Nonarchimedean Analysis, Geometry, and Computation

该项目包括非阿基米德（p进）分析和几何对算术代数几何问题的若干应用，包括理论和计算性质。获奖研究的分支包括p进微分方程理论的新技术，该理论先前已在计算机科学中得到应用；例如，PI的一个计算zeta函数的算法在密码学文献中被广泛引用。拟议的活动包括培训研究生具备几种能力，这将促进美国知识基础的增强；增加纽约市和洛杉矶低收入家庭学生参加充实活动的机会；为寻求数学教育职业的美国本科生提供新的培训机会；开源数学研究软件的开发并致力于交互式开源课程材料，包括数学软件新课程的介绍。毫不夸张地说，自2010年以来，完美曲面空间理论引发了算术几何的一场革命，以前所未有的速度迅速发展；然而，为了保持这一进展速度，对这一学科基础的深刻改进至关重要。进一步的工作还需要充分发挥完美曲面空间的潜力，以加深我们对朗兰兹对应所指示的几何和表征论对象之间关系的理解；特别是，这将需要深刻的洞察力，以便使迄今为止p-adic Hodge理论的本地构建全球化。另外，由p进分析驱动的计算进步已经并将继续对算术几何对象及其相关的l函数的研究产生变革性影响，为经验观察开辟了广阔的新领域。其净效应是将数论带回其作为经验驱动学科的根源，从而导致基于实验预测的新一代定理（与二次互易性和素数定理等结果的历史发展相呼应）。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。

This project consists of a number of applications of non-archimedean (p-adic) analysis and geometry to problems in arithmetic algebraic geometry, of both theoretical and computational nature. Offshoots of the awarded research include new techniques in the theory of p-adic differential equations, which has previously found applications to computer science; for instance, one of the PI's algorithms for computing zeta functions is widely cited in the cryptography literature. The proposed activities include training of graduate students in several capacities, which promotes enhancement of the US knowledge base; increased access to enrichment activities for low-income students in New York City and Los Angeles; new training opportunities for US undergraduates seeking careers in mathematics education; development of open-source software for mathematics research; and work on interactive open-source curricular materials, including the introduction on a new course on mathematical software. It is hardly an overstatement to assert that the theory of perfectoid spaces since 2010 has triggered a revolution in arithmetic geometry, with rapid advances coming at a previously unknown pace; however, deep improvements in the foundations of the subject are vital in order to sustain this rate of progress. Further work is also needed to fully realize the potential of perfectoid spaces to deepen our understanding of the relationship between geometric and representation-theoretic objects indicated by the Langlands correspondence; in particular, this will require deep insights in order to globalize the hitherto local constructions of p-adic Hodge theory. Separately, computational advances driven by p-adic analysis have had, and will continue to have, a transformative effect on the study of arithmetic-geometric objects and their associated L-functions, by opening up vast new territories for empirical observation. The net effect is to bring number theory back to its roots as an empirically driven subject, thus leading to a new generation of theorems based on experimental predictions (echoing the historical development of such results as quadratic reciprocity and the prime number theorem).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.