Kyan Valencik's DIMACS REU Webpage
| Name: | Kyan Valencik |
|---|---|
| Email: | kv340@scarletmail.rutgers.edu |
| Institution: | Rutgers University-New Brunswick |
| Project: | About mutations of Laurent Polynomials |
About My Project
We are examining the mutation graph of the potential of projective three-space \(x+y+z+\frac{1}{xyz}\).
Abstract: In mirror symmetry, there is believed to be a correspondence between Fano varieties and Laurent polynomials. In their paper Maximally Mutable Laurent Polynomials, Coates et al. proved that the 98 deformation classes of Fano three-manifolds with very ample anticanonical bundle correspond one-to-one with certain mutation classes of Laurent polynomials. Here, a mutation is a change of variables which changes a Laurent polynomial \(f\) into another Laurent polynomial \(f'\). For example, \(x \mapsto \frac{x}{(1+x/y}, y \mapsto \frac{y}{(1 + x/y)}\) changes \(x + y + \frac{1}{xy}\) to \(y + \frac{1}{xy} + \frac{2}{y^2} + \frac{x}{y^3}\). The "mutation graph" is the graph whose vertices are all Laurent polynomials produced in this way, and edges are the mutations. In our project we will examine the mutation graph of the Laurent polynomial \(x+y+z+\frac{1}{xyz}\) corresponding to projective three-space \(\mathbb{P}^3\).
Our Team
- Mentors: Chris Woodward, Dennis Hou (PhD Student), Yiyang Liu (PhD Student)
Weekly Summaries
Week 0 (Before the program)
I read chapters 1-3 of Introduction to Cluster Algebras by Fomin, Williams, and Zelevinsky. This proved helpful in viewing mutations as both an algebraic and combinatorial phenomenon.
Week 1
After settling into the program, I met with Professor Woodward and my other mentors to discuss our project goals. I was most interested to learn that our work has implications to the subject of mirror symmetry, a conjectural relationship between string theory and algebraic geometry. After the first meeting, I met with Dennis and Yiyang to discuss more about mirror symmetry (and learn it was way beyond my depths). I met with Dennis and Yiyang again the next day to unpack definitions 1.5 and 1.6 of the Coates paper (hereby called MMLP24) to better understand the theory behind laurent polynomial mutations. I began developing a Sage program which would allow me to mutate the potential function and graph its mutations as convex polytopes. On friday, I presented my program to our entire team and shared my idea of extending known mutations of the \(\mathbb{P}^2\) potential to \(\mathbb{P}^3\). Yiyang, Dennis, and I explored this further but were not able to generalize more mutations. However, in the process we came up with a lot of ideas that will help us understand the mutation graph once we have more data. It's been a long and productive week, but I enjoyed every part of it!
Week 2
My program for mass computing Laurent polynomial mutations and displaying the mutation graph is now up and running! Now that I can easily generate mutations, we need to consider equivalences between mutations up to monomial change of basis. After brainstorming the most optimal way to do this, we found a breakthrough paper titled "Normal Forms Convex Polytope" which describes an exact algorithm to find a "normal form" for Laurent polynomials and begin the classification of mutations. I Professor Woodward also assigned me to start reading Toric Varieties by Cox to better understand the algebraic geometric structures behind mutations of Laurent Polynomials and their corresponding polytopes.
Week 3
Much of this week was spent optimizing the mutation algorithm with the Laurent normal form process in mind. After much troubleshooting, I was able to parallelize the mass mutation program and produced over 70 unique mutations classes. However, these mutations were reaching magnitudes of up to 70 terms so I developed a new graph which would abstract away the polynomial itself and instead focus on its polytope data (volume, # vertices, edge lengths). Miraculously, this change revealed that the Markov tree was embedded into our graph. This generated many questions that I hope to explore in Week 4.
Week 4
This week was spent optimizing the program, looking for patterns in mutation tree (specifically the Markov branch), and coming up with conjectures.
Week 5
I finally got access to the Math Department compute server! This massively increased the depth of the mutation graph (which no longer can be described as a tree since we found cycles!). Right after meeting with the group, I noticed a pattern between cyclic mutations, of which I was able to quickly derive a first lemma about the mutation graph.
Week 6
This week, I made it my goal to understand Proposition 7.3 of MMLP24 and present it to the group. I was given an informal lecture by Yiyang about Toric geometry found in later chapters of Cox, which proved immensely helpful for understanding the proposition. In the process of studying the proof on my own, I found a way to extend this result about 2D mutations to our 3D case, and was able to procedurally generate mutations preserving the "Markov structure" of their edge lenghts!
Week 7
This week I ran many computations to check the robustness of my new Markov mutations method. I realized in the process that the conjecture I had settled on corresponded exactly to a factorization trick I noticed early in the project which allowed for nice cancellations which preserve Laurentness under mutation.
Week 8
Along with preparing for and giving my final presentation, when working with Yiyang I came to a massive realization that the Markov pattern corresponds exactly with mutations in the mutation graph for \(\mathbb{P}^2\).
References & Links
Here are some relevant resources:- Maximally Mutable Laurent Polynomials, Coates, Kasprzyk, Pitton, Tveiten. - arXiv.org