02/01: Kurt Vinhage
Entropy Rigidity for Anosov Flows in Dimension Three
Abstract: In the 80’s, A. Katok proved that for a geodesic flow on a negatively curved surface, coincidence of its entropy with respect to the Liouville measure with its topological entropy is equivalent to that surface being hyperbolic. The Katok Entropy conjecture states that similar conclusions should hold in higher dimensions as well. In this talk, I will discuss recent work, joint with Jacopo de Simoi, Martin Leguil and Yun Yang, which extends the scope of the rigidity phenomenon to Anosov flows in three dimensions. Time permitting, I will discuss ongoing progress in the dual question of entropy flexibility
be a non-degenerate indefinite quadratic form in 
variables. In the mid 80’s, Margulis proved the Oppenheim conjecture, which states that if 
and 
. In some cases, homogeneous forms of higher degree exhibit the same behavior if the number of variables is large enough in terms of the degree and if the group preserving the form is large enough, then the set of values at integral points can be studied from the point of view of homogeneous dynamics.