Talk

A recent manuscript of OpenAI asserts that Landau–Siegel zeros do not exist, a problem that has been open since the work of Landau, Page, and Siegel in 1935. The notion of exceptional zeros is even older, dating back to the early 1900's in work of Gronwall, Landau, and Hecke. The proof is an interpolation determinant argument of a kind familiar from transcendence theory, and its only analytic input is a classical inequality for the logarithmic derivative of $L(s,\chi)$. In these two talks I will present the argument in full, arranged as a direct proof.

We discuss a recent construction of a family of compact hyperbolic manifolds whose homology torsion grows exponentially with respect to volume. This shows that the upper bounds on torsion growth obtained by Bader, Gelander, and Sauer are asymptotically sharp. The construction is based on arithmetic hyperbolic manifolds of simplest type and uses right-angled Coxeter groups together with retraction techniques inspired by Bergeron–Haglund–Wise. A key ingredient is the existence of sufficiently many “independent” totally geodesic submanifolds, which produce large torsion in homology. We outline the main ideas of the construction and discuss the geometric mechanisms behind exponential torsion growth.

Musical intervals which sound pleasant, consist of frequencies that are rational multiples of each other. In joint work with Malte Leimbach and Oliver Fürst, we show that the sound of a generic drum (i.e. the spectrum of the Laplacian on a Riemannian manifold) contains no such harmonies at all. Our theorem has implications for a physics-inspired research programme by Alain Connes and Walter van Suijlekom in noncommutative geometry. In my talk, I will explain our result and the key ideas in this research programme, and how they are related.

A commutative ring spectrum is called even periodic if its homotopy groups vanish in odd degrees and are (at least locally) pairwise isomorphic in even degrees. Such rings are quite far from the usual setting of algebraic geometry, based on static rings concentrated in a single homotopy degree, or even the standard setup of derived geometry, based on connective rings. Yet algebro-geometric spaces that are locally of this form abound in applications to homotopy theory, especially chromatic theory. In this talk, we will discuss the operation of even periodization, which universally approximates a derived stack with an even periodic one, and discuss some applications this unassuming construction has to the Hahn-Raksit-Wilson even filtration, prismatic cohomology, and more.

K-theory, initially conceived in the context of algebraic geometry, can be understood as an invariant of stable categories. In this talk, I will explain how the recent viewpoint of K-theory as an invariant of dualisable categories interacts with and benefits the study of K-theory of schemes and analytic spaces, especially through examples of descent results in their context.