Chromatic Symmetric Functions: Solving Algebraic Conjectures Using Graph Theory

许多过程在很大程度上取决于自然物体固有的对称性，以及这些物体在空间中的运动如何影响它们；例如，确定亚原子粒子的能量状态并预测它们之间的转换，或者从随机噪声中区分通信信号。研究这类物体的一种方法是通过旋转扰动它们，并分析由此产生的相互作用。由于对称性，一些旋转会产生与其他旋转相同的结果。这些对称性可以用对称函数和多元多项式来编码，这些多项式在变量的任何排列下都是固定的。对不同对象的研究可以通过考虑给定代数结构或理论背景下的对称函数来表示。例如，对于亚原子粒子和量子物理，我们考虑福克空间，对于信号识别，我们考虑随机矩阵理论。本研究的长期目标是通过图论来研究对称函数。图在研究离散系统及其关系建模时经常出现。图论是离散数学中研究最多的领域之一，该研究的主要方法将是根据图及其结构来解释对称函数，以应用更广泛的知识。一个起点将是色对称函数X_G，它是目前图和对称函数之间唯一的主要连接。这个程序将扩展我们的知识，帮助我们理解X_G编码关于图的信息，并将X_G推广到基于k理论和其他现代代数思想的更强形式。此外，该程序将创建基于图的结构来表示代数对象，如LLT多项式和Schur函数的体积，从而为代数组合学中众所周知的难题（如解释麦克唐纳多项式）和确定对称群表示的分支规则提供新的攻击途径。本研究的长期愿景是，这两条路径将收敛并相互建立：色对称函数已经在特殊情况下与LLT多项式相关，而LLT多项式又被用于一般对称函数理论，以研究量子力学中量子态之间的粒子运动。因此，本研究的目的是建立一个桥梁，连接两个良好的研究领域的数学从双方建立。这样的桥梁将极大地推进两个领域的成果和方法，因为一个领域的技术可以很容易地应用于另一个领域。参加本课程的学生将在离散数学的两个主要领域的前沿学习有价值的技能，并立即应用于学术界或计算机科学行业的职位。

Many processes depend largely on the innate symmetries of natural objects, and how the movement of these objects in space affects them; for example, determining the energy states of a subatomic particle and predicting transitions between them, or distinguishing communication signals from random noise. One way to study such objects is to perturb them by rotations and analyze the resulting interactions. Because of symmetry, some rotations will produce an identical result to others. These symmetries may be encoded by symmetric functions, multivariate polynomials that are fixed under any permutation of the variables. The study of different objects can then be represented by considering symmetric functions within the context of a given algebraic structure or theory. As examples, for subatomic particles and quantum physics we consider the Fock space, and for signal identification we consider random matrix theory. The long-term goal of this research is to study symmetric functions through the theory of graphs. Graphs arise frequently in research to model discrete systems and their relationships. Graph theory is one of the most studied areas of discrete mathematics, and the main approach to this research will be to interpret symmetric functions in terms of graphs and their structure to apply a much broader range of knowledge. A starting point will be the chromatic symmetric function X_G, currently the only major connection between graphs and symmetric functions. This program will expand our knowledge helping us to understand the information X_G encodes about graphs, and to generalize X_G to a stronger form based on K-theory and other ideas from modern algebra. Additionally, this program will create graph-based constructions to represent algebraic objects such as LLT polynomials and plethysms of Schur functions, thus providing new avenues of attack for notoriously difficult problems in algebraic combinatorics such as interpreting Macdonald polynomials, and determining branching rules for representations of symmetric groups. The long-term vision of this research anticipates that these two paths will converge and build upon each other: chromatic symmetric functions are already related in special cases to LLT polynomials, which in turn are used in general symmetric function theory to study the movement of particles between quantum states in quantum mechanics. Thus, this research aims to form a bridge connecting two well-studied areas of mathematics by building from both sides. Such a bridge will greatly advance results and approaches in both fields, as techniques from one may be easily applied to the other. Students taking part in this program will learn valuable skills at the forefront of two major areas of discrete mathematics with immediate applications to positions in academia or the computer science industry.