18.217 Notes
For the fall 2026 term, 18.217 will focus on positroids.
For last year's course notes, please see
Ilani Axelrod-Freed's
website. Written notes will be posted quickly after each lecture, and typed notes will
follow at a later date, for general reference. Please reach out with any comments/corrections. I also made a Discord Server:
https://discord.gg/J9HAVVRXa for people to informally chat about the class and
share combinatorics resources!
Wednesday, September 9th
We began with some general motivation for studying positroids in various areas of mathematics,
and followed with a brief plan for the topics to be covered this term. We also discussed some
thoughts on AI and the grading structure for the course. The final half of class gave us an
introduction to various presentations of the grassmannian and ended with
a hint towards Plucker coordinates and Plucker relations.
Written Notes: here
Typed Notes: coming soon
Friday, September 11th
We first introduced definitions of Schubert cells and Schubert varieties
from the persepctive of pivots in the RREF of points in the grassmannian. Then, we conncted back to Plucker coordinates
and described how computing maximal minors gives us a less efficnent but more combinatorially rich
interpretation for pivots in points of the grassmannian. The lecture concluded with an introduction to matroidal stratification.
Written Notes: here
Typed Notes: coming soon
Monday, September 14th
We reviewed Schubert cells in terms of and Youngs diagrams, k-subsets, and dimension vectors obtained from the RREF of some
point in the grassmannian; we then discussed Bruhat order on the Schubert cells, which looks different from the familiar
partial order on permutations. For the remainder of the lectire, we introduced Pluker relations and gave numerous equivalent
definitions of matroids.
Written Notes: here
Typed Notes: coming soon
Wednesday, September 16th
Written Notes: here
Typed Notes: coming soon
Friday, September 18th
Written Notes: here
Typed Notes: coming soon
Monday, September 21st
Written Notes: here
Typed Notes: coming soon
Wednesday, September 23rd
Written Notes: here
Typed Notes: coming soon
Contact:
Email 1: ef110@wellesley.edu
(general & professional)