-injective (in all degrees) whenever dim(M) is at least 3 and M is either simply-connected or its handle dimension is at most dim(M) - 2.

This represents joint work with Ulrike Tillmann.

 

Given a finite set X, is an easy exercise to show that a binary operation * from XxX to X which is injective in each variable separately, and which is also associative, makes (X,*) into a group. Hrushovski and others have asked what happens if * is only partially associative - do we still get something resembling a group? The answer is known to be yes (in a strong sense) if almost all triples satisfy the associative law. In joint work with Tim Gowers, we consider the so-called `1%' regime, in which we only have an epsilon fraction of triples satisfying the associative law. In this regime, the answer turns out to be rather more subtle, involving certain group-like structures which we call rough approximate groups. I will discuss these objects, and try to give a sense of how they arise, by describing a somewhat combinatorial interpretation of partial associativity.