The Mukai conjecture for spherical varieties

Fano varieties are one of the fundamental building blocks of algebraic varieties, and their investigation is a central question in birational geometry. The Mukai conjecture concerns their geography, predicting a relationship between the Picard rank and the divisibility of the anticanonical divisor.
In this talk, we will discuss this question and survey the two main proof strategies in the literature: the study of rational curves on Fano varieties and decomposability properties of the anti-canonical divisor. We will then focus on spherical varieties, a large class of normal varieties with a group action that generalises toric, flag, and symmetric varieties. Here, these strategies can be combined into a third approach, relating the Mukai conjecture to the complexity of pairs in birational geometry. By using properties of this complexity, we then complete the proof. If time permits, we will conclude with what can be said about this third approach in general.
This is joint work with Giuliano Gagliardi and Heath Pearson. An effort will be made to keep the talk accessible.