The property of a variety or ring being F-split (under mild conditions, equivalently F-pure) is an extremely powerful condition. Perhaps most famously, in the 1980s, these techniques were applied in the study of Schubert varieties, and continue to be actively used in the study of algebraic groups. On the other hand, in the 1970s, these techniques were used to prove fundamental results about rings of invariants by reductive groups. These methods anticipated the fundamental ideas behind the tight closure theory. In the 1990s, it was discovered that there is a precise dictionary between some of the notions coming from the minimal model program, and invariants defined by variants of Frobenius splitting and tight closure theory (a correspondence that is still not fully understood). Some of these methods are also related to the study of vector bundles in characteristic p, another active area of research which has had numerous applications.
This conference that will take place at the University of Michigan, Ann Arbor, May 17th--May 22nd, 2010. The organizing committee consists of: M. Blickle (Universitat Duisburg-Essen), M. Brion (Universite de Grenoble), F. Enescu (Georgia State University), S. Kumar (University of North Carolina at Chapel Hill), M. Mustata (University of Michigan), K. Schwede (University of Michigan). The conference will focus on Frobenius splitting and related notions, methods, and applications to the following important areas of mathematics: the representation theory of algebraic groups, commutative algebra, and higher dimensional algebraic geometry. The conference will bring researchers together and stimulate communication between the various groups (communication which previously has been somewhat limited). It is expected that this conference will impact the mathematical community in a number of ways. Firstly, by exposing researchers to new potential applications of their own work and also to different points of view, the meeting will inspire new communication, collaboration and research. The participants of the conference will have different backgrounds, and thus many of the talks will necessarily be focused at a non-expert audience. Therefore, secondly, the talks given will be suitable for young mathematicians, especially graduate students and junior faculty. Finally, we also expect to attract other established researchers interested in learning about these techniques.
. Frobenius splitting is a method used in mathematics by several different groups of researchers, primarily in algebra and geometry. Most prominently, Frobenius splitting is applied to distinct problems in the fields of Commutative Algebra, Representation Theory and Algebraic Geometry. The conference attracted greater interest, from each of these fields, than we originally anticipated with nearly 100 participants (we had expected closer to 75). The primary purpose of this conference was to encourage interaction between these different groups of mathematicians, interaction which had previously been notably lacking. To accomplish this goal, during the first half of the conference we had a series of presentations accessible to a more general mathematical audience, giving broad surveys of how Frobenius splitting is employed in each of the various fields. These survey presentations were given by experts in each field, and they were very effective in bridging the differences in points of view of the people in different groups. In addition, this part of the conference added a significant educational component to our meeting, by offering an accessible introduction to this active area of research to the graduate students and other young researchers. The second half of the conference was devoted to presentations of cutting-edge research, which the audience (both the junior and senior members) were now prepared to appreciate. Overall the conference was a success. Participants from different fields learned about potential applications of their own work and about existing tools in related fields. We expect that the conference will lead to a more active interaction of the mathematical communities that make use of the technique of Frobenius splitting. We also held an open problems session in which the participants were able to propose questions related to their own research and obtain ideas and feedback from the audience. We hope that this will inspire new research directions. Based upon the feedback from students and junior researchers, we also feel that the educational component of the conference has been successful. Finally, there is a public website where additional information on the conference can be found, including detailed notes taken from many of the presentations. http://sites.google.com/site/frobeniussplitting/home
