Modular forms and Galois representations in finite characteristic

数学中一些最重要的定理是那些连接不同领域、概念或观点的定理。在代数论中，许多这样的定理都是以互易定律的名义提出的。互惠法的总体框架是朗兰兹计划。PI的许多工作都涉及到由互易定律引起的问题，特别是在有理数的绝对伽罗瓦群的线性、mod预演的背景下。在这种情况下，J-P猜想出了一个互惠定律。受Serre猜想和相关问题的启发，PI研究了Galois表示的模形式和变形之间的同余。未来，PI将继续研究和开发工具来处理这些互易定律和相关问题，这些互易定律和相关问题关系到自同构形式和伽罗瓦表示之间的复杂关系。建立互易定律的任何进展都有一个非常深远的影响，整个数论都能感受到。圆周率互易定律的工作包括提出新的想法和技术，这些新的想法和技术涉及到许多数学中心领域，如上面提到的那些，并可能对数论在密码学等具有巨大实际用途的领域的应用有用。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