International Research Fellowship Program: Permutation Groups and Model Theory

国际研究奖学金计划：排列群和模型理论

• 批准号: 0853293
• 负责人: Paul Baginski
• 金额: $ 13.61万
• 依托单位: Baginski Paul
• 依托单位国家: 美国
• 项目类别: Fellowship Award
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0853293&HistoricalAwards=false
• 关键词: International; Research; Fellowship; Program; Permutation

0853293BaginskiThis award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The International Research Fellowship Program enables U.S. scientists and engineers to conduct nine to twenty-four months of research abroad. The program's awards provide opportunities for joint research, and the use of unique or complementary facilities, expertise and experimental conditions abroad.This award will support a twenty-four-month research fellowship by Dr. Paul Baginski to work with Dr. Tuna Altinel at the University of Lyon in France.Permutation groups provide the mathematical abstraction of the symmetries of a specified object. While finite permutation groups have long been examined, the current project uses model-theoretic methods to extend this mathematical theory to several classes of infinite permutation groups with finitary properties. In particular, the project focuses on the strong model-theoretic properties of finite Morley rank (fMR) and countable categoricity. In the case of finite groups, many insights about permutation groups occurred toward the end of the classification program for finite simple groups. In the infinite case, model theorists have pursued an analogous classification program for simple groups of finite Morley rank for nearly thirty years. The current status of the classification prompted two prominent researchers, Alexandre Borovik and Gregory Cherlin, to signal in 2007 that significant progress on infinite permutation groups of finite Morley rank is imminent. The proposed project has begun by addressing one of Borovik and Cherlin?s questions concerning the fundamental model theoretic interactions between the permutation group and the set upon which it acts, equipped with a relational ?footprint? of the group action. The investigation proceeds in the fMR case toward analysis of primitive permutation groups, generic n-transitivity, and other related problems. In parallel, the project considers these same problems, but with countable categoricity in place of finite Morley rank. Whereas groups of fMR generally resemble algebraic groups over algebraically closed fields, countably categorical groups tend toward infinite-dimensional vector spaces over finite fields. Taken together, groups with the properties of finite Morley rank or countable categoricity encompass many familiar examples of infinite permutation groups which appear across mathematics. Advances in the fMR or countably categorical settings can be expected to have broad applicability within model theory and other fields such as number theory, algebraic graph theory and abstract geometry. The project features international cooperation, with involvement from researchers from the USA, France, the United Kingdom, Germany, and elsewhere, and the results would have rapid and wide circulation. Furthermore, implications of this research could easily reach beyond mathematics, since permutation groups are used in the applied study of symmetries of many objects, from crystals to the arithmetic curves that underlie much of modern cryptography. Due to their size, some of these objects are effectively ?infinite?. For example, the world-wide-web, when considered as a set of points (webpages) connected by lines (hyperlinks), forms a nearly impossible system for full-scale analysis. While computationally infinite, such objects are naturally finite. In this case, it may be fruitful to analyze their symmetries using infinite permutation groups with finitary properties, such as the ones featured in this research.

0853293 Baginski该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。国际研究奖学金计划使美国科学家和工程师能够在国外进行9到24个月的研究。 该计划的奖项提供了合作研究的机会，以及使用独特或互补的设施，专业知识和国外的实验条件。该奖项将支持保罗·巴金斯基博士与法国里昂大学的Tuna Altinel博士合作进行为期24个月的研究奖学金。排列群提供了特定对象对称性的数学抽象。虽然有限置换群已经被研究了很长时间，但目前的项目使用模型论方法将这一数学理论扩展到几类具有有限性质的无限置换群。特别是，该项目侧重于有限莫利秩（fMR）和可数范畴的强模型理论属性。在有限群的情况下，许多关于置换群的见解出现在有限单群分类程序的末尾。在无限情况下，模型理论家们对有限莫利秩的单群进行了类似的分类，已经有将近30年了。分类的现状促使两位著名的研究者亚历山大·博罗维克（Alexandre Borovik）和格雷戈里·切尔林（Gregory Cherlin）在2007年发出信号，表明有限莫利秩的无限置换群即将取得重大进展。拟议的项目已经开始解决博罗维克和切尔林之一？的问题有关的基本模型理论之间的相互作用的置换群和设置后，它的行为，配备了一个关系？脚印？的集体行动。调查进行fMR情况下对分析的原始置换群，一般的n-传递性，和其他相关问题。平行地，该项目考虑了这些相同的问题，但用可数范畴代替有限的莫利秩。fMR群一般类似于代数闭域上的代数群，而可数范畴群则倾向于有限域上的无限维向量空间。总的来说，具有有限莫利秩或可数范畴性质的群包含了许多常见的出现在数学中的无限置换群的例子。在fMR或可数范畴设置的进展可以预期在模型论和其他领域，如数论，代数图论和抽象几何中具有广泛的适用性。该项目以国际合作为特色，有来自美国、法国、英国、德国和其他地方的研究人员参与，其结果将迅速广泛传播。此外，这项研究的意义可以很容易地超越数学，因为置换群被用于许多对象的对称性的应用研究，从晶体到现代密码学的基础算术曲线。由于它们的大小，其中一些物体是有效的？无限？例如，万维网，当被认为是一组由线（超链接）连接的点（网页）时，形成了一个几乎不可能进行全面分析的系统。虽然计算上是无限的，但这些对象自然是有限的。在这种情况下，它可能是富有成效的分析他们的对称性使用有限性质的无限置换群，如在本研究中的特色。