Modular Cocycles, Explicit Class Field Theory, and Quantum Designs

模块化共循环、显式类场论和量子设计

• 批准号: 2302514
• 负责人: Gene Kopp
• 金额: $ 21.92万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-15 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302514&HistoricalAwards=false
• 关键词: Modular; Cocycles; Explicit; Class; Field

Recent research developments by the PI and others have revealed a connection between major open problems in number theory and quantum information theory. In number theory, Hilbert's 12th Problem (one of a famous list of 23 problems proposed in 1900) and the related Stark conjectures (formulated in the late 1970s) concern finding more explicit expressions for certain abstract structures involving algebraic numbers. In quantum information theory, Zauner's conjecture (1999) predicts the existence of highly regular geometric configurations called SICs, which describe quantum measurements. The research project involves investigating a connection between the Hilbert/Stark problems and Zauner's conjecture to provide new insights in both areas. Results from the project will allow for faster computation of SICs, which have potential applications to quantum state tomography as well as classical compressed sensing for radar. The project will also support a graduate student's involvement, allow the PI and his students to disseminate their work through conferences and seminars, support a research seminar in number theory at the PI's institution, and support outreach work by the PI to high school students.In the project, the PI will prove new results on complex analytic modular cocycles to refine the statement of the Stark conjectures for real quadratic fields and (through joint work with Appleby and Flammia) to conjecturally construct symmetric informationally complete positive operator-valued measures (SIC-POVMs or SICs) in every dimension. The project will develop the theory of (generalized, multiplicative) analytic modular cocycles and their "real multiplication values," reformulating (and proving results towards) the Stark conjectures in a language similar to the language of modular forms rather than L-functions. It will produce a geometric interpretation of explicit class field theory for real quadratic fields through structures generalizing SICs. Connections will be explored to Eisenstein and Shintani cocycles as studied by Charollois, Dasgupta, Greenberg, Hill, Sczech, and Solomon and to p-adic rigid meromorphic cocycles as studied by Darmon, Pozzi, and Vonk. Connections to quantum field theory will also be explored, and generalized beta integral relations from the mathematical physics literature will be applied to modular cocycles and their real multiplication values.This project is jointly funded by the Algebra and Number Theory in the Division of Mathematical Sciences and the Established Program to Stimulate Competitive Research (EPSCoR).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.

PI和其他人的最新研究发展揭示了数字理论中的主要开放问题与量子信息理论之间的联系。在数字理论中，希尔伯特的第十二个问题（1900年提出的23个著名列表之一）和相关的stark猜想（1970年代后期制定）涉及发现对某些涉及代数数字的某些抽象结构的更明确表达。在量子信息理论中，Zauner的猜想（1999）预测了称为SICS的高度规则几何构型的存在，它描述了量子测量。 该研究项目涉及调查希尔伯特/史塔克问题与Zauner的猜想之间的联系，以在这两个领域提供新的见解。该项目的结果将允许对SICS进行更快的计算，这些计算具有对量子状态断层扫描以及对雷达的经典压缩感的潜在应用。 The project will also support a graduate student's involvement, allow the PI and his students to disseminate their work through conferences and seminars, support a research seminar in number theory at the PI's institution, and support outreach work by the PI to high school students.In the project, the PI will prove new results on complex analytic modular cocycles to refine the statement of the Stark conjectures for real quadratic fields and (through joint work with Appleby和Flammia）在每个维度上都可以构建对称信息的完全正面操作员价值措施（SIC-POVMS或SICS）。该项目将发展（广义，乘法的）分析模块化共生及其“实际乘法价值”的理论，以类似于模块化形式的语言而不是l功能的语言的语言进行重新划分（并证明结果）。它将通过概括SICS的结构来对实际二次场的显式阶级字段理论产生几何解释。通过Charollois，Dasgupta，Greenberg，Hill，Sczech和Solomon和Solomon和Solomon的研究，将探索Eisenstein和Shintani Cocycles，以及由Darmon，Pozzi和Vonk研究的P-辅助僵硬的杂物性循环。还将探索与量子场理论的联系，并将来自数学物理学文献的广义beta整体关系应用于模块化的共体及其实际乘法值。该项目由数学科学划分和既定计划的代数和数字理论共同资助，以刺激有竞争力的研究（EPSCOR）。使用基金会的智力优点和更广泛的影响评估标准进行评估。