Program on Motivic Invariants and Singularities

本提案是为了支持圣母大学数学中心的动机不变量和奇点项目，该项目包括本科生暑期学校（2013年5月21日至25日），研究生和博士后暑期学校（2013年5月27日至31日），会议（2013年6月3日至7日），以及整个项目的杰出讲座系列。本科暑期学校将包括三个迷你课程：p进数和p进积分、热带几何、统计学习理论和奇点。高级暑期学校将包括五个迷你课程，分别是d模和消失循环、Donaldson-Thomas不变量和动机米尔诺纤维、单形猜想、动机积分和纳什猜想。会议将汇集动力集成和奇点理论相关领域的顶尖研究人员。此外，由Jan Denef主持的杰出系列讲座将向广大听众传播与该计划相关的令人兴奋的话题。会议的网站是：http://nd.edu/~cmnd/programs/mis2013/Since由M. Kontsevich于1995年创建，动机积分是一个与代数几何、奇点理论、数论和模型理论相关的快速发展的学科。这个理论在很多不同的领域都有应用，比如麦凯对应，极小模型程序中的奇点，以及朗兰兹程序中出现的轨道积分的研究。虽然在这方面已经做了大量的工作并取得了显著的进展，但仍有一些根本问题有待解决。这样一个问题就是单性猜想，它预测了p进积分和动力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Donaldson-Thomas理论的联系。鉴于最近的多重发展，这是一个很好的时机来开设一个关于动机整合和奇点理论之间联系的课程。在动机整合已经达到一定成熟的同时，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，这将导致在重要问题（如单形猜想）上取得进展，并形成新的问题。

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.