Vorträge in der Woche 06.05.2019 bis 12.05.2019

Montag, 06.05.2019: Arithmetic and geometry of algebraic surfaces: the place of elliptic fibrations and K3 surfaces

Cecilia Salgado (Universidade Federal do Rio de Janeiro)

In this talk I will go over recent progress on the arithmetic of algebraic surfaces, review some contributions of the study of elliptic fibrations to it and present open problems summarizing the state of the art. K3 surfaces have a special place in the classification of algebraic surfaces: they lie in an intermediate spot, being not too simple (as rational surfaces) nor too complicated (as surfaces of general type). They are ubiquitous in mathematics, having been studied by differential and algebraic geometers, dynamicists, arithmeticians, analysts and also by physicists. A special feature of K3 surfaces is that they are the only class of surfaces that might admit more than one elliptic fibration, with section, that is not of product type. The classification of such different fibrations has been the object of several papers since the 90's. I plan to outline a new approach to this problem and, if time allows, discuss arithmetic and geometric applications of the method.

Dienstag, 07.05.2019: Induced representations and the Mackey machine

Claudius Kamp

Dienstag, 07.05.2019: Positive Rekurrenz autoregressiver Prozesse

Valentin Bolz (Uni Tübingen)

Donnerstag, 09.05.2019: Witten conjecture for Mumford's kappa classes

Prof. Dr. Renzo Cavalieri (Colorado State University)

Kappa classes were introduced by Mumford, as a tool to explore the intersection theory of the moduli space of curves. Iterated use of the projection formula shows there is a close connection between the intersection theory of kappa classes on the moduli space of unpointed curves, and the intersection theory of psi classes on all moduli spaces. In terms of generating functions, we show that the potential for kappa classes is related to the Gromov-Witten potential of a point via a change of variables essentially given by complete symmetric polynomials, rediscovering a theorem of Manin and Zokgraf from '99. Surprisingly, the starting point of our story is a combinatorial formula that relates intersections of kappa classes and psi classes via a graph theoretic algorithm (the relevant graphs being dual graphs to stable curves). Further, this story is part of a large wall-crossing picture for the intersection theory of Hassett spaces, a family of birational models of the moduli space of curves. This is joint work with Vance Blankers (arXiv:1810.11443) .