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.
The research investigated in this project is the theory of cluster algebras, a framework unifying seemingly unrelated topics in mathematics and physics. The PI focused on problems ranging from measuring how distances change as deform a surface, think of a balloon or a doughnut, to calculating partition functions of interest to string theorists studying particle configurations. The PI’s research takes advantage of inherent symmetries in such theories to provide explicit and simple formulas where previous calculations would have involved complicated recursive processes. One of the outcomes of this research were new formulas (written as a special kind of rational function known as a Laurent polynomial) for certain elements of a cluster algebra in terms of a visual combinatorial model. This model involved counting perfect matchings, which are distinguished collection of arcs in a network of nodes and arcs such that every node is incident to exactly one arc. Previous work of the PI with Ralf Schiffler and Lauren Williams in 2009 had provided such formulas for cluster variables, which are the generators of a cluster algebra, in the case of a cluster algebra generated by curves on a surface. The work done on this grant, with Schiffler and Williams, extended these formulas to other elements, namely those associated to closed loops, and then proved that using these new elements, one could build a basis for such an algebra. A second outcome of this research was the study of cluster algebras of interest to string theorists and the use of perfect matchings to obtain formulas for cluster variables in such examples. Here, instead of working on a surface, one considers an infinite tesselation of the plane known by physicists as a brane tiling. The PI worked with undergraduates I. Jeong, S. Zhang, M. Leoni, S. Neel, and P. Turner, and provided new formulas for several such tilings. This has led to discussions and collaborations with physicists that are on-going. This research has been disseminated all over the world, as the PI has given plenary and invited talks in Scotland, Germany, Chicago, Mexico, San Diego, Berkeley, Canada, Washington D.C., Providence, Iceland, and Houston. The PI has also organized numerous workshops and conferences on this subject, including an AMS (American Mathematics Society) section meeting in Colorado, an AMS mathematical research community in Utah, a software development conference in Minnesota, and an MSRI (Mathematical Sciences Research Institute) summer graduate school in California. Additionally, this project has included teaching a semester-long topics course for graduate students with notes publicly available on the PI’s website, and the PI has mentored two Ph.D. Students and numerous REU (Research Experience for Undergraduates) students during the summers.
