Name: Sam Panitch sampan9 (at) sas.upenn.edu My bedroom University of Pennsylvania Mutations of Polynomials

We define a notion of mutation among polynomials (check the intro presentation below). A polynomial is 0-mutable it is mutation equivalent to a molynomial, meaning it can be turned into a polynomial of a single term through a series of mutations. In this project, we will explore the notion of k-mutability. We will search for an adequate definition that generalizes the notion of 0-mutability, as well as finding the equivalence classes of 0-mutable polynomials. Finally, we will explore the connections to cluster algebras, which are structures that generalize the notion of mutation.

## Research Log

### Before Program Start

I read through a couple of introductory sources on cluster algebras.

### Week 1

On Tuesday, I attended the REU program introduction. Later in the day, I met with my mentor to discuss future meetings and the project.

Throughout the week, I continued reading sources on cluster algebras. I also began a reading of some algebraic geometry. I built a presentation on one of the sources I was reading to show my group.

### Week 2

On Monday, Anna and I gave a brief talk on our project for the summer

I gave a talk on Cluster Algebras to my group, and improved my understanding of them greatly.

At the end of the week, I began writing code to perform mutations of polynomials, and I also attended a talk on AI by Lydia Chilton.

### Week 3

Throughout the week I worked a lot on my code. I made a ton of progress, and can both expand polynomials using the notion of mutation we have defined.It also handles drawing of the convex hull of the lattice points. In the process, I quickly showed that mutations can be used to translate polynomials.

### Week 4

I did a lot more reading on algebraic geometry, toric varieties, and cluster algebras. On Thursday, I gave a brief presentation on the Grassmannian.

This week also involved a lot of code polishing and refinement. I got polynomial reduction to mostly work, and Anna and I added support for polygon mutation, and created a github repository.

On Tuesday, I attended a talk based around graph theory where I learned about the number of disjoint triangles in co-triangle-free graphs.

### Week 5

This week consisted of lots of bug finding and toric variety reading.I also optimized the code significantly, so that it runs a lot faster.

I also seriously started considering how to classify 0-mutable polynomials. One question of interest here is whether or not a 0-mutable polynomial would ever have to be expanded before it could be reduced. Understanding this is critical for an algorithm that can efficiently reduce.

On Monday, I attended a talk by Lenda Zdeborova.

### Week 6

I gave a talk about algebraic genus to my group on Tuesday.

I spent a lot of time thinking about the reduction/expansion problem, unfortunately to no avail.

On Thursday, we found a recent paper very related the work we are doing. In it, they prove an equivalence of 0-mutable polynomials with something called "rigid maximally mutable" polynomials, but they prove this connection with a high-level geometric construction. It is of interest to find a combinatorial proof.

I attended a talk about scientific writing to prep for the final paper, and Rebecca Wright spoke about privacy in the modern world.

### Week 7

I thought a lot about proving the equivalence between Rigid Maximally Mutable polynomials and 0-mutable polynomials, with no success.

After meeting with the group on Wednesday, we agreed it would be wise to carefully work out a lot of examples. In the process, I uncovered a possible counterexample to the paper. Upon contact with the authors, it was found that there was an additional condition required for the theorem to be true, irreducibility of the polynomial.

There were talks about graduate school applications, ethics in research, and removing algorithmic bias.

### Week 8

I gave a presentation on dimension of an affine variety to my group.

Professor Woodward suggested taking on a small project to have a good result to present. I took on a further investigation of Rigid Maximally Mutable Polynomials. In particular, I looked at the relationship between the mutations that can be defined on a reducible polynomial and its factors, but after some thought, I found that nothing particularly interesting could be said.

On Friday, there was a talk by Cythia Rudin that explored the interpretability of machine learning methods.

### Week 9

There was a fun puzzle solving talk on Tuesday.

Mostly this week was spent preparing my final presentation and paper.I really enjoyed this experience and am extraordinarly grateful for everyone who made it possible.