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

Grynkiewicz, David J, Pasadena CA

Investigators

Abstract

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.