Alexander Postnikov: 18.217 Combinatorial Theory

18.217 M.I.T. Fall 2026

18.217 Combinatorial Theory: Positroids

Class meets: MWF 1-2 pm Room 32-141 First class: Wednesday, February 9

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%

IMPORTANT NOTE: In the past, grading in 18.217 was mostly based on problem sets. Because of AI, this is no longer a reliable method of students' evaluation. This year, there will still be problems sets. But they will carry a much lower weight. The problems sets will be given for students' practice and entertainement. The grades will mostly rely on in-class evaluations and students' participation.

Lectures: TBA

last updated: September 3, 2026