Program on Motivic Invariants and Singularities

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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.