2.00 pm, Seminar Hall Hecke algebras and the Langlands program Manish Mishra RuprechtKarlsUniversitat IHeidelberg. 201115 Abstract Given an irreducible polynomial f(x) with integer coefficients and a prime number p, one wishes to determine whether f(x) is a product of distinct linear factors modulo p. When f(x) is a solvable polynomial, this question is satisfactorily answered by the Class Field Theory. Attempts to find a nonabelian Class Field Theory lead to the development of an area of mathematics called the Langlands program. The Langlands program, roughly speaking, predicts a natural correspondence between the finite dimensional complex representations of the Galois group of a local or a number field and the infinite dimensional representations of real, padic and adelic reductive groups. I will give an outline of the statement of the local Langlands correspondence. I will then briefly talk about two of the main approaches towards the Langlands program  the type theoretic approach relying on the theory of types developed by BushnellKutzko and others; and the endoscopic approach relying on the trace formula and endoscopy. I will then state some of my results involving these two approaches.
