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

组合数论是一个发展迅速的数学领域，特别是在零和与反问题领域，由于许多不同起源的新兴方法的成功引入(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、等周法、最小零和、指数和和其他分析技术，以及序列的集合划分)，它的进步是显著的。虽然这些方法和其他方法最近已经成功地解决了该领域的许多重要的公开问题和猜想(如Kemnitz猜想)，但加法理论的基础仍有相当大一部分仍未解决。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受限和集的集合对(元素来自任意阿贝尔群)的结构(例如，对于素数阶群，存在现已建立的鄂尔多斯-海尔布伦猜想，但对于复合序群，这种性质的已知要少得多)。对于不表示零、最小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，存在类似的状态，作为某个子序列(可能具有固定长度)的和。该项目向David J.Grynkiewicz提供MPS杰出国际博士后研究奖学金(MPS-DRF)，以解决这类问题，并将零和理论和逆加法理论中的新兴方法分散和交织在一起，努力帮助创造更强大的技术。合作研究将主要在西班牙加泰罗尼亚理工大学与奥里奥尔·塞拉一起进行，但也将需要与法国的路易斯·加拉多、乔治·格雷科斯和弗朗索瓦·亨内切特、中国的高卫东、奥地利的阿尔弗雷德·格罗尔丁格和印度的R.Thangadurai进行较短的合作访问。G.Grekos组织的一个零和研讨会，有更多的研究人员和学生参加，也将在相应的合作访问期间在法国举行。这种研究的重要性不仅在于它的应用(从零和理论和加法理论得到的结果，从更好地理解Krull域中的非唯一因式分解，到其和具有小Haar度量的集合对的结构，到关于对角线和二次型的范围的更多信息，甚至是关于部分差集的参数范围的更完整的知识)，也不仅在于其内在的数学价值，而且在于这样一个事实，即用于回答零和加法理论的问题的方法和技术，就像数学之外的许多其他科学领域一样，存在于如此广泛的数学范围内。此外，由于这项研究的国际背景，该项目将促进美国研究人员与国际同行之间的持久联系。

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.