Combinatorics for commutative algebra

On last Friday, December 10th 2021, I saw a wonderful talk by Dr. Anna Weigandt of MIT, about the Castelnuovo-Mumford regularity of matrix Schubert varieties.

The art of giving math talks

Dr. Weigandt’s talk was extremely well delivered and was really instructive for me on the art of giving talks. The primary takeaway for me on the art of giving talks was the clear line she drew from a question in commutative algebra directly to a combinatorial question. While I love counting for counting’s sake, she explained how the story she was telling was about a numerical invariant of matrix Schubert varieties, and how it could be computed using inversion numbers and a newly-defined statistic on permutations, the Rajchgot index. This also helped me feel more motivation for the study of combinatorics in relation to commutative algebra.

Inversions and the Rajchgot index

The main mathematical result which I took away from the talk was the following:

Theorem The Castelnuovo-Mumford regularity of the matrix Schubert variety $X_\omega$ is $\operatorname{raj}(\omega) - \operatorname{inv}(\omega)$.

There are certainly other results which turn a question in algebra into a question in combinatorics, but I don’t think I have ever seen the line drawn so clearly. I hope that in the future I can keep this example in mind to help frame other results.