SM: Graph Models in Nanotechnology, Photonics, Chemistry, and Other Areas

KuchentDMS-0648786这一系列的教程和三个研讨会是英国剑桥艾萨克·牛顿数学科学研究所一个为期一个月的项目的一部分。该计划汇集了一大批对图形模型感兴趣的国际数学家和物理学家，这些图形模型出现在具有高度实践和理论重要性的广泛不同的领域。在过去的几十年里，由于数论、代数、概率论和谱几何的需要，对图形的分析得到了迅速的发展。这一领域最近经历了大量新的重要应用、新方法和新模式.出现了新的对象，如量子图(量子网络)或分数上的微分算子，它们提供了复杂系统的简化模型，并有助于研究困难的问题，如Anderson局部化和量子混沌。所有这些发展都激发了其他科学和数学领域的专家对分析图表和所用方法的多样性的兴趣。该计划从一个辅导工作坊开始，在那里年轻的研究人员和学生被介绍关于图形、量子图形和分形图的基本分析。随后的两个研讨会致力于在纳米技术、化学、光学、微电子和其他领域的量子图形和图形模型上进行频谱分析和量子混沌。第三堂课讨论图形和分形图分析的代数方面。除了参加研讨会外，还安排了知名研究人员和年轻研究人员的长期访问。近年来，由于许多新的应用席卷了广泛的领域：计算机图形学、信息技术、互联网研究、纳米技术、微电子学、超导、光学、化学和材料科学，图形分析经历了急剧的增长。这些领域的研究人员遇到的问题与数学家在数论、组合学、微分方程式和谱论等领域所面临的问题类似。该计划促进了数学和科学领域不同领域的研究人员之间迫切需要的交流与合作。因此，它加速了这一新领域的进展，这一领域在纳米技术、光子学和材料科学等关键科学技术领域得到了广泛应用。它还有助于提高美国研究人员在这些领域的参与度和知名度。为初级研究人员和研究生提供了重要的培训，包括专门设计的教程、研讨会和开放问题会议。计划出版一本论文集。该赠款支持该项目的美国参与者，主要是美国学生、博士后、初级教师、女性和其他代表性不足的群体。

KuchmentDMS-0648786 The series of a tutorial and three workshops is a part of asemester-long program at the Isaac Newton Institute forMathematical Sciences in Cambridge, UK. The program assembles alarge international group of mathematicians and physicistsinterested in graph models arising in widely disparate areas ofhigh practical and theoretical importance. Analysis on graphshas been developing very quickly in the last decades, due toneeds of number theory, algebra, probability theory, and spectralgeometry. This area has experienced recently a surge of newimportant applications, new methods, and new models. Newobjects, such as quantum graphs (quantum networks) ordifferential operators on fractals, have emerged, which providesimplified models of complex systems and help with studyingdifficult issues such as, for instance, Anderson localization andquantum chaos. All these developments have sparked the interestof experts in other areas of sciences and mathematics in analysison graphs and in the diversity of methods employed. The programstarts with a tutorial workshop where young researchers andstudents are introduced to the basics of analysis on graphs,quantum graphs, and fractals. Then two workshops follow devotedto spectral analysis and quantum chaos on quantum graphs andgraph models for nanotechnology, chemistry, optics,microelectronics, and other areas. The third workshop addressesalgebraic aspects of analysis on graphs and fractals. Besidesworkshop participation, longer term visits of both establishedand young researchers are arranged. Analysis on graphs has experienced sharp growth in recentyears, due to many new applications that sweep throughout a widescientific landscape: computer graphics, information technology,Internet studies, nanotechnology, microelectronics,superconductivity, optics, chemistry, and material science. Researchers in these areas meet problems that are similar tothose faced by mathematicians in areas like number theory,combinatorics, differential equations, and spectral theory. Theprogram facilitates the urgently needed communication andcooperation among researchers in different areas of mathematicsand sciences. It thus accelerates progress in this new area thathas extensive applications in crucial scientific and technologyfields, such as nanotechnology, photonics, and material science. It alsos help to increase participation and visibility of USresearchers in these fields. Significant training for juniorresearchers and graduate students is provided, includingspecially designed tutorials, seminars, and open problemssessions. A proceedings volume is planned to be published. Thegrant supports US-based participants in the program, mostly USstudents, postdocs, junior faculty, women, and otherunderrepresented groups.