LEAPS-MPS: Applications of Algebraic and Topological Methods in Graph Theory Throughout the Sciences

该奖项全部或部分根据2021年美国救援计划法案（公法117-2）资助。图形用于表示整个科学领域的复杂网络;例如，生物学，化学，计算机科学和工程学都使用图形网络来模拟现实生活中的现象。图的使用允许以简洁的方式对网络中大量实体之间的关系进行建模，并且这些关系可以通过图的结构属性来理解。图的这种结构性质往往是定量的，本项目的研究目标是研究大量的定量图度量。研究结果将应用于数学科学之外的复杂网络，以改进网络模型。该项目的教育目标是通过创造机会，增加数学专业的学术整合和社会整合，以增加本科数学专业的持久性。来自代表性不足群体的数学专业学生将被鼓励参与，目标是增加这些人的数量，他们可以成为未来科学劳动力的榜样。这个项目的一个主要目标是使用各种代数和拓扑方法研究基于距离的图不变量，以改进图描述的模型。一些当前的观点不考虑图的对称性，这是已知的影响某些应用程序;在这个项目中的第一个重点领域是进一步开发图不变量，以考虑这种对称性的图。PI将创建表示这些不变量的q-类似物的生成函数。该项目的第二个重点领域是使用嵌入技术来获得图不变量的新属性和边界。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Graphs are used to represent complex networks throughout the sciences; for example, biology, chemistry, computer science, and engineering all use graphical networks to model real-life phenomena. The use of graphs allows for a concise way to model the relationships among a large number of entities in a network, and these relationships can be understood through the structural properties of graphs. Such structural properties of graphs are often quantitative, and the research goal of this project is to study numerous quantitative graph measures. The research results will be applied to complex networks, which arise outside of the mathematical sciences, in order to improve network models. The educational goal of this project is to increase the persistence in the undergraduate mathematics major by creating opportunities that increase both the academic integration and social integration of mathematics majors. Mathematics majors from underrepresented groups will be encouraged to participate with the goal of increasing the number of such individuals who could serve as role models for the scientific workforce of the future. A major goal of this project is to study distance-based graph invariants using various algebraic and topological methods in order to improve models which the graph describes. Some current viewpoints do not account for the symmetries of a graph, which are known to affect certain applications; the first area of focus in this project is to further develop graph invariants to account for such symmetries of graphs. The PI will create generating functions that represent q-analogs of these invariants. The second area of focus in this project is to use embedding techniques to obtain new properties and bounds on graph invariants.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.