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进展揭示了数论和量子信息论中主要开放问题之间的联系。在数论中，希尔伯特第 12 个问题（1900 年提出的著名的 23 个问题之一）和相关的斯塔克猜想（在 20 世纪 70 年代末提出）关注的是为涉及代数数的某些抽象结构寻找更明确的表达式。在量子信息论中，Zauner 的猜想（1999）预测存在称为 SIC 的高度规则的几何配置，它描述了量子测量。 该研究项目涉及调查希尔伯特/斯塔克问题和佐纳猜想之间的联系，以提供这两个领域的新见解。该项目的结果将允许更快地计算 SIC，这在量子态断层扫描以及雷达的经典压缩感知方面具有潜在的应用。该项目还将支持研究生的参与，允许 PI 及其学生通过会议和研讨会传播他们的工作，支持 PI 所在机构的数论研究研讨会，并支持 PI 向高中生进行的推广工作。在该项目中，PI 将证明复杂解析模余循环的新结果，以改进实二次域的 Stark 猜想的陈述，并且（通过与 Appleby 和 Flammia 的联合工作）在每个维度上推测地构建对称信息完整的正算子值测度（SIC-POVM 或 SIC）。该项目将发展（广义、乘法）解析模余循环理论及其“真实乘法值”，用类似于模形式语言而不是 L 函数的语言重新表述（并证明结果）斯塔克猜想。它将通过推广 SIC 的结构，对实二次场产生显式类场论的几何解释。将探索与 Charrollois、Dasgupta、Greenberg、Hill、Sczech 和 Solomon 研究的 Eisenstein 和 Shintani 余循环以及 Darmon、Pozzi 和 Vonk 研究的 p-adic 刚性亚纯余循环的联系。还将探索与量子场论的联系，并将数学物理文献中的广义贝塔积分关系应用于模余循环及其实乘值。该项目由数学科学部代数与数论和该奖项反映了 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。