Termine

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.

The idea of six functors formalisms originates in Grothendieck's work on duality for étale cohomology of schemes. Much more recently, a simple and powerful definition of this structure was given using the theory of higher categories. This has greatly improved our ability to work with such structures. However it does not simplify our task of constructing six functor formalisms, and in fact apriori it makes it much harder. Nevertheless, work of Liu-Zheng formalized the most important construction principle, which goes back to the original work of Artin, Grothendieck and Verdier on the six functor formalism on étale cohomology. I will explain a new approach to this construction principle which is joint work with Bastiaan Cnossen and Tobias Lenz. To do this we recast the problem as that of computing a certain universal (infinity,2)-category, which we then do by combining methods from parametrized and (infinity,2)-category theory.