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Ergodic Theory Meeting

Thursday 20th April 2017

All talks will be in Room B3.03 in the Zeeman Building (Mathematics Institute).

Schedule

1:30pm-2:30pm Vadim Kaloshin (Maryland)

Title: On convex planar billiards, Birkhoff Conjecture and whispering galleries
Abstract: A mathematical billiard is a system describing the inertial motion of a point mass inside a domain with elastic reflections at the boundary. In the case of convex planar domains, this model was first introduced and studied by G.D. Birkhoff, as a paradigmatic example of a low dimensional conservative dynamical system.
A very interesting aspect is represented by the presence of 'caustics', namely curves inside the domain with the property that a trajectory, once tangent to it, stays tangent after every reflection. Besides their mathematical interest, these objects can explain a fascinating acoustic phenomenon, known as "whispering galleries", which can be sometimes noticed beneath a dome or a vault.
The classical Birkhoff conjecture states that the only integrable billiard, i.e., the one having a region filled with caustics, is the billiard inside an ellipse. We show that this conjecture holds near ellipses. This is based on joint works with A. Avila-De Simoi and with Sorrentino.

2:45pm-3:45pm Holly Krieger (Cambridge)

Title: Height functions in complex dynamics
Abstract: I will discuss the role of height functions in complex and arithmetic dynamics, with a focus on applications to problems in unlikely intersections. This includes a dynamical analogue of the André-Oort conjecture proved by myself, Ghioca, Nguyen, and Ye, as well as recent work on bounded height theorems analogous to those of Bombieri-Masser-Zannier. Time permitting, I will also discuss recent work on reduction of dynatomic curves and connections to the dynamical uniform boundedness conjecture.

TEA

4:15pm-5:15pm Frederic Paulin (Orsay)

Title : On the ergodic theory of group actions on trees
Abstract : In this talk, I will study the variational principle and the exponential mixing properties of Gibbs measures under the discrete time geodesic flow on the quotient of a locally finite simplicial tree by a lattice. Applications will be given to error terms in equidistribution and counting problems in non-Archimedean local fields. This is joint work with Anne Broise and Jouni Parkkonen.


For people who arrive in time we will meet in the Common Room around 12:00 to go for lunch in University House (cafeteria style food). We will also go to a restaurant for dinner in the evening.

This is part of the LMS Scheme 3 funded network of collaborative meetings.