RTG: Electronic Computational Homotopy Theory Research Community

传统上，学术研究社区是围绕大学内的院系建立的，要求参与者在物理上接近。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，在线社区将在21世纪的研究中发挥越来越大的作用。该项目将发展在线关系和人际基础设施，目标是创建一个自我维持的纯数学在线研究社区。这包括研究生培训、促进合作和建立社区。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。该计划将：*向持份者证明网上研究社区是进行数学研究的可行和有效的手段；*尝试不同类型的在线课程，以确定哪种课程最有效地吸引参与者，并对他们的长期职业生涯产生积极影响；以及*确定在线社区是否比传统部门更有效地接触某些受众，特别是美国公民、国民和永久居民，以及历史上在数学科学领域代表性不足的群体。电子计算同伦理论（eCHT）研究社区将开展在线课程，阅读研讨会和研究研讨会，以及本科研究计划。每年，该项目将资助几名研究生和博士后。在来自不同机构的高级人员的指导下，这些学生和博士后将管理eCHT的活动。学生和博士后将在一个协作的在线环境中接受高级人员的建议和指导。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对几何物体结构的代数不变量的研究。这个理论的一部分涉及到这些不变量的有效计算，既适用于一般类，也适用于非常特殊的情况。一个中心的例子是球的稳定同伦群的计算。eCHT研究团体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，这是同伦理论中越来越重要的工具。该项目的参与者将从一个特别大的人才库中抽取。该奖项反映了美国国家科学基金会的法定使命，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。

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.