The PI will pursue three related projects, which highlight the interplay between algebraic combinatorics, representation theory, and tropical geometry. The first project involves a study of cluster algebras, defined by Fomin and Zelevinsky, with an eye towards proving the positivity conjecture that has been open since the founding of this field in 2001. The second project is an exploration of critical groups of graphs, also known as sandpile groups, which were independently introduced by researchers in diverse fields such as graph theory, dynamical systems, electrical networks, and arithmetic geometry. The third project considers various objects from algebraic geometry, including linear systems and Jacobians, and examines their analogues for metric graphs, also known in the literature as quantum graphs or abstract tropical curves. In addition to a number of intrinsic questions arising in each of these fields, the topics of these three projects exhibit intriguing connections to other areas in both pure and applied mathematics. Some of these subjects are Teichmuller theory, number theory, and geometric combinatorics, as well as mathematical physics, combinatorial optimization, and mathematical biology.
At its heart, algebraic combinatorics involves counting, but this enumeration typically is done while keeping track of certain data. This is similar to the census, where it is more useful to obtain a detailed breakdown including demographic information rather than simply a single number stating the number of Americans. A common theme throughout the PI's research is the use of such enumeration techniques to provide new approaches for solving problems in other areas of mathematics. For example, in the theory of cluster algebras, certain geometric formulas arise through a process called seed mutation. However, these same expressions can be computed instead by counting, as long as one knows what features for which to look. The PI will study more phenomena like this, where complicated expressions can be reduced to more concrete calculations. The above topics naturally lend themselves to computational projects and undergraduate research. For instance, the PI plans to use the open source math software Sage with students to get more of them interested in these topics, while creating computational packages for other researchers. This work may also lead to the discovery of new combinatorial patterns motivating further research.
