Modular forms and Galois representations in finite characteristic

数学中一些最重要的定理是把不同的领域、概念或观点联系起来的定理。在代数数论中，许多这样的定理被称为互性定律。互易律的一般框架是朗兰兹纲领。PI的大部分工作都是处理由互易律引起的问题，特别是在绝对伽罗瓦理性群的线性、模表示的背景下。在这种情况下，互惠律是由J-P推测出来的。Serre。在Serre猜想和相关问题的激励下，PI研究了模形式和伽罗义表示变形之间的同余。未来，PI将继续研究和开发工具来处理这些互惠定律以及与自同构形式和伽罗瓦表示之间复杂关系有关的相关问题。建立互惠定律的任何进展都会对整个数论产生非常广泛的影响。PI在互易律方面的工作涉及到新的思想和技术，这些思想和技术影响了数学的许多核心领域，就像上面提到的那样，并且可能对数论在密码学等领域的应用有用，这些领域有很大的实际用途。PI还希望通过让学生参与研究项目来传播知识

ABSTRACTSome of the most important theorems in mathematics are those whichconnect up different fields, concepts or view-points. In algebraicnumber theory many of such theorems go under the name of reciprocitylaws. The general framework for reciprocity laws is the Langlandsprogram. Much of the PI's work deals with issues that arise fromreciprocity laws especially in the context of linear, mod prepresentations of the absolute Galois group of the rationals. Areciprocity law in this case has been conjectured by J-P. Serre.Motivated by Serre's conjecture and related issues the PI has studiedcongruences between modular forms and deformations of Galoisrepresentations. The PI will continue to study and develop tools toapproach such reciprocity laws and related questions that have abearing on the intricate relationship between automorphic forms andGalois representations in the future.Any progress towards establishing reciprocity laws has a very broadimpact that is felt all across number theory. The work of the PI onreciprocity laws involves coming up with new ideas and techniques thatimpact many central areas of mathematics, like those mentioned above,and potentially might be useful to applications of number theory inareas like cryptography that are of great practical use. The PI alsoexpects to disseminate knowledge by involving students in researchproject