Graphs in spectral analysis of complex systems

本项目主要研究图的光谱分析和图在更复杂系统光谱分析中的应用等几个相互关联的问题。大多数问题都是从量子图作为量子混沌学模型的使用中获得灵感的，量子混沌学是数学物理学的一个分支，涉及在经典极限下表现出混沌动力学的量子系统的特性。该项目解决了量子图和相关系统的几个开放问题，可以在3年的时间框架内解决。利用图作为模型已经取得了一些显著的成功，特别是在研究谱统计的普适性和研究图特征函数的节点统计方面。然而，该项目仍有许多重要的未解问题和一些极有希望的方向。第一个问题是关于泛型图的谱是否简单的基础问题。此外，最近关于图特征函数的节点域数目的结果暗示了迹公式的存在，并将这个数目与图的周期轨道的性质联系起来。PI研究了谱参数变化时开图（带无限引线的图）上零点的动态变化。其目的是量化零的数量发生变化的事件。PI还使用一种特殊类型的图作为模型，研究长周期轨道（比图的大小还要长）内的相关性。最后，PI正在寻找将图解求和方案扩展到新的应用问题的方法，并寻找统一几个现有图解方案的代数结构。考虑了在组合学和计算分子生物学中可能的直接应用。该项目解决了几个问题的频谱性质的图，我们目前的理解是不够的。它还使用图形作为模型来探索已知实验答案但缺乏数学证明的问题，并且探索甚至实验答案都未知的问题。也就是说，PI致力于一个非常有前途的想法，即为图上的节点计数建立一个“跟踪公式”。如果成功，这将是一个突破性的发展，第一个将量子态结构（微观结构）与图上经典闭合轨迹结构（宏观结构）联系在一起的公式。到目前为止，类似的公式只适用于系统的量子能量，任何关于系统对应量子态的精细结构的信息都是向前迈出的一大步。PI还利用他在广泛的研究领域（数学物理、图形学和组合学）的经验，寻找与量子输运相关的量的半经典评估背后的统一代数结构。代数结构可以帮助形式化未来问题和不同物理量的计算；现在，每一种情况下的计算都是从第一原理开始的。PI使用图形来对与其他更复杂的系统（如量子空腔）相关的问题进行更深入的数学理解。因此，该项目研究的问题与介观物理和工程直接相关。对这些问题的回答是解开量子效应的关键，而量子效应发生在当今芯片制造商所能达到的尺度上。该项目还考虑了组合学和计算分子生物学的应用。量子图也为新研究者和学生提供了一个完美的教育媒介。PI正在编写关于量子图的介绍性文本，并在项目中使用更容易的任务作为研究生和本科生的研究项目。

The project focuses on several inter-related questions of spectral analysis of graphs and the use of graphs in spectral analysis of more complicated system. Most questions draw inspiration from the use of quantum graphs as models for quantum chaology, a branch of mathematical physics concerned with the properties of quantum systems that in the classical limit exhibit chaotic dynamics. The project addresses several open problems on quantum graphs and related systems that can be tackled within a 3-year time frame. Using graphs as models has already resulted in some notable successes, in particular in studying the universality of spectral statistics and in studying the nodal statistics of graph eigenfunctions. There remain, however, many important unanswered questions and some extremely promising direction that are addressed by the project. The first is a foundation-type question of whether the spectrum of a generic graph is simple. Further, recent results on the number of nodal domains of graph eigenfunctions hint at the existence of a trace formula, connecting this number with the properties of the periodic orbits of the graph. The PI studies the dynamics of zeros on an open graph (a graph with infinite leads attached) when the spectral parameter is varied. The aim is to quantify the events in which the number of zeros changes. The PI also studies the correlations within the set of long periodic orbits (longer than the graph size) using a special type of graphs as a model. Finally the PI is looking at ways to extend the diagrammatic summation schemes to new applied questions and also searches for algebraic structures unifying several existing diagrammatic schemes. Possible direct applications to questions in combinatorics and computational molecular biology are considered.The project addresses several questions of spectral properties of graphs where our current understanding is insufficient. It also uses graphs as a model to explore questions for which the experimental answer is known but mathematical proofs are lacking and, furthermore, to explore questions for which even experimental answers are yet unknown. Namely, the PI works on a highly promising idea of establishing a "trace formula" for the nodal count on graphs. If successful, this would be a break-through development, a first formula to tie together the structure of the quantum state (microscopic structure) with the structure of the classical closed trajectories on the graph (macroscopic structure). So far, similar formulae were only available for the quantum energies of a system, and any information about the fine structure of the corresponding quantum state of the system is a great step forward. The PI also builds on his experience in a wide variety of research areas (mathematical physics, graphs and combinatorics) to search for the unifying algebraic structures behind semiclassical evaluation of the quantities related to the quantum transport. Algebraic structures can help to formalize calculations in future questions and for different physical quantities; right now the calculations are performed in each case starting from the first principles. The PI uses graphs to achieve deeper mathematical understanding in questions that are pertinent to other, more complicated systems, like quantum cavities. Thus the questions studied in the project have direct relevance to mesoscopic physics and engineering. Answering them holds the key to the quantum effects which happen on the scales that are within reach of today's chip manufacturers. Applications to combinatorics and computational molecular biology are also considered in the project. Quantum graphs also serve as a perfect educational medium for new researchers and students. The PI is writing an introductory text on quantum graphs and uses easier tasks within the project as graduate and undergraduate research projects.