Moduli of Varieties

Algebraic geometry is concerned with the study of the geometry of the zeroes of polynomials. It is a central topic with strong connections to all branches of mathematics. A fundamental and challenging problem in algebraic geometry is to classify all algebraic varieties. After fixing some discrete invariants, it remains to understand all continuous families with the same discrete invariants. A moduli space is a variety whose geometry simultaneously captures all continuous families with the same invariants. One of the most famous and heavily studied example of a moduli space is the moduli space of compact Riemann surfaces of genus g. Recently, there has been rapid and exciting progress in extending this to all dimensions.

This collaboration is focused on improving our understanding of the geometry of moduli spaces in a range of contexts. The aim of the collaboration is to determine appropriate invariants to construct the corresponding moduli spaces and to study their rich geometry. We will look at the case of positive characteristic, K-trivial varieties, algebraic foliations, K-stability of Fano varieties, and generalizations of K-stability to other contexts.