In Wednesday October 2 activities started at 9:15 am at the Fields Institute. We had three lectures in the morning, by Hillel Furstenberg, Elon Lindenstrauss and Shimon Brooks.
Furstenberg’s lecture focussed on the following conjecture: for almost all x in the unit interval we have that Hdim(Ax)+Hdim(Bx) is at least 1, where Ax is the closure of the orbit of x under multiplication by 2 (modulo the integers) and Bx is the closure of the orbit of x under multiplication by 3 (Hdim is Hausdorff dimension). He showed how this conjecture would follow from another conjecture regarding fractals in the plane.
Lindenstrauss explained the notion of rigidity for orbits and for invariant measures in a dynamical system. Rigidity can happen on the presence of several dynamical conditions (higher rank semi-group actions) or under restrictions on the entropy of the system (maximality, positivity). The lecture covered classical results (such as Furstenberg’s rigidity of orbits, or Ratner’s classification of measures on unipotent dynamics) and explained some recent developments.
Brooks discussed Quantum Unique Ergodicity for quasimodes. This roughly means the following: you have a surface of negative curvature and look at quasimodes — approximate eigenfunctions for the laplacian operator. The question is whether or not these approximate eigenfunctions are equidistributed as the eigenvalue grows. The main theme was to understand to what extent large eigenspaces can be responsible for failure in equidistribution, and it seems that this phenomenon is easier to detect in the context of quasimodes rather than eigenfunctions.
There were no lectures in the afternoon, but there was a very nice field trip in the evening: the symposium banquette, held at the Integral House. Among the appetizers we had some delicious pistachio pizza, salmon crepes and a certain sup which I could not identify (but was definitely great). Our host was James Stewart, the famous mathematician and math textbook writer. It turns out that Stewart is a really charismatic and approachable person — he even gave us a tour through his Integral House!