Termine

A line field on a manifold is a smooth and continuous assignment of a tangent line to each point on the manifold. The projective span of a smooth manifold is the maximal number of linearly independent line fields. After motivating the study of this fascinating numerical invariant, I will not only give a necessary condition for the existence of linearly independent line fields on (almost) complex manifolds, but I will also explain that this condition is additionally sufficient in certain cases. Finally, we will apply these necessary and sufficient conditions to obtain a refinement of the Schwarzenberger condition that dictates which cohomology classes can be the Chern classes of a complex vector bundle (with prescribed line bundle splitting properties) over complex projective space. This is joint work with Nikola Sadovek (Dresden) based on [arXiv:2411.14161] (recently accepted for publication in Mathematische Annalen).

Abstract: Calabi-Yau manifolds form an important class of complex manifolds which admit Ricci-flat Kähler metrics thanks to Yau's solution of Calabi's conjecture. As such, in terms of the Berger classification, Calabi-Yau manifolds constitute building blocks of Riemannian manifolds with special holonomy included in SU(n). These play an important role in string theories compactifications. Calabi-Yau manifolds are also the objects of the mirror symmetry conjecture. In this talk, we describe a gluing constructions for families of Ricci-flat Kähler metrics on crepant resolutions and on polarized smoothings of Calabi-Yau manifolds with isolated conical singularities as obtained by Hein-Sun and we obtain asymptotic expansions in terms of the parameters of degeneration.

This talk concerns a variant of the Schottky problem, which asks to classify Jacobians among all abelian varieties. In characteristic p, there is a rich extra structure to consider. Namely, in characteristic p, abelian varieties can be partitioned into so-called Ekedahl-Oort strata, within which all abelian varieties have isomorphic p-torsion group schemes. From this point of view, it is fruitful to investigate which p-torsion group schemes can occur as the p-torsion of the Jacobian of a (specific type of) curve. In this talk, we treat the 2-torsion of curves in characteristic 2 that admit a separable double cover to another curve. Through an analysis of the first De Rham cohomology, we prove that the p-torsion of a double cover of an ordinary curve is determined by the ramification breaks of the cover. This generalizes a result by Elkin and Pries, where the base curve is the projective line, so that the covers are hyperelliptic curves. When the base curve is not ordinary, we establish bounds on the Ekedahl-Oort type of the cover.

We present a novel global-in-time variational approach to gradient flows and doubly nonlinear equations in (reflexive) Banach spaces. It is based on the De Giorgi's principle, which states that solving a gradient flow is equivalent to being a null-minimizer of a suitable energy functional among all trajectories sharing the same initial position. As for the similar Brezis-Ekeland-Nayroles (BEN) principle (which applies only to a convex framework), finding a minimizer for such functional is not difficult in general, but proving that the minimum is zero poses a real challenge. In the BEN formulation, the task has been accomplished by Ghoussoub, resorting to the tool of self-dual Lagrangians. Our approach allows to extend the analysis to nonconvex energies, directly dealing with the De Giorgi's functional, and it relies on a convexification of the problem in spaces of measures exploiting the so-called superposition principle. The validity of the null-minimization is then recovered by a careful application of the Von Neumann minimax theorem, and by employing the "backward boundedness" property of the dual Hamilton-Jacobi equation. The talk is based on a joint work with A. Pinzi and G. Savaré.

In this talk, I will recall the notions of C*- and W*-superrigidity for groups. I will then present examples of non-amenable groups that can be completely reconstructed from their reduced C*-algebras, but not from their group von Neumann algebras. These groups are constructed as infinite direct sums of amalgamated free product groups and provide the first known examples of non-amenable groups exhibiting this behavior. This is joint work with Juan Felipe Ariza Mejía and Ionut Chifan.

In this talk we survey several results on biconservative submanifolds, with particular emphasis on biconservative hypersurfaces in Euclidean spaces that are invariant under the action of SO(p+1)×SO(q+1). Using the Poincaré?Bendixson Theorem, we analyze the dynamics of their profile curves in the orbit space and establish the existence of infinitely many complete SO(p+1)×SO(q+1)-invariant biconservative hypersurfaces in Euclidean spaces.

Conformal-biharmonic maps arise as critical points of the conformal-bienergy functional, a second-order functional obtained by modifying the classical bienergy functional through curvature terms motivated by conformal geometry. In dimension four, this functional enjoys conformal invariance, making it particularly relevant in the study of variational problems in conformal geometry. In this talk we focus on stability properties of conformal-biharmonic maps. We investigate the conformal-biharmonic stability of the identity map of compact Einstein manifolds with non-negative scalar curvature and show that its conformal-biharmonic index coincides with the classical harmonic index, with a remarkable exception in the case of the four-dimensional sphere. We also study conformal-biharmonic hypersurfaces in space forms and compute the index and nullity for hyperspheres in spheres. Our results reveal new stability phenomena and emphasize the geometric differences between biharmonic and conformal-biharmonic variational theories.