Joel Kamnitzer, McGill University
The top-heavy conjecture and the topology of (real) matroid Schubert varieties
Suppose we are giving a spanning set S in a vector space V and we consider all subspaces of V spanned by subsets of S. The top-heavy conjecture states that the number of dimension k subspaces is less than or equal to the number for codimension k subspaces. This elementary statement was first conjectured by Dowling and Wilson in 1975 and resisted any proof for 40 years. Finally though, it was resolved by Huh and Wang in 2017, and partially led to Huh's 2022 Fields Medal. I will outline the details of the proof, which relies on the study of the topology of a beautiful space called a matroid Schubert variety. Finally, I will discuss our own contribution to this subject, which is the study of the topology of the real locus of this space (which unfortunately does not lead to the proof of any famous conjecture).
MC 5501