Conferências Magnas - Elon Lindenstrauss
15 de agosto às 16:30
Universidade Federal Fluminense
Título: Quantum unique ergodicity and homogeneous dynamics
The quantum unique ergodicity conjecture of Rudnick and Sarnak states that on a compact hyperbolic manifold Laplacian eigenfunctions become equidistributed in the semiclassical limit. A particularly interesting testing ground of this conjecture are arithmetic surfaces (and higher dimensional arithmetic manifolds) where the conjecture (and close analogues of it) have been established in several (but not all) of the most interesting cases, one of the approaches being via homogeneous dynamics. I will discuss this connection, as well as recent work with S. Brooks on behavior of quasimodes that gives evidence that degeneracies in the spectrum play less of a role in this question as was previously thought.
20 de agosto às 17:00
Instituto Nacional de Matemática Pura e Aplicada
Título: Homogeneous dynamics and diagonalizable actions
Algebraic actions on quotient spaces of algebraic (or Lie) groups G/Gamma give highly interesting dynamical systems that are in many cases intimately linked with number theory. Already in the 1950’s Cassels and Swinnerton Dyer observed implicitely (in a dual language) that higher rank diagonal groups have interesting rigidity properties; this was explicitly brought to the fore by Furstenberg in 1967. These delicate regidity properties are still far from being understood; I will survey some of the progress made in this direction, particularly regarding the classification of invariant measures.