This 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.
