Activities This Week

In this talk I will discuss a necessary and sufficient condition for denseness of horopherical orbits in the non-wandering set of a higher-rank homogeneous space $G / \Gamma$, for a Zariski dense discrete subgroup $\Gamma < G$, possibly of infinite covolume. In rank one this condition (established in this setting by Eberlein and Dal’bo) implies in particular that the horospherical subgroup acts minimally on the non-wandering set if and only if the discrete group $\Gamma$ is convex co-compact. In contrast, we show that Schottky groups in higher-rank can support non-minimal horospherical actions. This distinction between rank-one and higher-rank is due to the role that Benoist’s limit cone plays in the analysis. Based on joint work with Hee Oh.

In this talk I will discuss a necessary and sufficient condition for denseness of horopherical orbits in the non-wandering set of a higher-rank homogeneous space $G / \Gamma$, for a Zariski dense discrete subgroup $\Gamma < G$, possibly of infinite covolume. In rank one this condition (established in this setting by Eberlein and Dal’bo) implies in particular that the horospherical subgroup acts minimally on the non-wandering set if and only if the discrete group $\Gamma$ is convex co-compact. In contrast, we show that Schottky groups in higher-rank can support non-minimal horospherical actions. This distinction between rank-one and higher-rank is due to the role that Benoist’s limit cone plays in the analysis. Based on joint work with Hee Oh.

Polynomial equations and polynomial maps are central objects in modern mathematics, and understanding their geometry and singularities is of great importance. In this talk, I will pitch the idea that polynomial maps can be studied by investigating analytic properties of regular measures pushed-forward by them (over local and finite fields). Such pushforward measures are amenable to analytic and model-theoretic tools, and the rule of thumb is that singular maps produce pushforward measures with bad analytic behavior. I will discuss some results in this direction, as well as some applications to group theory and representation theory. In particular, I plan to mention some recent results on local integrability of Harish-Chandra characters.

Based on joint projects with R. Cluckers, I. Glazer, J. Gordon and S. Sodin.