Talk

The talk plans to present an introduction to the eponymous conjecture predicting a mysterious link between rational points on an elliptic curve defined over rational numbers,
and analytic properties of the associated Hasse-Weil L-function. Some recent progress will be discussed.

Given any smooth 4-manifold bounding a Seifert manifold, the Seifert action on its boundary can be used to define their boundary Dehn twists. If the given 4-manifold is simply-connected, this Dehn twist is always topologically isotopic to the identity, but usually not smoothly isotopic, making it a very nice potential example of exotic diffeomorphisms. For some choices of small enough Brieskorn spheres and some choices of their fillings, it has been shown that their boundary Dehn twists are infinite-order exotic. In this talk, we prove that for any Brieskorn homology sphere bounding a positive-definite 4-manifold, their boundary Dehn twists are always infinite-order exotic, using both the equivariant Seiberg-Witten homology of Baraglia-Hekmati and the family Seiberg-Witten theory. This is a joint work with JungHwan Park and Masaki Taniguchi.

As a consequence of his proof of rigidity of the category of localising motives, Efimov has constructed refinements of localising invariants. Such refined invariants often contain a lot more information than the original ones. For example, the refined $TC^-$ of the rational numbers is not a rational spectrum; it contains very subtle p-complete information as well. In this talk, we'll explain how to compute it after base change to $ku$. The computation involves a surprising connection to q-de Rham cohomology, and in partcular, to the question of whether there exists a "q-Hodge filtration" on q-de Rham cohomology. This is joint work with Samuel Meyer.

A major open problem in topology are (rational) injectivity results about assembly in $K$- and $L$-theory, e.g. the Borel and Novikov conjectures.

Malkiewich showed that given a ring R and finite group $G$ there exists a so called coassembly map going from the K-theory of the group ring $R[G]$ to the homotopy fixed points of the K-theory spectrum of $R$ equipped with the trivial action. Moreover the composition of the assembly map with this coassembly map agrees with the norm map. This implies that for a finite group G, the assembly map is rationally and $K(n)$-locally injective. In recent work with Alex M\"uller and Holger Reich, we generalize this fact to arbitrary additive invariants of either stable or Poincaré categories, while also allowing for twisted coefficients. In particular, Malkiewich's theorem is true more generally for $THH$, $TC$, Grothendieck-Witt and $L$-theory. The fundamental insight is that the content of the theorem is fundamentally about the interaction of the algebraic theory of G-actions with the algebraic theory of $E_\infty$-monoids. The general nature of this approach allows one to ask the question if there exists a generalization to more general groups, using the framework of dualizable categories.

We meet for a final gathering of the QFT Workshop in the MPIM Lecture Hall: There will be no official program, so it is very much appreciated if some participants would like to (informally) share their progress or some interesting ideas, observations and open questions that came up during their stay.

We outline mod l non-vanishing of Hecke L-values in self-dual families over imaginary quadratic fields. This includes the vanishing of the mu-invariant of Rubin’s p-adic L-function. (Joint with W. He, S. Kobayashi and K. Ota.)