Abstract: We shall survey possible definitions of a wonderful model for a topic arrangement and discuss their existence and properties.

Abstract: I will explain tropical cohomology which is a tool to recover Hodge theoretic information of families of complex algebraic varieties from tropical varieties. Tropical varieties which arise from matroids satisfy a version of Poincaré duality for this cohomology theory. I will also explain some combinatorial consequences of this phenomenon.

Abstract: Demailly (2012) showed that the Hodge conjecture is equivalent to the statement that any (p,p)-dimensional closed current with rational cohomology class can be approximated by linear combinations of integration currents. Moreover, the statement that all strongly positive currents with rational cohomology class can be approximated by positive linear combinations of integration currents can be viewed as a strong version of the Hodge conjecture (1982). In this talk, I will explain the construction of a current which does not verify the latter statement on a toric variety, where the Hodge conjecture is known to hold. The example belongs to the family of ‘complex tropical currents’, which we extend their framework to toric varieties, discuss their extremality properties, and express their cohomology classes as recession fans of their underlying tropical varieties. Finally, the counterexample will be the tropical current associated to a 2-dimensional balanced subfan of a 4-dimensional toric variety, whose intersection form does not have the right signature in terms of the Hodge index theorem. This is a joint work with June Huh.

Abstract: A lattice polytope is called spanning if its lattice points generate the lattice. In this talk, we discuss current work in project to show that the Ehrhart h∗-vector of a spanning lattice polytope has no gaps. This result would generalize a recent theorem by Blekherman, Smith, Velasco and would imply a polyhedral version of the Eisenbud-Goto conjecture. We also discuss applications for non-spanning lattice polytopes. This is joint work with Johannes Hofscheier and Lukas Katthän.