International Research Fellowship Program: Permutation Groups and Model Theory

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。国际研究奖学金项目使美国科学家和工程师能够在国外进行9至24个月的研究。该计划的奖励为联合研究提供了机会，并利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Paul Baginski博士与Tuna Altinel博士在法国里昂大学进行为期24个月的研究。置换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理论方法将这一数学理论扩展到具有有限性质的无限置换群的几类。特别地，该项目侧重于有限Morley秩（fMR）和可数范畴的强模型理论性质。在有限群的情况下，关于置换群的许多见解出现在有限单群分类程序的末尾。在无限的情况下，模型理论家已经为有限莫雷秩的简单群追求了一个类似的分类程序近三十年。这种分类的现状促使两位杰出的研究人员Alexandre Borovik和Gregory Cherlin在2007年发出信号，表示有限Morley秩的无限排列群即将取得重大进展。拟议的项目已经开始解决Borovik和Cherlin？关于置换群和它所作用的集合之间的基本模型理论相互作用的问题，配备了关系足迹？在集体行动中。在fMR的情况下，对原始置换群、一般n-传递性和其他相关问题的分析进行了调查。同时，该项目考虑了这些相同的问题，但用可数的范畴代替有限的Morley秩。而fMR群通常类似于代数闭域上的代数群，可数范畴群倾向于有限域上的无限维向量空间。综上所述，具有有限莫雷秩或可数范畴性质的群包含了许多在数学中出现的熟悉的无限置换群的例子。fMR或可数分类设置方面的进展有望在模型理论和其他领域（如数论、代数图论和抽象几何）中具有广泛的适用性。该项目的特点是国际合作，有来自美国、法国、英国、德国和其他国家的研究人员参与，研究成果将迅速广泛传播。此外，这项研究的意义可以很容易地超越数学，因为排列群被用于许多物体对称性的应用研究，从晶体到现代密码学基础的算术曲线。由于它们的大小，其中一些物体实际上是“无限的”。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，但这样的对象自然是有限的。在这种情况下，使用具有有限性质的无限置换群来分析它们的对称性可能会很有成效，例如本研究中的无限置换群。

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.