NSF-BSF Combinatorial Set Theory and PCF

该项目支持美国服务不足地区(阿巴拉契亚)的高级数学研究和培训，方法是利用NSF-BSF计划的资金，在PI和以色列本古里安大学的一名同事之间建立国际合作伙伴关系。我们将开发一个横向协作结构，将这两个机构连接起来，使我们能够将我们的专业知识结合起来。使用我们在大流行期间开创的教学模式，我们将让两国的学生参与以高级人员的数学研究为中心的合作研究。该项目的研究部分以PCF(可能余终性)理论为中心，这是谢拉开发的一系列工作，用于分析集合论中的基本算法。谢拉通过使用这些技术在集合论和其他领域取得了令人惊叹的结果，但即使在三十年后，PCF理论在奇异不动点上的全部含义仍未被完全理解。我们的项目通过继续PI和他的以色列同行最近的工作来解决这一差距，重点是绘制出PCF理论在基数算术和无限Ramsey理论中施加限制的程度。这个奖项反映了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。

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.