Course information:

MAT-280 Section 001

CRN 36583

Lectures: 1:10-2 pm MWF, TLC 3212

Office hours: W 2-3 pm in MSB 3228 (subject to change at the beginning of the quarter)

Course description: This course will survey some of the rich combinatorics that has emerged in the last 25 years from the theory of total positivity. Classically, a matrix is totally positive if every square submatrix has positive determinant. Suprisingly, the set of totally positive matrices in SL(n) form a topological open ball. The notion of total positivity was generalized by Lusztig and Postnikov, and gives rise to many topologically simple, combinatorially interesting spaces such as the positive Grassmannian. In a more algebraic direction, total positivity lead to the development of cluster algebras, which are commutative rings endowed with an intricate combinatorial structure. The course will cover total positivity for SL(n) and for the Grassmannian; a generalization of the positive Grassmannian inspired by high energy physics called the amplituhedron; and, time permitting, connections to cluster algebras.

Prerequisites: 245, 250AB. 

Grading: Grades in the course are determined by three main components.

1. Lecture attendance (30%)

   • Lectures may occasionally need to be rescheduled, held online, or taught by a guest lecturer. Any irregularities will be communicated via email and posted here.

2. Exercises (30%): 2 exercises are due on Canvas every week except for the final two weeks. Weekly due date to be determined on first day of class.

   • A running list of exercises is here. You can choose any two problems (which you have not previously done) to turn in each week. Ideally, you would choose problems relevant to recent lectures, but if you need to brush up on previous content, it's also fine to do exercises for older lectures.

   • These exercises are intended to help you keep up with, and digest, the lecture content. As such, I ask that you spend time sincerely engaging with the problems, and try to sometimes pick the harder ones. When you get stuck, try to talk to me or your classmates before turning to the internet. It is fine to use the internet as a resource, but I ask you to do so judiciously, to help you learn more efficiently rather than to avoid learning.

   • Your solutions should be legible and clear. You are encouraged to collaborate with other students in the class for all steps of problem-solving up until writing, which you should do alone. Please list the students you collaborated with, and any sources which substantively contributed to your final solution. 

   • There are no extensions. You will be graded on what you turn in on the weekly due date.

3. Project (40%): you will choose a topic related to the course and will either

   • write a 5-10 page expository paper, due on Thursday December 11 at 11:59 pm

or

3. •  give a 20-30 min presentation during the final week of classes.

You have until October 31 to decide which option you'd like. More details on the project, including the grading criteria, will be posted later. See References below for some possible topics.

Plan for the next few lectures

• 9/24: Totally positive part of GL(n). Lindström-Gessel-Viennot lemma and path matrix parametrization of positive part.

• 9/26: More parametrizations for totally positive GL(n), connections to reduced words for (w_0, w_0). Positivity tests from double wiring diagrams.

• 9/29: Totally nonnegative part of GL(n). Definition of cells, parametrizations from graphs, indexing set for cells. Positivity tests.

• 10/1: The Grassmannian. Plucker coordinates and relations, the totally nonnegative Grassmannian, positroid cells. Embedding totally nonnegative GL(n) into TNN Gr(n, 2n).

• 10/3: Planar bipartite graphs in a disk, almost perfect matchings. The boundary measurement map.

Total positivity

• Total positivity: tests and parametrizations (Fomin, Zelevinsky)

• Total positivity, Grassmannians, and networks (Postnikov)

• Totally nonnegative Grassmannian and Grassmann polytopes (Lam)

• Variations on a theme of Kasteleyn, with application to the TNN Grassmannian (Speyer)

• Positroids and Schubert matroids (Oh)

• The positive tropical Grassmannian, the hypersimplex and the m=2 amplituhedron (Lukowski, Parisi, Williams)

• Positive configuration space (Arkani-Hamed, Lam, Spradlin)

The amplituhedron

• The m=1 amplituhedron and cyclic hyperplane arrangements (Karp, Williams)

• Decompositions of amplituhedra (Karp, Williams, Zhang)

• The m=2 amplituhedron (Bao, He)

• The positive Grassmannian, amplituhedra, and clusters (Williams)

• The amplituhedron BCFW triangulation (Even-Zohar, Lakrec, Tessler)

Additional topics:

• A formula for Plücker coordinates associated with a planar network (Talaska)

• Parametrizations of flag varieties (Marsh, Rietsch)

• Network parametrizations for the Grassmannian (Talaska, Williams)

• Positroid varieties: juggling and geometry (Knutson, Lam, Speyer)

• The twist for positroid varieties (Muller, Speyer)

• Parity duality for the amplituhedron (Galashin, Lam)

• Amplituhedra and origami (Galashin)