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.