Prochainement

Lundi 27 janvier 14:00-15:00 Valentina Franceschi (LJLL)
Minimal bubble clusters in the plane with double density

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Lieu : IMO ; salle 3L15.

Résumé : We present some results about minimal bubble clusters in the plane with double density. This amounts to find the best configuration of $m\in \mathbb N$ regions in the plane enclosing given volumes, in order to minimize their total perimeter, where perimeter and volume are defined by suitable densities. We focus on a particular structure of such densities, which is inspired by a sub-Riemannian model, called the Grushin plane.
After an overview concerning existence of minimizers, we focus on their Steiner regularity, i.e., the fact that their boundaries are made of regular curves meeting at 120 degrees. We will show that this holds in a wide generality.
Although our initial motivation came from the study of the particular sub-Riemannian framework of the Grushin plane, our approach works in wide generality and is new even in the classical Euclidean case.

Minimal bubble clusters in the plane with double density  Version PDF

Lundi 3 février 14:00-15:00 Burglind Jöricke (MPIM)
Fundamental groups, slalom curves and extremal length

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Lieu : IMO ; salle 3L8.

Résumé : We define the extremal length of elements of the fundamental group of the twice punctured complex plane and give effective estimates for this invariant. The main motivation comes from 3-braid invariants and their application, for instance to effective finiteness theorems in the spirit of the Geometric Shafarevich Conjecture over Riemann surfaces of second kind.

Fundamental groups, slalom curves and extremal length  Version PDF

Lundi 10 février 14:00-15:00 Andreas Juhl ( Humboldt-Universität Berlin)
Singular Yamabe problem, residue families and conformal hypersurface invariants

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Lieu : IMO ; salle 3L8.

Résumé : We describe recent progress on constructions of natural conformally invariant differential operators which are associated to hypersurfaces in Riemannian manifolds. The constructions rest on the solution of a singular version of the Yamabe problem. We outline two basic approaches. The first rests on conformal tractor calculus (Gover-Waldron) and the second generalizes the notion of residue families (introduced by the author) which involves the Feffermann-Graham Poincaré-Einstein metric. We prove the equivalence of both methods. Both constructions are curved analogs of symmetry breaking operators in representation theory (Kobayashi). Among many things, this naturally leads to a notion of extrinsic Q-curvature which generalizes Branson’s Q-curvature. The presentation will describe work of Gover-Waldron, Graham, Juhl-Orsted and others.

Singular Yamabe problem, residue families and conformal hypersurface invariants  Version PDF

Passés

Lundi 20 janvier 14:00-15:00 Mihajlo Cekic (LMO)
Resonant spaces for volume-preserving Anosov flows

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Lieu : IMO ; salle 3L8.

Résumé : Recently Dyatlov and Zworski proved that the order of vanishing of the Ruelle zeta function at zero, for the geodesic flow of a negatively curved surface, is equal to the negative Euler characteristic. They more generally considered contact Anosov flows on 3-manifolds. In this talk, I will discuss an extension of this result to volume-preserving Anosov flows, where new features appear : the winding cycle and the helicity of a vector field. A key question is the (non-)existence of Jordan blocks for one forms and I will give an example where Jordan blocks do appear, as well as describe a resonance splitting phenomenon near contact flows. This is joint work with Gabriel Paternain.

Notes de dernières minutes : Report de la séance annulée du 16/12/19.

Resonant spaces for volume-preserving Anosov flows  Version PDF
Lundi 13 janvier 14:00-15:00 Mateus Sousa (Munich)
Fourier uniqueness pairs

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Lieu : IMO ; salle 3L8.

Résumé : Given a collection of functions where the Fourier transform is well defined, we call a pair of sets (A, B) a Fourier uniqueness pair if every function that vanishes on the set A with a Fourier transform vanishing on the set B has to be identically zero. In case A coincides with B, we call it a Fourier uniqueness set. In this talk we will review the long history of problems involving Fourier uniqueness pairs and present some new results concerning Fourier uniqueness pairs consisting of sets of powers of integers.

Fourier uniqueness pairs  Version PDF