The Bruhat order on clans, and applications to local structure of symmetric orbit closures

I will describe a result of mine which explicitly determines the ``Bruhat order'' (or closure containment order) on the orbits of the symmetric subgroup $GL_p \times GL_q$ on the flag variety $GL_{p+q}/B$, relative to their parametrization by combinatorial objects known as ``clans''.  I will give some applications of this theorem, surveying some recent work of myself joint with Alexander Woo and Alexander Yong.  Specifically, I will relay how the theorem can be used to study:

1.  The local structure and singularities of such orbit closures (via ``patch ideals'');
2.  Cohomological invariants (via Gr\"{o}bner geometry of ``matrix orbit closures"); and
3.  Local isomorphisms of different orbit closures whose combinatorial parameters are related by pattern embeddings (via ``Mars-Springer varieties").