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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.

Given a Riemannian manifold, the geodesic X-ray transform of a symmetric tensor field is defined by the line integrals of the latter over geodesics, and it is a central object in geometric inverse problems. Broadly speaking, one is interested in recovering information about the unknown tensor field from its X-ray transform, and the extent to which this is possible depends heavily on the underlying geometry. In this talk, we focus on the geodesic X-ray transform in the geometric setting of two-dimensional asymptotically hyperbolic manifolds, which are non-constant curvature generalizations of hyperbolic space. This setting is interesting in part due to its connections to theoretical physics, specifically the AdS-CFT Correspondence. The geodesic X-ray transform on symmetric tensor fields of positive rank has a natural nullspace, implying that such a tensor field cannot be uniquely recovered from its X-ray transform. On asymptotically hyperbolic surfaces we propose gauge representatives modulo the nullspace of the transform to be reconstructed from the data, by proving a ``transverse traceless-conformal-potential'' decomposition. We then use our tensor decompositions to provide range characterizations of the geodesic X-ray transform in the special case of 2-dimensional hyperbolic space, as well as to develop reconstruction procedures. Based on joint work with François Monard and Yuzhou Zou.

By showing that there is a topologically free boundary action, we prove that the outer automorphism groups of the topological full groups of the Cuntz?Krieger groupoids are C*-simple, meaning that their reduced C*-algebras are simple (this is an extreme case of non-amenability). These groups generalise the outer automorphism groups of the Higman?Thomson groups, so our result applies, in particular to, Out(V_n), where V_n is a Higman?Thompson group. This is joint work with Xin Li and Takuya Takeishi.