Some of the most important theorems in mathematics are those which connect up different fields, concepts or view-points. In algebraic number theory many of such theorems go under the name of reciprocity laws. The general framework for reciprocity laws is the Langlands program. Much of the PI's work deals with issues that arise from reciprocity laws especially in the context of linear, mod p representations of the absolute Galois group of the rationals. A reciprocity law in this case has been conjectured by J-P. Serre. Motivated by Serre's conjecture and related issues the PI has studied congruences between modular forms and deformations of Galois representations. The PI will continue to study and develop tools to approach such reciprocity laws and related questions that have a bearing on the intricate relationship between automorphic forms and Galois representations in the future.
Any progress towards establishing reciprocity laws has a very broad impact that is felt all across number theory. The work of the PI on reciprocity laws involves coming up with new ideas and techniques that impact many central areas of mathematics, like those mentioned above, and potentially might be useful to applications of number theory in areas like cryptography that are of great practical use. The PI also expects to disseminate knowledge by involving students in research project
