Time: Tuesday, August 13, from 2pm to 4pm (Hanoi time).

Venue: Offline at Room 301, Building A5, Institute of Mathematics, combined with zoom. The online participants can join Zoom from 1:45pm onwards.

Abstract: Let $G$ be a linear algebraic group acting on an affine variety $V$, all are defined over a field $k$. In 2005, G. Roehrle et al proposed a geometric approach to study completely reducible subgroups due to J.-P. Serre via cocharacter closedness of rational orbits $G(k).v$. Here we say that the rational orbit $G(k).v$ of a rational point $v$ is cocharacter closed if this orbit contains the limit point (if exists) along any cocharacter of $G$. Now assume further that $k$ is a valued field, we may endow $G(k)$ and $V(k)$ with the $v$-adic topology induced from that of the base field $k$. The aim of this talk is to discuss the relationship between the cocharacter closedness and Hausdorff closedness of rational orbits, as well as the cocharacter closure and Hausdorff closure of these ones.