Special Year in Number Theory and Combinatorics

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Proposal Number: 0412622Proposal Title: Special Year in Number Theory and CombinatoricsReceived on 12/17/03Principal Investigator: F. GarvanCO-PI(s): Miklos BonaNeil WhiteRichard CrewInstitution: University of FloridaABSTRACT The 2004-2005 Special Year at the Department of Mathematics, Universityof Florida is in Number Theory and Combinatorics and features twoparallel programs. The Number Theory part of Special Year 2004-05 will be highlighted by two conferences: (i) An International Conference on Additive Number Theory in November 2004, and (ii) An International Conference on Arithmetic Geometry in February 2005. The highlight of the Combinatorics part of the Special Year Program will be (iii) A Workshop on Automated Deduction in Geometry in September 2004, and (iv) The Third International Conference on Pattern AvoidingPermutations in March 2005. There are plans to publish the refereedproceedings of these conferences. Topics for these conferences have been chosen from areas of our research strengths but involve broader themes. The conferences will be augmented by lectures throughout the year on the latest trends in research by eminent visiting mathematicians, as well as History Lectures which should appeal to students. The Additive Number Theory Conference will involvethree main themes: partitions and q-series; Goldbach, Waring type problems, and sieves; and basis and density questions. The Arithmetic Geometry Conference will be devoted to p-adic methods in arithmetic and algebraic geometry. Automated Deduction in Geometry Workshop will be a forum to exchange ideas and views, to present research results and progress, and to demonstrate software tools on the intersection between geometry and automated deduction. The Conference on Pattern Avoiding Permutations is on the exact and asymptotic enumeration and structural analysis, of permutations with subsequence conditions. This area is connected to the theory of Young tableaux, labeled trees, partially ordered sets, and planar maps, as well as sorting and searching. The Special Year Program in Number Theory and Combinatorics will be of interest even to researchers outside of mathematics and therefore will strengthen ties with other disciplines. Number Theory is the study of the properties of whole numbers and is the oldest branch of mathematics. Topics such as partitions, elliptic curves, and cryptography, covered by our conferences in number theory will be of interest to physicists and those in computer science. Combinatorics is the study of discrete collections of objects, and how they can be arrangedand counted efficiently. The areas covered by the conferences in combinatorics will be of appeal to computer and engineering sciences. Number Theory and Combinatorics are ideally suited to attract the attention of non-experts and students to significant developments in mathematics. Thus the History Lectures in these areas will inform graduate and undergraduate students as to how fundamental ideas and concepts developed, and this in turn will help in attracting talented students to take to more advanced study in mathematics.