Cluster algebras, critical groups, and tropical curves

PI将进行三个相关的项目，突出代数组合学、表示理论和热带几何之间的相互作用。第一个项目涉及对由Fomin和Zelevinsky定义的簇代数的研究，着眼于证明自2001年该领域成立以来一直开放的正性猜想。第二个项目是探索图的关键群，也被称为沙堆群，它们是由图论、动力系统、电子网络和算术几何等不同领域的研究人员独立引入的。第三个项目考虑了代数几何中的各种对象，包括线性系统和雅可比矩阵，并研究了它们在度量图中的类似物，在文献中也称为量子图或抽象热带曲线。除了这些领域中出现的一些内在问题外，这三个项目的主题与纯数学和应用数学的其他领域都有有趣的联系。其中一些学科是Teichmuller理论，数论，几何组合，以及数学物理，组合优化和数学生物学。代数组合学的核心是计数，但这种枚举通常是在跟踪某些数据的同时进行的。这与人口普查类似，在人口普查中，获得包括人口统计信息在内的详细分类比简单地说明美国人数量的单个数字更有用。在PI的研究中，一个共同的主题是使用这种枚举技术为解决其他数学领域的问题提供新的方法。例如，在聚类代数理论中，某些几何公式是通过称为种子突变的过程产生的。然而，这些相同的表达式可以通过计数来计算，只要知道要查找哪些特征。PI将研究更多这样的现象，在这些现象中，复杂的表达式可以简化为更具体的计算。上述主题自然适合于计算项目和本科生研究。例如，PI计划与学生一起使用开源数学软件Sage，让更多的学生对这些主题感兴趣，同时为其他研究人员创建计算包。这项工作也可能导致新的组合模式的发现，从而推动进一步的研究。

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.