New and Classical Ideas in Zero-Sum and Additive Theory: International and Collaborative Postdoctoral Research

零和理论和加法理论的新经典思想：国际合作博士后研究

• 批准号: 0502193
• 负责人: David Grynkiewicz
• 金额: --
• 依托单位: Grynkiewicz, David J
• 依托单位国家: 美国
• 项目类别: Fellowship Award
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-09-01 至 2007-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0502193&HistoricalAwards=false
• 关键词: New; Classical; Ideas; Zero; Sum

Combinatorial Number Theory, particularly in the areas of zero-sums and inverse problems, is a rapidly developing area of mathematics whose progress has advanced noticeably due to the successful introduction of many newly emerging methods with varied and disparate origins (including the polynomial method, linear algebraic techniques, Galois Theory, the use of combinatorial covers, the isoperimetric method, minimal zero-sums, exponential sums and other analytic techniques, and the partitioning of sequences into sets). While these and other methods have succeeded recently in solving many important open problems and conjectures from the field (such as the Kemnitz Conjecture), there remain considerable portions of the foundation of additive theory still unsolved. Only under very restricted circumstances do we fully understand the structure of a pair of sets (with elements from an arbitrary abelian group) with a small cardinality sumset or restricted sumset (for instance, for prime order groups there is the now established Erdos-Heilbronn conjecture, but much less of this nature is known for composite order groups). A similar state exists concerning the structure of sequences (with terms from an abelian group) that do not represent zero, represent zero minimally, or only represent a small number of elements, as a sum of some subsequence (with possibly fixed length). This project provides an MPS Distinguished International Postdoctoral Research Fellowship (MPS-DRF) to David J. Grynkiewicz in order to address these types of questions and to disperse and interweave the emerging methods in zero-sum and inverse additive theory in an effort to help create yet stronger techniques.The collaborative research will be conducted principally at the Polytechnical University of Catalonia in Spain along with Oriol Serra, but will also entail shorter collaborative visits to work with Luis Gallardo, Georges Grekos, and Francois Hennecart of France, Weidong Gao of China, Alfred Geroldinger of Austria, and R. Thangadurai of India. A zero-sum workshop organized by G. Grekos, involving more researchers and students, will also occur during the corresponding collaborative visit in France. The importance of such research lies not just in its applications (results from zero-sum and additive theory have ranged from better understanding of non-unique factorizations in Krull Domains, to the structure of pairs of sets whose sum has small Haar Measure, to increased information about the range of diagonal and quadratic forms, and even to more complete knowledge about the range of parameters for partial difference sets), nor just in its intrinsic mathematical value, but also in the fact that the methods and techniques employed and developed to answer questions from zero-sum and additive theory, like many other areas of science outside of mathematics, lie across such a broad range of mathematics. Additionally, due to the international setting for the research, the project will foster lasting ties between US based researchers and their international counterparts.

组合数论，特别是零和反问题，是数学中一个迅速发展的领域，由于成功地引入了许多不同起源的新兴方法，其进展显著（包括多项式方法，线性代数技术，伽罗瓦理论，组合覆盖的使用，等周方法，最小零和，指数和和其他分析技术，以及将序列划分为集合）。虽然这些方法和其他方法最近成功地解决了许多重要的开放问题和来自该领域的问题（如Kemnitz猜想），但添加剂理论的基础仍有相当大的部分未解决。只有在非常有限的情况下，我们才能完全理解具有小基数和集或限制和集的一对集合（元素来自任意阿贝尔群）的结构（例如，对于素数阶群，现在已经建立了Erdos-Heilbronn猜想，但对于复合阶群，这种性质就少得多了）。关于不表示零、最低限度地表示零或仅表示少量元素的序列（具有来自阿贝尔群的项）的结构，也存在类似的状态，作为某个子序列（可能具有固定长度）的和。该项目为大卫·J·格林凯维奇提供了MPS杰出国际博士后研究奖学金（MPS-DRF），以解决这些类型的问题，并将零和逆加法理论中的新兴方法分散和交织，以帮助创建更强大的技术。合作研究将主要在西班牙加泰罗尼亚理工大学进行，沿着Oriol Serra，但也需要与法国的路易斯·加利亚多、乔治·格雷科斯和弗朗索瓦·亨内卡特、中国的高卫东、奥地利的阿尔弗雷德·格罗丁格和R.印度的唐加杜赖。G.组织的零和研讨会。Grekos，涉及更多的研究人员和学生，也将在法国相应的合作访问期间发生。这种研究的重要性不仅在于其应用（零和和和加法理论的结果从更好地理解Krull域中的非唯一因子分解，到其和具有小Haar测度的集合对的结构，到增加关于对角形式和二次形式的范围的信息，甚至到关于偏差集的参数范围的更完整的知识），也不仅仅是在其内在的数学价值，而且在这样一个事实，即方法和技术的使用和发展，以回答问题，从零和和加法理论，像许多其他领域的科学以外的数学，谎言，跨越这样一个广泛的数学。此外，由于研究的国际背景，该项目将促进美国研究人员与国际同行之间的持久联系。