The interleaving distance is a canonical pseudo-distance for persistence modules and plays an essential role in topological data analysis. After the pioneering work by Curry, Kashiwara and Schapira introduced the interleaving distance on the category of sheaves and studied its stability property. In this talk, I will explain that we can use this distance for sheaves to get a lower bound of the Hamiltonian displacement energy in a cotangent bundle through its stability with respect to Hamiltonian deformations. I would also like to explain our recent result on the completeness of the distance and its application to C^0-symplectic geometry. Joint work with Tomohiro Asano.

We will discuss real algebraicity of links and open books and how to obtain overtwisted structures in a real algebraic way.

Uniqueness and reconstruction in the three-dimensional Calderón inverse conductivity problem can be reduced to the study of the inverse boundary problem of a Schrödinger operator with a real potential. Due to the lack of a rigorous definition, the Born approximation has been relegated to a marginal place in the reconstruction problem. In this talk we will introduce the Born approximation for Schrödinger operators in the ball, which amounts to studying the linearization of the inverse problem. We first analyze this approximation for real and radial potentials in any dimension >2. We show that this approximation satisfies a closed formula that only involves the spectrum of the Dirichlet-to-Neumann map, and which is closely related to a particular moment problem.

Breathers are temporally periodic and spatially localized solutions of evolutionary PDEs. They are known to exist for integrable PDEs such as the sine-Gordon equation, but are believed to be rare for general nonlinear PDEs. When the spatial dimension is equal to one, exchanging the roles of time and space variables (in the so-called spatial dynamics framework), breathers can be interpreted as homoclinic solutions to steady solutions and thus arise from the intersections of the stable and unstable manifolds of the steady states. In this talk, we shall study the nonlinear Klein-Gordon equation and show that small amplitude breathers cannot exist (under certain conditions).

This is a conjecture on weighted estimates for the classical Fourier extension operators of harmonic analysis. In particular, let E be the extension operator associated to some surface, and g be a function on that surface. If we 'erase' part of Eg, how well can we control the 2-norm of the remaining piece? The Mizohata-Takeuchi conjecture claims some remarkable control on this quantity, involving the X-ray transform of the part of the support of Ef that we kept. In this talk we will discuss the basics and history of the problem, as well as some small progress. This is joint work with Anthony Carbery and Hong Wang.

I will present my recent joint work with J. F. de Bobadilla, proving that a family of isolated hypersurface singularities with constant Milnor number has constant multiplicity. The key idea is to endow the A'Campo model of "radius zero" monodromy with a symplectic structure. More precisely, I will construct a smooth manifold, fibered over an annulus, together with a fiberwise symplectic form such that the following holds. Over the outer circle ("positive radius"), we get the usual Milnor fibration. In turn, over the inner circle ("radius zero"), the symplectic monodromy has very simple dynamical properties.

The goal of the talk is to motivate and introduce the main questions regarding the Fourier restriction operator for the paraboloid, as well as to present some recent developments that we obtained in joint work with Camil Muscalu. We will describe some connections between the study of this operator and questions in dispersive PDEs and geometric measure theory. Once we set up the background and comment on the current tools employed to treat the main conjectures in this area, we will change our focus to a different framework that unifies the treatment of the linear and multilinear theory previously presented.

Extremal Kähler metrics were introduced by Calabi in the 80’s as a type of canonical Kähler metric on a Kähler manifold, and are a generalisation of constant scalar curvature Kähler metrics in the case when the manifold admits automorphisms. A natural question is when the blowup of a manifold in a point admits an extremal Kähler metric. We completely settle the question in terms of a finite dimensional moment map/GIT condition, generalising work of Arezzo-Pacard, Arezzo-Pacard-Singer and Székelyhidi. Our methods also allow us to deal with a certain semistable case that has not been considered before, where the original manifold does not admit an extremal metric, but is infinitesimally close to doing so.