Instructions for taking notes while reading the assigned pages:
Instructions for exercises:
(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.
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:
Draw all ordinary triangulations for C3, the triangle with one puncture. There should be fewer than 14.
Draw ten or so ordinary triangulations for C4, the 4-gon with one puncture.
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.
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.
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.
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.
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