RTG: Electronic Computational Homotopy Theory Research Community

传统上，学术研究社区是围绕大学内部的院系建立的，要求参与者离得很近。然而，现代电信技术使人们能够以新的方式进行富有成效的互动，在线社区在21世纪的研究中将发挥越来越大的作用。该项目将发展在线关系和人类基础设施，目标是创建一个自给自足的纯数学在线研究社区。这包括研究生培训、促进协作和建立社区。在线社区可以接触到新的受众，并利用传统社区无法获得的效率。这个项目将：*向利益相关者证明在线研究社区是进行数学研究的可行和有效的手段；*试验不同类型的在线项目，以确定哪些在线项目在吸引参与者并对他们的长期职业生涯产生积极影响方面最有效；以及*确定在线社区在接触某些受众方面是否比传统部门更有效，特别是对于美国公民、国民和永久居民以及历史上在数学科学中代表性不足的群体。电子计算同伦理论(ECHT)研究社区将举办在线课程、阅读研讨会和研究研讨会，以及本科生研究项目。每年，该项目将资助几名研究生和博士后助理。在来自不同机构的高级人员的监督下，这些学生和博士后伙伴将管理ECHT活动。学生和博士后伙伴将在协作的在线环境中由高级人员提供建议和指导。该项目的科学主题是计算同伦理论。现代同伦理论的一个方面是研究影响几何对象结构的代数不变量。这一理论的一部分涉及对这些不变量的有效计算，无论是对于一般类还是在非常特定的情况下。一个中心的例子是球面稳定同伦群的计算。ECHT研究界将专注于同伦理论中的这些类型的计算问题，并将支持机器计算，这是同伦理论中日益重要的工具。该计划的参与者将从一个特别大的群体中选出。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Traditionally, academic research communities are built around departments within universities, requiring participants to be in physical proximity. However, modern telecommunications technology allows people to interact productively in new ways, and online communities will play a growing role in research in the 21st century. This project will develop online relationships and human infrastructure, with the goal of creating a self-sustaining online research community in pure mathematics. This includes graduate student training, fostering collaboration, and building community. Online communities can reach new audiences and take advantage of efficiencies that are not accessible to traditional communities. This project will:* demonstrate to stakeholders that online research communities are a viable and effective means for conducting mathematical research;* experiment with different types of online programs to determine which are the most effective at engaging participants and positively impacting their long-term careers; and* determine whether online communities are more effective than traditional departments at reaching certain audiences, especially with regard to U.S. citizens, nationals, and permanent residents and with regard to groups that are historically underrepresented in the mathematical sciences.The electronic Computational Homotopy Theory (eCHT) research community will conduct online courses, reading seminars, and research seminars, as well as an undergraduate research program. Each year, several graduate students and postdoctoral associates will be funded by the project. Under the supervision of senior personnel from different institutions, these students and postdoctoral associates will manage the eCHT activities. Students and postdoctoral associates will be advised and mentored by the senior personnel in a collaborative online environment.The scientific theme of the project is computational homotopy theory. One aspect of modern homotopy theory is the study of algebraic invariants that inform on the structure of geometric objects. Part of this theory involves the effective computation of these invariants, both for general classes and in very specific cases. One central example is the computation of the stable homotopy groups of spheres. The eCHT research community will focus on these types of computational questions in homotopy theory and will support machine computation, a tool of growing importance in homotopy theory. The participants in the program will be drawn from an exceptionally large pool.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.