Benjamin Nill: Geometry of numbers and toric Fano varieties

Toric Fano varieties are among the most studied classes of toric varieties. They have an explicit correspondence with (certain) lattice polytopes that makes explicit classification results also in higher dimensions possible. A lattice polytope is a convex polytope whose vertices have integer coordinates. In this way, interesting invariants of toric Fano varieties (such as the anticanonical degree) have often an intuitive, combinatorial description (such as the volume). In this talk, I will present the current state of affairs for toric Fano varieties with canonical singularities, sketch a geometry-of-numbers proof of a sharp bound on their degrees, and explain the challenges that arise. If time allows, I will also discuss an open problem in the combinatorial mirror symmetry framework of Batyrev and Borisov.