"Automorphic forms and L-functions have a long tradition in number theory which can be traced back to the classical work of Jacobi, Dirichlet, and Riemann. They make hidden symmetries of integers visible and provide deep links to other branches of mathematics such as algebraic geometry, combinatorics, representation theory, ergodic theory, dynamical systems, and mathematical physics. Gergely Harcos, the researcher of the present proposal, intends to apply methods from spectral theory and representation theory to establish new bounds for automorphic forms and L-functions. Such bounds play a key role in the solution of difficult Diophantine problems addressing equidistribution phenomena, and they also shed light on deep conjectures such as the Grand Riemann Hypothesis.Gergely Harcos returned to Europe in 2006 with a Marie Curie Intra-European Fellowship after a decade of successful work in the United States. He fulfilled several goals set out in his earlier proposal. He introduced a new line of mainstream research at Rényi Institute and initiated a regular seminar on automorphic forms in order to build a collaborative network. He designed and taught new courses in number theory at Central European University for an international audience which further added to his impact. Gergely Harcos will stay at the host where he will be elected a permanent member soon. The present project serves as an organic continuation of his earlier Marie Curie proposal."