In Tuesday October 1 the activities started in the morning, at 9:30 at the Fields Institute. We had two lectures in this morning session, the first by Nalini Anantharaman and the second by Jean Bourgain. Both of these lectures were focused on studying the eigenvalues of certain operators.
Anantharaman first recalled a classical theorem about how eigenfunctions of the laplacian distribute on manifolds with ergodic geodesic flow. Using this as inspiration, she described some recent work on a graph-theoretical analogue where the eigenvalues of the adjacency matrix play the role of the spectrum of the laplacian.
Bourgain discussed some progress towards a spectral gap theorem on the compact case. He used diverse techniques such as diophantine approximation, quite different to the classical hyperbolic approach.
After lunch, the afternoon session started at 1:30 pm. There were three lectures, by Masaki Tsukamoto, Manfred Einsiedler and Anatole Katok.
Tsukamoto considered the dynamics occurring on the space of Brody curves (1-Lipshitz holomorphic maps) when the complex numbers act by translations. The goal was to estimate the mean dimension of this action, and indeed, he gave good estimates in terms of more classical invariants.
Einsiedler used methods of ergodic theory (both in the real and p-adic case) to prove equidistribution of integer points in expanding spheres. Actually, he proved much more than this; the pair consisting of an integral vector and its complementary integer lattice gives a point in a suitable parameter space, and these points equidistribute.
Katok discussed the very general problem of completely describing all invariant measures on (reasonable) dynamical systems. Such a general problem is out of reach at present, but if we impose more conditions such as positive entropy, then some progress can be made. Katok also presented some ideas to attack the case of entropy zero.
The first lecture was given by Dmitry Jakobson, and the topic was spectral theory in the context of dynamical systems. Using very down-to-earth examples, Jakobson succeeded in making accessible to undergraduate students concepts such as the Quantum Unique Ergodicity problem (indeed, he proved a baby-case during his lecture!).
The second speaker was Ralf Spatzier. This beautiful lecture, of completely elementary nature, made understandable (even to high school students) the notion of a dynamical system and periodic orbits. He first explained that a dynamical system on a finite set always has periodic orbits. Later, after explaining the intermediate value theorem and the notion of continuity, he proved that every discrete-time dynamical system on the interval [0,1] with continuous transition function must have a fixed point. He used these example to illustrate more general principles about existence of fixed points and periodic orbits.
The panel discussion had the following participants (from left to right in the table): Elon Lindenstrauss, Ralf Spatzier, Zach Paikin (moderator), Dmitry Jakobson, Peter Sarnak. The topics discussed covered a variety of questions raised by the moderator, the audience and even some questions sent by twitter to the moderator during the discussion! Although it might not be in the records, the panelists were open to discuss with undergraduate and high school students after the panel. In particular, after the panel discussion, the Fields medalist Elon Lindenstrauss approached a kid sitting in front of me to give him a more detailed explanation about his question on Benford’s Law; this explanation was too long to be given during the panel discussion. It is very inspiring to see how professional mathematicians are so approachable, and are open to share their knowledge even with non-specialists.
As a side note which is completely irrelevant to the Symposium, 1st of October was my 25th birthday. I am not from Toronto, but fortunately I found a friend from Kingston who was at the Fields institute for the thematic program on Calabi-Yau varieties. We went for a beer after the panel discussion, which was a nice opportunity to get some insight of the non-mathematical activities in Toronto.