Mario Garcia-Fernandez (ICMAT, Madrid) will speak in the geometry seminar on Tuesday 24 March 2020 at 11ham in the Salle des Profs. Mario’s title is Gravitating vortices with positive curvature and his abstract is below.

I will overview recent work with Chengjian Yao and Vamsi Pingali in arXiv:1911.09616, where we give a complete solution to the existence problem for gravitating vortices with positive topological constant c>0, as introduced in arXiv:1510.03810. Our main result establishes the existence of solutions provided that a GIT stability condition for an effective divisor on the Riemann sphere is satisfied. To this end, we use a continuity path starting from Yang’s solution with c=0, and deform the coupling constant α towards 0. A salient feature of our argument is a new bound S(g) \geq c for the curvature of gravitating vortices, which we apply to construct a limiting solution along the path via Cheeger-Gromov theory.

Sobhan Seyfaddini (Université Paris Diderot) will speak in the geometry seminar on Tuesday 26 November 2019 at 11h15am in the Salle de Profs. Sobhan’s title is Barcodes and Hamiltonian homeomorphisms and his abstract is below.

Hamiltonian homeomorphisms are those homeomorphisms of a symplectic manifold which can be written as uniform limits of Hamiltonian diffeomorphisms. One difficulty in studying Hamiltonian homeomorphisms (particularly in dimensions greater than two) has been that we possess fewer tools for studying them. For example, (filtered) Floer homology, which has been a very effective tool for studying Hamiltonian diffeomorphisms, is not well-defined for homeomorphisms. We will show in this talk that using barcodes and persistence homology one can indirectly define (filtered) Floer homology for Hamiltonian homeomorphisms. This talk is based on joint projects with Buhovsky-Humiliére and Le Roux-Viterbo.

Yakov Eliashberg (Stanford University) will speak in the geometry seminar on Thursday 2 April 2020 at 11am in the Salle de Profs. Yakov’s title and abstract are to be announced.

Hanne Van Den Bosch (Universidad de Chile, Santiago de Chile) will speak in the geometry seminar on Tuesday 17 December 2019 at 11am in the Salle de Profs. Hanne’s title is Dirac operators describing Graphene Quantum Dots and her abstract is below.

Low energy electronic excitations in graphene, a two-dimensional lattice of carbon atoms, are described effectively by a two–dimensional Dirac operator. For a bounded flake of graphene (a quantum dot), the choice of boundary conditions determines various properties of the spectrum. Several of these choices appear in the physics literature on graphene. For a simply connected flake and a family of boundary conditions, we obtain an explicit lower bound on the spectral gap around zero. We can also study the effect of the boundary conditions on eigenvalue sums in the semiclassical limit.
This is joint work with Rafael Benguria, Edgardo Stockmeyer (PUC, Chile), and Søren Fournais (Aarhus).