Program on Motivic Invariants and Singularities

这项提议是为了支持圣母大学数学中心的动机不变量和奇点计划，该计划包括一个本科生暑期学校(2013年5月21日至25日)、一个研究生和博士后暑期学校(2013年5月27日至31日)、一个会议(2013年6月3日至7日)以及贯穿整个计划的杰出讲座系列。本科生暑期班将由三门小型课程组成，分别是p-进位数和p-进位积分、热带几何以及统计学习理论和奇点。高级暑期班将包括五门关于D-模和零圈、Donaldson-Thomas不变量和Motivic Milnor纤维、Monodromy猜想、Motivic积分和Nash猜想的迷你课程。这次会议将汇集与动机整合和奇点理论相关领域的顶尖研究人员。此外，由简·德内夫主持的杰出讲座系列将向广大观众传播与该节目有关的令人兴奋的话题。会议的网站是：http://nd.edu/~cmnd/programs/mis2013/Since自1995年M.Kontsevich创立以来，动机积分一直是与代数几何、奇点理论、数论和模型理论相联系的一门迅速发展的学科。该理论在一系列不同的主题中得到了大量的应用，如McKay对应、最小模型程序中的奇点以及朗兰兹程序中的轨道积分的研究。虽然在这方面已经做了很多工作并取得了显著进展，但一些根本问题仍然悬而未决。这样的问题就是Monodromy猜想，它预言了p-进积分和动量积分与更经典的奇点不变量之间的联系。最近对这个话题的进一步兴趣来自于与唐纳森-托马斯理论的联系。鉴于最近的多重发展，现在是一个很好的时机来制定一个关于动机整合和奇点理论之间联系的计划。虽然动机整合已经达到一定的成熟度，但一些新的有趣的研究方向正在被发现。通过将来自不同但相关领域的专家聚集在一起，该计划可以实现富有成效的思想交流，从而在重要问题(如Monodromy猜想)和提出新问题方面取得进展。

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.