Activities This Week

Chow’s theorem says that a compact analytic submanifold of projective space is algebraic. A generalization of this result (proved with Starchenko) says that a complex analytic subvariety of affine space which in addition is definable in some o-minimal structure over the reals is algebraic.

In these 2-3 talks I will speak about the history of this problem, several related and less known results, and the reasons to expect the theorem to be true (not a full proof).

Some minimal background on definability in structures, as well as basics of o-minimality will be assumed.

Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space. In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.