NSF-BSF Combinatorial Set Theory and PCF

This project supports advanced mathematical research and training in an underserved area of the United States (Appalachia) by leveraging funding from the NSF-BSF program to build an international partnership between the PI and a colleague at Ben-Gurion University in Israel. We will develop a horizontal collaborative structure bridging the two institutions to allow us to combine our expertise. Using a pedagogical model we pioneered during the pandemic, we will involve students in both countries in collaborative research centered on the mathematical research of the senior personnel in the grant. The research component of the project is centered on PCF(Possible CoFinality) theory, a body of work developed by Shelah to analyze cardinal arithmetic in set theory. Shelah obtained fantastic results in set theory and other fields by using these techniques, but even after three decades, the full implications of PCF theory at singular fixed points are not fully understood. Our project addresses this gap by continuing the recent work of the PI and his Israeli counterpart, with a focus on mapping out the extent to which PCF theory imposes constraints in cardinal arithmetic and infinitary Ramsey theory. 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.