Young tableaux, beloved of combinatorialists, tolerated by
representation theorists and geometers, seem at first glance
to be an unruly combinatorial set. I'll define a simplicial complex
in which they index the facets, and slightly more general objects
(Buch's ``set-valued tableaux'') label the other interior faces.
The theorem that says we're on a right track: This simplicial complex
is homeomorphic to a ball. I'll explain why this is surprising,
useful, and shows why Buch didn't discover the exterior faces too.
Finally, I'll explain how algebraic geometry forced these definitions
on us (or, ``How I made my peace with Young tableaux''). This work
is joint with Ezra Miller and Alex Yong.