Nonarchimedean Analysis, Geometry, and Computation

这个项目包括非阿基米德分析和几何在算术代数几何问题上的一些应用，既有理论性质的，也有计算性质的。获奖研究的分支包括p-进微分方程式理论中的新技术，该理论以前已在计算机科学中得到应用；例如，PI的一种计算Zeta函数的算法在密码学文献中被广泛引用。拟议的活动包括对研究生进行多方面的培训，以促进增强美国的知识基础；为纽约市和洛杉矶的低收入学生提供更多获得丰富知识的机会；为寻求数学教育职业的美国本科生提供新的培训机会；开发数学研究的开源软件；以及编写互动式开源课程材料，包括介绍一门新的数学软件课程。可以毫不夸张地说，自2010年以来的完美拟态空间理论引发了算术几何的革命，取得了前所未有的快速进步；然而，为了保持这一进步速度，对该学科基础的深刻改进是至关重要的。还需要进一步的工作来充分认识完美拟态空间的潜力，以加深我们对朗兰兹对应所指示的几何对象和表示论对象之间的关系的理解；特别是，这将需要更深刻的洞察力，以便使p-进Hodge理论的迄今为止的局部构造全球化。另外，由p元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的影响，为经验观察开辟了广阔的新领域。最终的结果是使数论回归其作为经验驱动的学科的根源，从而导致基于实验预测的新一代定理(呼应了诸如二次互易和素数定理等结果的历史发展)。该奖项反映了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 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.