Termine

Abstract:In this work, we study the integral algebraic K-theory groups of ring spectra whose homotopy groups are polynomial algebras in a single variable, including Morava K-theories and their two-periodic counterparts. Using a filtration on topological cyclic homology that we call the orbit filtration, we prove results on the cardinalities of these groups. When the coefficient field is the algebraic closure of Fp, we determine these cardinalities completely. For Morava K-theories, we combine these methods with the motivic filtration on topological cyclic homology to determine the cardinalities of the integral algebraic K-theory groups in every degree not congruent to 0 or 1 modulo 2p-2. This is joint work with Gabriel Angelini-Knoll.

I will discuss of various curvature or spectral constraints implying that a complete Riemannian manifold of dimension 3 is diffeomorphic to the Euclidean space.

In their seminal work, Xiao-Zhu have constructed exotic Hecke correspondences between the special fibers of different Shimura varieties, which do not necessarily lift to the generic fibers. As a consequence, they have constructed geometric realizations of the geometric Jacquet-Langlands correspondence, and verified generic instances of the Tate conjectures for certain Shimura varieties. Crucially, they only address Shimura varieties with good reduction, i.e., with hyperspecial level at a fixed prime. In these two talks, I will explain how to generalize their methods when allowing bad reduction (more precisely, when the level at a fixed prime is very special, which exists when the local group is quasi-split, but not necessarily unramified). In the first talk, I will explain how to construct exotic Hecke correspondences in these ramified situations between PEL type Shimura varieties. Combined with the "spherical" part of the categorical local Langlands correspondence, these yield new instances of the geometric Jacquet-Langlands correspondence. The key is to resolve the integral models by the splitting models of Pappas-Rapoport. In the second talk, I will define splitting versions of affine Deligne-Lustzig varieties, and explain how to study their irreducible components. Combining this with the aforementioned geometric Jacquet-Langlands correspondence, this can be used to verify generic instances of the Tate conjecture for certain Shimura varieties at very special level.

Abstract: We propose a method of capturing the ends of open manifolds by means of bordisms approaching infinity. This perspective allows us to relate moduli spaces of open manifolds with moduli spaces of their compact exhaustions, and their "end-bordisms" by means of a fibre sequence, whih yields a delooping of a fibre sequence constructed by Cardim. In order to set this up, we also study a categorical construction of the category of open manifolds from the category of compact manifolds.

In joint work with Macpherson, Steinhorn and Wolf on multidimensional asymptotic classes, we introduced ring-measurable structures. In these, every definable set is assigned a "size" in an ordered ring, subject to additivity and Fubini-type conditions on definable families. This generalises both the measurable structures of Macpherson and Steinhorn and the asymptotic counting in pseudofinite structures. We asked there whether ring-measurability implies simplicity. In ongoing work with Macpherson and Steinhorn, we show that the generic triangle-free graph (the Henson graph H_3), which is not simple, is not ring-measurable: counting configurations in two different ways yields polynomial identities with no admissible common solution. I will explain the framework and sketch the argument. I will then discuss the generic bowtie-free graph, which defines copies of H_3 that are not fully embedded, so that the argument does not transfer directly.