Course information:

MAT-280 Section 001

CRN 36583

Lectures: 1:10-2 pm MWF, Bainer 1128

Office hours: W 4-5 pm, F 2-3pm in MSB 3228

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 Sunday at 11:59pm until 11/23.

   • 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.

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)

For your final project, you should dive in to some of the research that has been done in total positivity in the last 25 years. You should base your project off of a research paper in the field; it's fine to also use some expository papers as references (or use more than one research paper). If you have a paper in mind and are not sure if it's "in the field", come talk to me. If you can't figure out a paper to read, come talk to me.

The audience for your paper/talk is your classmates. In particular, you can assume your audience is familiar with anything we've covered in the course, and overall has familiarity with algebraic combinatorics.

The research paper you choose likely has too much content to cover in a 25 minute talk or 5-10 page paper. You should decide what to cover carefully--you do not necessarily need to discuss the main result of the paper. For example, maybe the paper is in very broad generality, and you would like to discuss a special case with the nicest combinatorics. The most important thing is to craft a good narrative for the results that you do cover, which is comprehensible to someone who hasn't read the paper. If you're having trouble deciding on scope, come talk to me.

You can, and almost certainly should, present things differently from the authors of the paper. First, your goals are different from theirs: they need to prove results rigorously, you need to communicate results comprehensibly to me and your classmates. Second, you should present things as they make the most sense to you, which is not necessarily the same as what made the most sense to the authors. Third, if you are covering just a portion of their results, you can probably tailor notation, definitions, etc. to exactly what you're discussing. 

Talk guidelines:

• Duration: 25 minutes (including questions). If not many people opt to give talks and you would prefer more time, let me know. Practice your talk beforehand!

• Format: your choice. Think hard about whether slides or chalk will suit your topic better. You will turn in either your notes for the chalk talk or your slides. Handwritten slides/hand-drawn figures are fine as long as they are clear.

• Grading: 

  • • 50% delivery: slide/chalkboard use is effective; concepts are well-motivated and illustrated using examples and figures where appropriate; pacing facilitates understanding; results are correctly attributed.

    • 50% content: content is well-chosen and is mostly material that has not been covered in lecture; thought has been put into which details to include (because they improve understanding) and which to omit (because they are technical/not illuminating); what is presented makes logical sense and is free from mathematical errors.

Paper guidelines:

• 5-10 pages+ references, prepared using LaTeX. Hand-drawn figures are fine as long as they are clear. 

• Grading: 

  • 50% writing: writing is clear, readily understandable to someone in the target audience, and free from typos; formatting is consistent and conforms to standard math writing conventions; concepts are well-motivated and illustrated using examples and figures where appropriate; results are correctly attributed.

  • 50% content: content is well-chosen and is mostly material that has not been covered in lecture; thought has been put into which details to include (because they improve understanding) and which to omit (because they are technical/not illuminating); what is presented makes logical sense and is free from mathematical errors. Includes at least one proof or proof sketch.

What actually happened in lecture

• 9/24: [notes] Totally positive and totally nonnegative matrices. Lindström-Gessel-Viennot lemma and statement of path matrix parametrization of TP matrices. 

• 9/26: [notes] Proof of path matrix parametrization. Connection to elementary Jacobi matrices.

• 9/29: [notes] More parametrizations for totally positive GL(n), connections to reduced words for (w_0, w_0). Positivity tests from double wiring diagrams, briefly.

• 10/1: [notes] Totally nonnegative part of GL(n). Definition of cells, proof that cells cover TNN GL(n).

• 10/3: [notes] More on TNN GL(n). Proof that cells are disjoint, semi-algebraic description. 

• 10/6: [notes] The Grassmannian, Plucker coordinates and relations. The totally nonnegative Grassmannian, positroid cells.

• 10/8: [notes] Planar bipartite graphs in a disk, almost perfect matchings, dimer partition functions.

• 10/10: [notes] Almost perfect matchings vs. flows. The boundary measurement map.

• 10/13: [notes] Moves on graphs that preserve boundary measurements. Reduced plabic graphs. Trips and trip permutations. Fundamental theorem of reduced plabic graphs.

• 10/15: [notes] Decorated permutation and Grassmann necklace of a positroid. Bijection between decorated permutations and Grassmann necklaces.

• 10/17: [notes] Adding and removing a bridge.

Plan for the next few lectures

• 10/2o: Adding and removing a lollipop. Proof of parametrization theorem.

• 10/22: How to describe the closure of a positroid cell. Posets for various indexing objects. Oh's theorem on positroids.

• 10/24: More indexing objects (Le-diagrams, certain Bruhat intervals). Another description of positroids.

• 10/27: Positroid polytopes, positroidal subdivisions of the hypersimplex.

• 10/29: The positive Dressian, and positive tropical Grassmannian.

• 10/31: The amplituhedron