Colloquium[ Edit ]
Moderator: Ron Rosenthal
I will discuss some recent results on minimal actions of general countable groups. In particular I will describe a new property of such minimal actions called the DJ property which is defined in terms of the notion of disjointness of actions and explain how it is related to an old question of Furstenberg on the algebra spanned by the minimal functions on a group. All concepts above will be explained.
In 2008 Agol showed that a 3-manifold with a certain condition on its fundamental group is virtually fibered, i.e. it has a finite covering that is a surface bundle over the circle. A few years later it was shown by Agol and Wise that the fundamental groups of most 3-manifold satisfy Agol's condition, i.e. most 3-manifodls are virtually fibered. We will outline a proof of Agol's theorem following an approach taken by myself and Kitayama.
Nonholonomic mechanics concerns with mechanical systems whose velocity is constrained. If these velocity constraints are linear, they define k-planes at every point of the configuration space of the system. In more complex situations further constraints appear: the movement of the system not only has to be tangent to these k-planes, but must obey conditions in which tangent vectors to the trajectories have constant length, or satisfy other, in general nonlinear, relations. This equips kinematics of nonholonomic mechanical systems with various geometric structures. These are: vector distributions on manifolds, their symmetry groups, differential invariants, associated exterior differential systems, Cartan connections, etc.
In the lectures we will discuss these geometric structures in simple examples of existing (or possible to exist) mechanical systems. We will concentrate on systems whose kinematics is described by a low dimensional parabolic geometry i.e. a geometry modeled on a homogeneous space G/P, with G being a simple Lie group, and P being its parabolic subgroup. The considered systems will include a movement of ice skaters on an ice rink, rolling without slipping or twisting of rigid bodies, movements of snakes and ants, and even movements of flying saucers. Geometric relations between these exemplary nonholonomic systems will be revealed. An appearance of the simple exceptional Lie group G2 will be a repetitive geometric phenomenon in these examples.