Hannah Markwig
Eberhard Karls Universität Tübingen
Counting bitangents of quartic curves --- arithmetic, real, tropical
Already Plücker knew that a smooth complex plane quartic curve has exactly 28 bitangents. Bitangents of quartic curves are related to a variety of mathematical problems. They appear in one of Arnold's trinities, together with lines in a cubic surface and 120 tritangent planes of a sextic space curve. Together with his group, Bernd Sturmfels has revived an interest in these topics and provided computational approaches and new constructions. In this talk, we review known results about counts of bitangents under variation of the ground field. Special focus will be on counting in the tropical world, and its relations to real and arithmetic counts. We end with new results concerning the arithmetic multiplicity of tropical bitangent classes, based on joint work in progress with Sam Payne and Kris Shaw.

Günter M. Ziegler
Freie Universität Berlin
Realization spaces of polytopes: Dimensions, naive guesses, singularities and special examples
The space of all realizations of a convex polytope can described as an explicit intersection of real quadrics. While the "Universality Theorem" says that there exist realization spaces that are extremely complicated, most examples met in practice turn out to be smooth manifolds, and the "naive guess" for their dimensions seems to be usually correct.
I will argue that it pays off to look at specific examples - the most exciting ones will be two 4-dimensional polytopes, namely the bipyramid over a prism, and the 24-cell, whose realization space is non-trivial, but also still mysterious - and poses computational challenges.
(Joint work with Laith Rastanawi and Rainer Sinn.)