Cluster Algebras with Applications to Arithmetic Complexity

As part of a course taken at the University of Melbourne (Advanced Discrete Mathematics) I wrote these notes which attempt to give background on cluster algebras and algebraic complexity theory. Subsequently, I attempt to combine these two theories in hopes of new novel discoveries.

However, it turns out this has already been attempted by Sergey Fomin, Dima Grigoriev, and Gleb Koshevoy in their paper Subtraction-Free Complexity, Cluster Transformations, and Spanning Trees. Thus, the combining part of the notes mainly follow their paper, with a sprinkle of my own thoughts.

The notes can be found here, and/or you can read the abstract below:

Abstract

This document is a short exposition of the theory of Cluster Algebras followed by an application to Arithmetic Complexity Theory, specifically Subtraction-Free Complexity.

This document was made as an assignment for the course Advanced Discrete Mathematics (MAST90030), taken at the University of Melbourne