Termine

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.

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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.

Abstract: We show that the normalized Kähler?Ricci flow on a compact Kähler manifold with semiample canonical bundle converges in the Gromov?Hausdorff topology to the metric completion of the twisted Kähler?Einstein metric on the canonical model, as conjectured by Song?Tian. This is based on joint work with Man-Chun Lee and Valentino Tosatti.

Condensed Mathematics is a relatively new approach to topology that facilitates working with algebraic structures equipped with a topology. In homotopy theory, we study all kinds of spaces using algebraic invariants, which are very often naturally endowed with a topology. In my first talk, I will introduce you to the world of condensed mathematics in the context of homotopy theory. I will explain the notion of a condensed homotopy type and how it is defined in the case of schemes. I will provide an overview of the information encoded in this invariant and in what sense it refines more classical invariants such as the étale homotopy type or the pro-étale fundamental group. My focus will be on the question of when a scheme is (not) condensed contractible, i.e., its condensed homotopy is (not) contractible. In my second talk, I will continue the study of condensed contractible schemes. The main goal will be to compute the condensed fundamental group of a Dedekind ring. More specifically, we will see that the scheme Spec(Z) is not condensed contractible, even though it is étale-contractible. This talk is based on joint work with Haine, Holzschuh, Lara, Martini, and Wolf, as well as on extended results from my dissertation.