3094 — Spring 2019, Research in cluster algebras

Instructions for taking notes while reading the assigned pages:

Correct any typos you see
Read slowly and carefully - verify all the given examples on your own
Write down vocab words and concepts that you’ve never seen before or have forgotten.
If the words or concepts are explained in this textbook, write down the definitions/explanations. Otherwise, look up all these vocab words/concepts (on the internet or your algebra textbook) and attempt to understand them.
Bring the questions about these new concepts to the meeting

Instructions for exercises:

Be ready to present to me
You can use any sources (book, online, or people - please credit your sources)
Hand-written is fine.

Notes on 4-people meeting (with Ana GE) week 12 Wed, April 10, 2019

Part 5 HW assigned during 3-people meeting week 10 Wed, March 17, 2019

(Note: we had a meeting between part 4 and part 5, but I forgot to write it down here)

Continue drawing tagged triangulations for D4 (there are fifty). The fourteen tagged triangulations for D3 are here: D3 associahedron
Keep in mind the following four families of surfaces: polygon (disk), polygon with one puncture, annulus, and polygon with two punctures.
Be more familiar with tagged triangulations: Keeping in mind the four families of surfaces,
- read Chapter 2 of Schiffler’s Lecture notes from CIMPA 2016 - I printed this for you
- watch two of lectures by Musiker from summer 2011 (Aug 8-12) at MSRI
  1. lecture 1 Mon: Intro to cluster algebras from surfaces
  2. lecture 2 Tues: tagged arcs and punctured surfaces
  3. This following video is less relevant, but it won’t hurt to watch: lecture 3 Wed: combinatorial formulas for cluster variables
Read the paper the Mutation Class of Dn Quivers by Vatne - I printed this for you.
- Ignore the words you don’t understand mentioned in the introduction section.
- Start with Sections 1 and 2 first, before Section 3.
Goal: write a joint report following Vatne’s paper, but discuss the surface model. For now, do type A (Sec 2) and D (Sec 3).
- You may use figures in which Vatne posted on Arxiv, see Vatne’s source code (Overleaf project)

Part 4, notes for 3-people meeting Week 7 Wed, March 6, 2019

Lecture: ordinary triangulations of Cn, an n-gon with one puncture, mutation of ordinary triangulation (when possible), associating a quiver to an ordinary triangulations of Cn, and examples in C3.
HW Part 4 (assigned during 3-people meeting Week 7 Wed, March 6, 2019:
1. Summarize in your notebook the following part from Brualdi Section 7.6: Let an be the number of triangulations of an (n+3)-gon. Prove the recurrence relation of an. Find the generating function of an. Prove an explicit formula for an.
  - Reference: Section 5.4. Seed patterns of type Dn of FWZ Intro to Cluster Algebras Chapters 4–5
  - A more difficult reference (to be used later): The number of elements in the mutation class of a quiver of type Dn by Buan and Torkildsen
2. Draw all ordinary triangulations for C3, the triangle with one puncture. There should be fewer than 14.
3. Draw ten or so ordinary triangulations for C4, the 4-gon with one puncture.
4. Later, you will be asked to adopt Brualdi Section 7.6’s proof but for the number of ordinary triangulations for Cn. Start thinking about this.
5. (added later) More stuff to read and watch on generating functions
  - See Lecture notes on the same topic as Brualdi 7.6 and Bona Sec 8.1.2.1 Catalan numbers (Given Week 7 Mon and Week 8 Mon, Wed, 2019)
  - Watch Lecture videos by TheTrevTutor: Generating Functions and Recurrence Relations and Generating Functions

Questions Part 3

During your two-people meeting, focus on the following:

Counting the number of quivers that are mutation equivalent to A3, A4 quivers, and An in general. Let a_n be the number of quivers of type An. Can you write a formula or a recurrence relation for this sequence?
Consider the connection with polygons, see Section 2.2 of https://arxiv.org/pdf/1608.05735.pdf. How many different triangulations of a polygon are there? You can google this for more info.
Count the number of quivers of type D4. Previously I asked you to think about shapes - but now please consider the directions of arrows as well. The number is much bigger, so you don’t necessarily need to draw all of them. Count the number of quivers of type D5 (again, don’t draw all of them).
The next step will be to consider the connection between Dn quivers and triangulations of a once-punctured polygon. I will explain this to you when we meet in person.

Note: At the end of your 2-people meeting, please write a joint email report to me to summarize your meeting. In your individual email report, you’ll continue to briefly summarize only what you did when you worked alone.

Reading Part 2

Do Exercise 2.2.2 (from Sec 2.2) from FWZ textbook.
Take notes on Sec 2.7 of FWZ textbook.
Attempt all the exercises. Note that the Keller’s App has a function which goes from quiver to exchange B-matrix.

Reading Part 1

Take notes on Sec 2.1 and 2.2 (pg 17-20) of FWZ textbook. Budget an hour or more for each of pages 18, 19, and 20.
Practice doing mutations (1 hour)
- Do several sequences of mutations starting from the quiver 1->2->3 by hand and then using Keller’s Quiver mutation in JavaScript
- Do several sequences of mutations starting from the quiver of shape D4 by hand and then using Keller’s Quiver mutation in JavaScript
Exercises from reading: Exercise 2.1.4 and Exercise 2.2.3.

Computing Part 1

Play with Keller’s Quiver mutation in JavaScript, either download or use it on the browser.
- Try to compute the mutation class for the quiver 1 -> 2 -> 3 (type A3) both by hand and using Keller’s app
- Try to compute the mutation class for the quiver (with 4 vertices) 1 -> 2 -> 3 -> 4 -> 1 (type D4) both by hand and using Keller’s app
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