Summer School: Titles and Abstracts

Tautological Intersection theory on Moduli Spaces of pointed
rational curves

by Renzo Cavalieri (Colorado State)

The goal of these lectures is to familiarize students with moduli spaces and illustrate how to study their geometry by studying the geometry of families of parameterized objects. We will use moduli spaces of rational pointed curves as our guiding examples. We will study several different modular compactifications of this moduli space, and their intersection theory. If time permits we will discuss the tropical counterparts of the spaces introduced and mention some natural correspondence theorems.
Cavalieri Notes and Exercises

Toric Varieties: Positivity and Syzygies

by Milena Hering (Edinburgh)

This lecture series is an introduction to toric varieties. We will cover affine toric varieties, toric varieties corresponding to a fan, toric varieties corresponding to a polytope, line bundles on toric varieties, positivity of toric line bundles (criteria for line bundles to be globally generated/nef/ample), and syzygies of embeddings of toric varieties.
Hering Notes and Hering Exercises

Hard Lefschetz properties and Hodge-Riemann relations in geometry, algebra, and combinatorics

by June Huh (Princeton)

We will give a broad overview of the Hard Lefschetz properties and the Hodge-Riemann relations in the theory of polytopes, complex manifolds, Coxeter groups, algebraic varieties and related objects. The Hard Lefschetz property and the Hodge-Riemann relations for matroid tropical varieties and their combinatorial applications will be explained in some detail (joint work with Karim Adiprasito and Eric Katz). In the other direction, we will explain how to produce interesting examples in algebraic geometry using tropical varieties that do not satisfy the Hodge-Riemann relations (joint work with Farhad Babaee). In both cases, a notion of convexity for tropical varieties will play an important role.
Huh Exercises