Abstracts of upcoming talks at the MPIM. For an overview see also the calendar.

## Rational homology cobordisms of plumbed manifolds and arborescent link concordance

We investigate rational homology cobordisms of 3-manifolds with non-zero first Betti number.

This is motivated by the natural generalization of the slice-ribbon conjecture to multicomponent links.

We introduce a systematic way of constructing rational homology cobordisms between plumbed 3-manifolds and concordances between arborescent links.

We then describe a sliceness obstruction based on Donaldson's diagonalization theorem that leads to a proof of the slice-ribbon conjecture for 2-component Montesinos' links up to mutation.

## Quadratic Relations in Feynman Categories

I will start with a brief introduction to Feynman categories, and then explore the consequences of

quadratic relations for the morphisms in Feynman categories.

In the first part of the talk, I show how this leads to cubical complexes and prove that in this

way both the complex for moduli spaces of curves and the cubical complexes for Cutkosky rules

and calculations for Outer space arise naturally from push-forwards.

In the second part, I study generalizations of the quadradtic relations, which naturally lead

## tba

## On Sequences of Integers of Quadratic Fields and Relations with Artin’s Primitive Root Conjecture

I will consider the integers $\alpha$ of the quadratic field $ \mathbb{Q} (\sqrt[]{d})$ with $d$ is a square-free integer. Using the embedding into $ \text{GL}(2,\mathbb{R})$ we obtain bounds for the smallest positive integer $\nu$ such that $\alpha^\nu\equiv 1\bmod p.$ More generally, if $\mathcal{O}_{f}$ is a number ring of conductor $f$, we study the first integer $n=n(f)$ such that $\alpha^n\in\mathcal{O}_{f}$. We obtain bounds for $n(f)$ and for $n(fp^{k})$.

## Doubling Properties of Solutions to Elliptic PDE

## New guests at the MPIM

## Representation theory learning seminar

## The Hilbert scheme of points of affine 3-space

We describe the scheme structure of the Hilbert scheme of points of affine 3-space,

in terms of representations of the Jacobi algebra of a quiver with potential. This exhibits the Hilbert scheme of points as the critical locus of a regular function on a smooth variety.

We discuss the torus action on the Hilbert scheme and its Euler characteristic.

## Hecke's integral formula and Kronecker's limit formula for an arbitrary extension of number fields

The classical Hecke's integral formula expresses the partial zeta function of real quadratic fields as an integral of the real analytic Eisenstein series along a certain closed geodesic on the modular curve. In this talk, we present a generalization of this formula in the case of an arbitrary extension E/F of number fields. As an application, we present the residue formula and Kronecker's limit formula for an extension E/F of number fields, which gives an integral expression of the residue and the constant term at s=1 of the``relative'' partial zeta function associated to E/F.

## On the periodicity of geodesic continued fractions

In this talk, we present some generalizations of Lagrange's periodicity theorem in the classical theory of continued fractions. The main idea is to use a geometric interpretation of the classical theory in terms of closed geodesics on the modular curve. As a result, for an extension F/F' of number fields with rank one relative unit group, we construct a geodesic multi-dimensional continued fraction algorithm to``expand'' a basis of F over the rationals, and prove its periodicity. Furthermore, we show that the periods describe the relative unit group.