Daniel Greb (Universität Duisburg-Essen) will speak in the geometry seminar on Tuesday 17 March 2020 at 11h am in the Salle des Profs. Daniel’s title is Projectively flat sheaves and characterisations of finite quotients of projective spaces and his abstract is below.

I will explain how to extend the classical characterisation of projective space among Kähler-Einstein Fano manifolds in terms of a Chern class (in)equality to the class of Fano varieties with Kawamata log terminal singularities. I will spend significant time on discussing the necessary tools, which range from analysis (harmonic metrics / Simpson correspondence) and classical differential geometry to algebraic geometry (local fundamental groups of klt singularities). This is joint work with Stefan Kebekus and Thomas Peternell, and partly with Behrouz Taji.

Mircea Petrache (Pontificia Universidad Catolica, Santiago de Chile) will speak in the geometry seminar on Friday 28 February 2020 at 11h am in the Salle N 08.08 (this is the room behind the elevators in the 8th floor of the NO building). Mircea’s title is Uniform measures and manifolds all of whose curvatures are constant and his abstract is below.

A uniform measure in Euclidean space R^d is a measure with respect to which balls B(x,r) with center x in the support, are assigned mass dependent of r and independent of the choice x. For example any invariant measure with respect to a subgroup of the isometry group of R^d is uniform, and called a homogeneous measure. However we also have a few exotic examples of non-homogeneous uniform measures, such as the volume measure of the “light cone” {x^2+y^2+z^2=w^2} in R^4.This class of measures was first studied by David Preiss as the crucial ingredient of his 1987 proof of the Besicovitch conjecture. The complete classification of uniform measures remains a difficult open problem, even restricted to ambient dimension d=2. I will detail the known classification of 1-dimensional uniform measures in R^d for general d, for which, in joint work with Paul Laurain, we show that they are constituted of disjoint unions of helices or of toric knots, or equivalently, of analytic curves all of whose curvatures are constant.

Eva Miranda (Universitat Politècnica de Catalunya) will speak in the geometry seminar on Tuesday 26 May 2020 at 11ham in the Salle des Profs. Eva’s title is The symplectic and contact geometry of forms with “singularities” and her abstract is below.

We will overview the study of symplectic and contact structures with singularities which appear modelling some problems in Celestial Mechanics and Fluid Dynamics and describe several applications to the study of their Hamiltonian (and Reeb) Dynamics.
In these motivating examples the singularities are associated to the line at infinity or collision set and are an outcome of regularization techniques. These singular symplectic structures ($b^m$-symplectic structures) can be formalized as smooth Poisson structures which are symplectic away from a hypersurface and satisfy some transversality properties. We can desingularize these structures associating a family of symplectic structures (with good convergence properties) to singular symplectic structures with even exponent (the so-called $b^{2k}$-symplectic structures) and a family of folded symplectic structures for odd exponent ($b^{2k+1}$-symplectic structures) yielding, in particular, topological constraints for their existence. This procedure generalizes to its odd-dimensional counterpart (joint work with Cédric Oms) and puts in the same picture different geometries: symplectic, folded-symplectic, contact and Poisson geometry. The applications of this “desingularization kit” include the construction of action-angle coordinates for integrable systems, the study of their perturbation (KAM theory) and the existence of periodic orbits away from the critical hypersurface. Time permitting, we will end up this talk proving an existence problem of contact structures with singularities on a given manifold and with some open questions concerning the study of Reeb Dynamics in the singular case, in particular, the existence of periodic orbits (Weinstein conjecture).