|Home Institution:||Rutgers University-New Brunswick|
|Project:||Mutations of Polynomials|
We are examining mutations of polynomials by studying cluster algebras and toric varieties.
Abstract: Mutations arise in mirror symmetry and so-called cluster theory when there are different natural coordinate systems on the same solution set. For example, consider the set of 2-dimensional subspaces of a 4-dimensional vector space. Taking two basis vectors we get a 2x4 matrix, and any two of the determinants of 2x2 matrices inside the resulting 2x4 matrix give coordinates for the set. The various coordinates systems are related by a mutation. We will explore this and other examples. More concretely a mutation of a polynomial f(x,y) in two variables x,y with factor h(x,y) is one obtained by replacing each monomial x^i y^j with x^i y^j h(x,y)^(l(i,j)) where l(i,j) is some power that depends in a linear way on the powers i,j. For example, a mutation is given by replacing y with y(1+x)^(-1) and multiplying by (1+x)^(-2). This mutation applied to f(x,y) results in the polynomial f'(x,y) = (1+x) + 2y + y^2. If two polynomials are related by mutations we say they are mutation-equivalent. Problem: classify mutation-equivalence-classes of polynomials. There is a conjecture of Petracci and collaborators about what the mutation-equivalence classes are. We will investigate a version of this conjecture.
It was an exciting first week of the program! I dived into the resources provided by my mentors, and have started to build a picture of the problems that we will be studying. I have become more familiar with what cluster algebras are, and have also started learning some algebraic geometry. During our group meetings we discussed some background material and potential questions to focus on throughout the summer.
It was a very busy week! At the start of the week we gave our project introductory presentation and had the chance to listen to everyone else's as well. I also gave a presentation within my group about some algebraic geometry necessary for the project. Throughout the week I focused on studying toric varieties and basic algebraic geometry. I have a better idea now of the math necessary to approach our research questions. I am looking forward to continuing learning more new math!
This week I started out with studying some more background material on cluster algebras. Then, I focused mainly on better understanding our main questions concerning mutations of polygons. I began with working with the definition of 0-mutable, and started writing a Macaulay2 program that applies mutations to polynomials. I would like to see all the mutations that can be applied to a given polynomial. In our group meetings we discussed the Grassmanian and other algebraic geometry. I will be giving a presentation about mutations of polygons next week.
Another week filled with polynomials, polygons, and algebraic geometry! I began this week studying mutations of polygons, and how they relate to mutations of polynomials. This helped my understanding of our research questions. I gave a presentation overviewing the topic, and worked through some examples on my own. There are a lot of interesting questions to be investigated in this area. Later in the week I also did some more research on algebraic geometry. I'd like to better understand the connection between varieties and mutations of polynomials.
I spent most of this week learning more albegraic geometry. I focused on the definition of the genus of a curve. This required knowing about projective space, the Hilbert function, and the Hilbert polynomial. At the end of the week we had a group meeting, where I presented what I was able to put together throughout the week. We also talked more about 0-mutable polynomials, and whether we can use code to determine whether a given polynomial is 0-mutable. For now, we are focusing on a specific trianlge, and are trying to classify all the polynomials that have this triangle as their convex hull.
This week I did a lot of reading about mutations of polynomials and mutations of polygons. Because there are so many sources with different definitions of mutation, I started putting together a document of all the different defintions and their comparisons. There was also a key paper called Mirror Symmetry and smoothing Gorenstein toric affine 3-folds in this area published this week. We began looking at this paper, and trying to understand the provided examples. We have needed to slightly revise the definition of mutation we are working with.
I split my time this week between learning some algebraic geometry and studying 0-mutable polynomials. I spent the first half of the week reading about morphisms. The two texts I found most helpful were Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry and Kollar's The structure of algebraic threefolds: an introduction to Mori's program. After giving my presentation on morphisms hapfway through the week, I switched my focus back to the newly published paper Mirror Symmetry and smoothing Gorenstein toric affine 3-folds. I focused on working through all the details of the examples and figuring out the possible mutation data for the given polynomials.
As we are nearing final presentations and final papers, I spent the week thinking about some small questions related to mutations of polynomials and how they relate to mutations of polygons. I worked on showing the equivalence of definitions of mutations in different papers, so that I could apply propositions from each paper.
The week was a great conclusion to the program! I spent the first half of the week putting together the final project presentation with my partner Sam. Wednesday morning we had a successful presentation, and then had the opportunity to watch everyone else's presentations the rest of the week. The second half of the week we also focused on finishing the final paper. It was a nice feeling to see our summer's work put nicely together. Thank you to everyone who made this experience possible!