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

LEAPS-MPS：代数和拓扑方法在图论中在整个科学领域的应用

• 批准号: 2313262
• 负责人: Lindsey-Kay Lauderdale
• 金额: $ 21.95万
• 依托单位: Southern Illinois University at Carbondale
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-03-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2313262&HistoricalAwards=false
• 关键词: LEAPS; MPS; Applications; Algebraic; Topological

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.

该奖项的全部或部分资金根据《2021 年美国救援计划法案》（公法 117-2）提供。图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，并且可以通过图的结构属性来理解这些关系。图的这种结构特性通常是定量的，该项目的研究目标是研究大量的定量图度量。研究成果将应用于数学科学之外出现的复杂网络，以改进网络模型。该项目的教育目标是通过创造增加数学专业的学术融合和社会融合的机会来提高本科数学专业的毅力。将鼓励来自代表性不足群体的数学专业学生参与，目标是增加可以成为未来科学劳动力榜样的此类人员的数量。该项目的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。当前的一些观点没有考虑图的对称性，已知这会影响某些应用；该项目的第一个重点领域是进一步开发图不变量以解释图的这种对称性。 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界限。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。