MATH 7410 — Topics in Combinatorics: Total Positivity (Fall 2025)
Course Description: We will discuss combinatorial aspects of recent developments in the theory of total positivity, focusing on the combinatorics and geometry of the totally positive Grassmannian. Tentative topics include:- Grassmannians, flag varieties, and their totally positive parts.
- (Affine) permutations, reduced words, and matroids.
- Positroid cells and varieties.
- Planar bipartite graphs and the dimer model.
- Statistical mechanics: electrical resistor networks and the Ising model.
Instructor: Pavel Galashin (udе.llenroc@nihsаlаg).
Time and Location: Tuesday & Thursday 2:55-4:10 PM in 230 Malott Hall
NOTE: I will be in Germany from Aug 24 to Sep 1. Our first class will occur on Tue, Sep 2.
Grading: Based on several homework problem sets.
Course materials:
The following sources will be relevant throughout the course:-
[Lam14] T. Lam, Totally nonnegative Grassmannian and Grassmann polytopes, Current Developments in Mathematics, 2014.
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[Pos06] A. Postnikov, Total positivity, Grassmannians, and networks, preprint, 2006.