Vorträge in der Woche 26.05.2025 bis 01.06.2025

Dienstag, 27.05.2025: L2-Betti numbers

Paul Vögele

Dienstag, 27.05.2025: Counting Weierstrass points on degenerating algebraic curves

Lucas Gierczak (Université d’Aix-Marseille)

Weierstrass points on algebraic curves are special points of high importance in algebraic geometry and arithmetic geometry. In this talk, we study how those special points behave when the algebraic curve degenerates to a nodal curve. To this end, we first explain why tropical geometry is a relevant formalism for studying degeneration questions. We then define a tropical analogue on metric graphs (seen as tropical curves) for Weierstrass points, and explore the properties of the so-called “tropical Weierstrass locus”. We also associate intrinsic weights to the connected components of this locus, and show that their total sum for a given metric graph and divisor is a function of few combinatorial parameters (degree and rank of the divisor, genus of the metric graph). Finally, in the case the divisor on the metric graph comes from the tropicalization of a divisor on an algebraic curve, we specify the compatibility between the Weierstrass loci.

Freitag, 30.05.2025: Kondensierte Ergodentheorie

Nick Ruoff (Uni Tübingen)

Aufbauend auf dem ersten Vortrag führen wir kondensierte abelsche Gruppen (und andere algebraische Strukturen) ein, die einerseits eine Einbettung lokalkompakter abelscher Gruppen erlauben, andererseits aber bessere homologische Eigenschaften haben. Das ermöglicht es, Konzepte aus der Ergodentheorie, die sich an homologischer und homotopischer Algebra orientieren, präzise zu machen, wie wir am Beispiel des "stetigen Cocycles" mithilfe stetiger Gruppenkohomologie sehen werden.

Freitag, 30.05.2025: Massless phases for the Villain model in d>=3

Wei Wu (New York University, Shanghai, PR China)

The XY and the Villain models are models which exhibit the celebrated Kosterlitz-Thouless phase transitions in two dimensions. The spin wave conjecture, originally proposed by Dyson and by Mermin and Wagner, predicts that at low temperature, spin correlations of these models are closely related to Gaussian free fields. I will review the historical background and present some recent progress for the Villain model in d>=3. Based on the joint work with Paul Dario (CNRS).