Oberseminare und sonstige Vorträge

In the 1980s, Witten introduced the Witten genus that assigns a modular form to each closed manifold with a string structure. Ando, Hopkins and Rezk showed that this refines to an E_infty-map MString -> TMF to the spectrum of topological modular forms. But their proof is very intricate. I will present a new approach to the string orientation via equivariant TMF, giving also an introduction to this theory.

Rational homotopy equivalence is a weakening of the usual notion of homotopy equivalence, that is easier to study algebraically in terms of commutative differential graded algebras (like the de Rham complex of a smooth manifold). The simplest rational homotopy invariant is the rational cohomology algebra, but there can in addition be so-called Massey products between triples or higher tuples of classes that can also distinguish rational homotopy types. Defining tensors on the cohomology of a space by multiplying triple or fourfold Massey products by a further cohomology class gives an object that has less dependence on choices than the Massey products themselves, making them easier to work with. On the other hand, for a closed oriented manifold, Poincare duality allows all Massey products to be recovered from these tensors. Moreover, suitable interpretations of these tensors can capture information about the rational homotopy type even when the Massey products are undefined, and they have nice functoriality properties. For closed k-connected manifolds of dimension up to 6k+2 (k > 0), these tensors (along with the cohomology algebra itself) suffice to determine the rational homotopy type. This is based on joint work with Diarmuid Crowley and Csaba Nagy.

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.

In this talk we compute the index homomorphism of even K-groups arising from a class in even KK-theory via the Kasparov product. Due to the seminal work of Baaj and Julg, under mild conditions on the C*-algebras in question, every class in KK-theory can be represented by an unbounded Kasparov module. We then describe the corresponding index homomorphism of even K-groups in terms of spectral localizers. This means that our explicit formula for the index homomorphism does not depend on the full spectrum of the abstract Dirac operator D, but rather on the intersection between this spectrum and a compact interval. The size of this compact interval does however reflect the interplay between the K-theoretic input and the abstract Dirac operator. Since the spectral projections for D are not available in the general context of Hilbert C*-modules we instead rely on certain continuous compactly supported functions applied to D to construct the spectral localizer. In the special case where even KK-theory coincides with even K-homology, our work recovers the pioneering work of Loring and Schulz-Baldes on the index pairing.