This proposal is for support of the Program on Motivic Invariants and Singularities at the Center for Mathematics at the University of Notre Dame, which consists of an Undergraduate Summer School (May 21-25, 2013), a Graduate and Post-doc Summer School (May 27-31, 2013), a conference (June 3 -7, 2013), as well as a Distinguished Lecture Series throughout the program. The Undergraduate Summer School will consist of three mini course on p-adic numbers and p-adic integration, tropical geometry, and statistical learning theory and singularities. The advanced Summer School will consist of five mini-courses on D-modules and vanishing cycles, Donaldson-Thomas invariants and the motivic Milnor fiber, the Monodromy Conjecture, motivic integration, and the Nash Conjecture. The conference will bring together top researchers in areas related to motivic integration and singularity theory. In addition, the Distinguished Lecture Series, to be delivered by Jan Denef, will disseminate to a wide audience an exciting topic related to the program. The website of the conference is: http://nd.edu/~cmnd/programs/mis2013/
Since its creation by M. Kontsevich in 1995, motivic integration has been a rapidly developing subject connected to Algebraic Geometry, Singularity Theory, Number Theory, and Model Theory. The theory found a lot of applications to a diverse set of topics such as the McKay correspondence, singularities in the Minimal Model Program, and the study of orbital integrals that appear in the Langlands program. While there has been a lot of work and notable progress in this direction, some fundamental problems are still open. Such a problem is the Monodromy Conjecture, which predicts a connection between p-adic and motivic integrals and more classical invariants of singularities. Further recent interest in the topic comes from connections with Donaldson-Thomas theory. In light of recent multiple developments, this is a good time to have a program on connections between motivic integration and singularity theory. While motivic integration has achieved a certain maturity, several new interesting research directions are being uncovered. By bringing together experts from different but related fields, the program can achieve a fruitful exchange of ideas that will result in progress on important problems (such as the Monodromy Conjecture) and in formulating new questions.
at the Center for Mathematics at Notre Dame that ran for three weeks in May - June 2013. As part of the program, there was a Summer School for undergraduates, a Summer School for graduate students and recent Ph.D.'s, and a conference. The grant was used for funding for travel and local expenses of young participants to the program. The major activities consisted of: 1) May 20-24, 2013: Undergraduate Summer School. The following mini-courses took place: "p-adic numbers and p-adic integration". Lecturer: Margaret M. Robinson (Mt. Holyoke College). This course was an introduction to Igusa zeta functions, whose poles are expected to be related to singularity invariants by the Monodromy Conjecture. "Statistical learning theory and singularities". Lecturer: S. Lin (Institute for Infocomm Research, A*STAR Singapore). Many difficult problems in machine learning are victims of the curse of singularities. As the big data becomes more important, the key mathematical issues need to be analyzed at a deeper level. In this course, we give a brief introduction to singular learning theory, a powerful geometric approach recently developed by Sumio Watanabe. No prior knowledge of statistics is required. "Tropical geometry". Lecturer: Dustin Cartwright (Yale University). Tropical geometry is a way of studying algebraic varieties through their tropicalizations, which are combinatorial and polyhedral objects. He gave an introduction to tropical geometry and its applications to singularities and enumerative problems. 2) May 27-31, 2013: Graduate and Postdoc Summer School The following mini-courses took place. "Monodromy Conjecture." Lecturer: F. Loeser (UPMC). This mini-course presented the history, development, and the state-of-the art of this conjecture. "Motivic integration". Lecturer: T. de Fernex (Utah). This was an introduction to motivic integration, arc spaces, and jet schemes. Various applications were presented on topics such as singularities of pairs and stringy Hodge numbers. "Nash Conjecture". Lecturer: J. Fernandez de Bobadilla (ICMAT-Madrid). Besides an introduction to the topic, this course surveyed the recent solution for the surface case of this conjecture, and the latest developments. "Donaldson-Thomas invariants and motivic Milnor fibers". Lecturer: B. Szendroi (Oxford University). DT invariants have been linked with motivic Milnor fibers by work of K. Behrend and others. The mini-course was an introduction to this topic. "D-modules and vanishing cycles". Lecturer: C. Sabbah (Ecole Polytechnique). This mini-course discussed D-module approaches to singularity theory. 3) June 3-7, 2013: Conference The following speakers participated: P. Aluffi (Talahasse), B. Bhatt (Michigan), M. Blickle (Mainz), A. Chambert-Loir (Rennes), R. Cluckers (Lille), T. de Fernex (Utah), L. Ein (UIC), M. Gonzalez Villa (Heidelberg), J. Gordon (UBC), E. Gorsky (Stony Brook), L. Halle (Stockholm), K. Kedlaya (UCSD), A. Libgober (UIC), F. Loeser (UPMC), L. Maxim (Wisconsin), J. Nicaise (Leuven), Ana Reguera (Valladolid), C. Schnell (Stony Brook), J. Sebag (Rennes), K. Tucker (Princeton), W. Veys (Leuven), V. Vologodsky (Oregon), U. Walther (Purdue).
