Modular forms and Galois representations in finite characteristic

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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