Nonarchimedean Analysis, Geometry, and Computation

非阿基米德分析、几何和计算

基本信息

批准号：
1802161
负责人：
Kiran Kedlaya
金额：
$ 33万
依托单位：
University of California-San Diego
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1802161&HistoricalAwards=false
关键词：
Nonarchimedean Analysis Geometry Computation

项目摘要

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.

该项目包括非阿基米德（p-adic）分析和几何在算术代数几何问题中的大量应用，包括理论和计算性质。获奖研究的分支包括 p 进微分方程理论中的新技术，该技术之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种能力的培训，从而促进美国知识基础的增强；为纽约市和洛杉矶的低收入学生提供更多参加丰富活动的机会；为寻求数学教育职业的美国本科生提供新的培训机会；开发数学研究的开源软件；并致力于交互式开源课程材料的研究，包括介绍数学软件的新课程。毫不夸张地说，自 2010 年以来的完美空间理论引发了算术几何的一场革命，其进步速度之快前所未有。然而，为了维持这种进展速度，该学科基础的深入改进至关重要。还需要进一步的工作来充分认识完美类空间的潜力，以加深我们对朗兰兹对应所表明的几何和表示论对象之间关系的理解；特别是，这将需要深刻的洞察力，以便将迄今为止的 p-adic Hodge 理论的局部构建全球化。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 L 函数的研究产生变革性影响，为经验观察开辟广阔的新领域。最终效果是使数论回归其根源，作为一个经验驱动的学科，从而产生基于实验预测的新一代定理（呼应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。