18.217 M.I.T. Fall 2026
18.217
Combinatorial Theory: Positroids
Class meets:
MWF 1-2 pm
Room 32-141
Instructor:
Alexander Postnikov
(office hours: Monday 2-3pm, Room 2-367)
TA:
Ryota Inagaki
(office hours: Tuesday 11am - 12pm, Room 2-390)
Synopsis:
Positroids connect combinatorics, algebraic geometry, representation theory,
convex geometry, tropical geometry, and theoretical physics.
We will discuss these topics with primary focus on their combinatorial aspects.
We will cover some of the following topics (as time allows):
-
Grassmannians and flag varieties, Schubert cells and Schubert
varieties, Richardson varieties
-
Toric varieties, fans, and polytopes
-
positive Grassmannian, positroid decomposition, and positroid varieties
-
Permutations, reduced decompositions, Bruhat order
-
pipe dreams, Young tableaux, Gelfand-Tsetlin patterns, Berenstein-Zelevinsky
patterns, honeycombs
-
Schur, Schubert, and Grothendieck polynomials, Demazure characters
-
Matroids, polymatroids, submodular functions, alcoved polytopes
-
Positroids, plabic graphs, non-crossing paths, matchings, L-diagrams
-
Octahedron recurrence, square moves, membranes, cluster algebras
- ...
Some prior knowledge of basics of algebraic geometry and Lie theory will be helpful
for this course.
Course Level: Graduate
Grading:
In-class evaluations 40%, participation 40%, problem sets 20%
Problem Sets:
Lectures:
last updated: October 9, 2026