The Geometry and Global Analysis of Arithmetic Manifolds

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The PI will use techniques from analytic number theory to investigate the global harmonic analysis and topology of arithmetic manifolds. His first proposed line of investigation is to apply Arthur's endoscopic classification of automorphic representations of the classical groups to the problem of estimating the multiplicities with which nontempered Archimedean representations occur at a given level. The PI aims to establish a number of cases of a conjecture of Sarnak and Xue on these multiplicities, and give sharp asymptotics for the growth rate of Betti number of arithmetic manifolds in congruence towers.The PI's second proposed investigation is into the concentration of higher rank eigenfunctions on compact locally symmetric spaces, and in particular the question of how large the L^p norms of such an eigenfunction or its restriction to various subspaces can be. The first step of this is to derive the correct 'convex bounds' for these L^p norms, which will be sharp in the case of non-negative curvature and for spectral clusters in the general case, by combining techniques from semiclassical analysis and representation theory. The second component is to improve these convex bounds (in particular, to produce a saving in the exponents) in as many cases as possible by introducing arithmetic amplification as in the work of Iwaniec and Sarnak.Arithmetic manifolds are central objects in number theory, as they encode information about objects which were first studied by the Greeks such as prime numbers and the solutions of polynomial equations in integers or rational numbers. They also allow one to study special cases of questions in geometry, analysis and mathematical physics using powerful tools from number theory. An example of this is the recent progress on the Quantum Unique Ergodicity conjecture of Rudnick and Sarnak, which helps us to understand the way in which quantum mechanics starts to resemble classical mechanics at high energies. The PI's proposed investigation of the concentration of eigenfunctions would improve our understanding of this phenomenon. In addition, the `non-tempered' automorphic forms that the PI will study play an important role in many problems of counting and dynamics, and the project may have interesting consequences in these areas.