RTG: Electronic Computational Homotopy Theory Research Community

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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.