Georgia Institute of Technology College of Sciences

This talk is an oral comprehensive exam in partial fulfillment of the requirements for a doctoral degree. To any topological surface we can assign a certain communtative algebra called a cluster algebra. A surface cluster algebra naturally records the geometry of the surface. The algebra is generated by arcs of the surface. Arcs carry a simplicial structure where the maximal simplices are triangulations. If you squint you can view a surface cluster algebra as a coordinate ring of decorated Teichmuller space with Penner's coordinate. Recent work from many authors has shown that the automorphisms of the surface cluster algebra which preserve triangulations arise from the mapping class group of the surface. But there are additional automorphisms that preserve meaningful structure of the cluster algebra. In this talk we will define surface cluster algebras and discuss future research toward understanding structure preserving automorphisms.

The thesis investigates a version of the sum-product inequality studied by Chang in which one tries to prove the h-fold sumset is large under the assumption that the 2-fold product set is small. Previous bounds were logarithmic in the exponent, and we prove the first super-logarithmic bound. We will also discuss a new technique inspired by convex geometry to find an order-preserving Freiman 2-isomorphism between a set with small doubling and a small interval. Time permitting, we will discuss some combinatorial applications of this result.