This Pan-American Advanced Studies Institutes (PASI) award, jointly supported by the NSF and the Department of Energy (DOE), will take place May 14-25, 2012 at the Centro de InvestigaciÃ³n en MatemÃ¡ticas (CIMAT) in Guanajuato, Mexico. Organized by Dr. Srikanth Iyengar of the Department of Mathematics at the University of Nebraska and Dr. Sarah J. Witherspoon of the Department of Mathematics at Texas A & M University, in collaboration with Dr. Ragnar-Olaf Buchweitz (Canada) and Dr. JosÃ© Antonia de la PeÃ±a (Mexico), the PASI will address issues in commutative algebra and its interactions with algebraic geometry, representation theory, and theoretical physics. The specific topics to be covered are: matrix factorizations and maximal Cohen-Macaulay modules; cluster algebras and cluster categories; cohomological theory of support varieties; and topological quantum field theories and Frobenius algebras.
This PASI will familiarize graduate students, postdocs, and early career faculty with emerging applications of commutative algebra. It will also provide an opportunity for building and fostering collaborations among researchers from various countries in the Americas. The organizers will encourage participants to organize a special session at an American Mathematical Society (AMS) meeting during the three-year period following the institute. Furthermore, they will continue to mentor those participants and those who become involved in the program at the Mathematical Sciences Research Institute-Berkeley, California (MSRI) in commutative algebra during the year 2012-2013. A wiki page will serve as a platform for dissemination of notes from the lecturers, and for discussions on problems and questions related to the lectures.
The PIs in collaboration with Ragnar-Olaf Buchweitz (Canad) and Jose Antonio de la Pena (Mexico) organized a Pan-American Advanced Study Institute titled Commutative Algebra and Its Interactions with Algebraic Geometry, Representation Theory, and Physics, at the Centro de Investigacion en Matematicas in Guanajuato, Mexico, May 14-25, 2012. The Institute was targeted at advanced graduate students, postdocs, and early career faculty in areas related to the topics of the Institute, with the goal of familiarizing them with emerging applications of these areas and interactions among them. The Institute featured five lecture series, each comprising four or five lectures, given by Luchezar Avramov, Osamu Iyama, Graham Leuschke, Ernesto Lupercio, and Daniel Murfet. There were three survey lecture series of two lectures each, a variety of research talks building on the lecture series, and shorter talks by some of the participants on their research. All lectures were at the advanced graduate level, and were supplemented by problem sessions and discussions. Intellectual Merit: The topics covered in the Institute are at the intersection of several important and rapidly developing areas of mathematics, each subject benefitting from advances in the others. At the heart of it all is commutative algebra that provides a guide book and some of the critical tools used in the seemingly diverse topics covered at the Institute. Broader Impacts: The Institute provided a venue for initiating collaborations among experts and mathematically younger researchers in commutative algebra, algebraic geometry, representation theory, and theoretical physics. It also provided a rare oppportunity for building and fostering collaborations among researchers from various countries in the Americas. Specifically, it helped recent PhDs, graduate students, and junior faculty to interact with experts in commutative algebra, algebraic geometry, and representation theory, to learn about current research in these fields and interactions between fields, to discuss ideas with one another, and to begin collaborations on future research. Continuing impacts of the Institute are not only the collaborations resulting from it, but also further opportunities for interacting and mentoring: Some of the organizers and speakers have continued to mentor many of the participants of the Institute, for example those involved in the program at the Mathematical Sciences Research Institute in commutative algebra during the year 2012-13. As well, two of the participants are organizing a follow-up session at an American Mathematical Society meeting in spring 2014.