Time： Every Thursday (13:00-15:00), From 2016-03-31 to 2016-06-16 

Venue: Classroom 82J12, Jiayibing Building, BICMR

Low-dimensional topology and dynamical systems are two active parts of mathematics that have always been connected. For example knot theory emerged in the 19th century from an attempt of Kelvin and Tait to understand vortex lines in fluid mechanics, and Thurston's program in the 70th (recently achieved by Agol) showed that the topology of 3-dimensional manifolds is very closely related to the dynamics of homeomorphisms of surfaces.

In this course we will review several examples of such connections, focusing on problems that involve vector fields on 2- and 3-manifolds (e.g., Poincaré-Bendixson Theorem, persistence of vortex lines, existence of sections for vector fields), with an emphasis on fundamental examples (geodesic flows, Lorenz flows, Anosov flows). We will address problems such as the existence of periodic orbits, study invariants measures. 

Finally we will devote our attention to two important (and mostly open) problems: the construction and classification of so-called Birkhoff sections for 3d vector fields (raised by Fried and Ghys), and the construction of invariants under volume-preserving diffeomorphisms (raised by Arnold). We will pay attention to make the course accessible to all students by making appropriate reminders.