Here is a survey of the first day at the Fields Medal Symposium 2013.
Activities started in the afternoon at 1:30 pm, at the Fields Institute. Walter Craig (Director of the Fields Institute) gave some brief opening remarks about this scientific activity. He recalled the goals of this symposium: to celebrate the deep contributions of the Fields medalist Elon Lindenstrauss to mathematics, to bring together leading researchers in the fields relevant to Lindenstrauss’s work, and to encourage young generations to get interested in mathematics (specially, during the Student Event).
The speakers of this afternoon were Alex Eskin, Shahar Mozes and Benjamin Weiss.
Eskin took as starting point the problem of counting periodic trajectories of balls in a billiard table up to bounded length (and avoiding repetitions in a suitable way). It turns out that establishing an asymptotic formula for this problem in full generality is complicated issue, and it leads to the study of dynamics in moduli spaces of surfaces with a distinguished regular 1-form.
Mozes discussed new equidistribution results for integral points on expanding bodies. Current developments in ergodic theory are put together in a very elegant way, and the final outcome is a sharp refinement of a classical result of W. Schmidt about integer points in expanding domains.
Weiss discussed entropy as a tool for distinguishing non-isomorphic dynamical systems. Roughly speaking, entropy is a measure of the amount of information per time in a dynamical system. There are several definitions for this concept and in favorable cases these different definitions agree, providing fine invariants for studying the isomorphism problem among dynamical systems.
The evening program started with the Public Opening at 7:00 pm, at the Isabel Bader Theatre (UofT). Details of the ceremony can be found here. The Public Opening was followed by a generous reception around 9:30, with refreshments and enjoyable discussions.