One of the fundamental results of PL topology on one hand, and the algebraic geometry of toric varieties on the other, is that any map in the respective categories (PL homeomorphism on the one hand, and birational map on the other) can be factorized into elementary moves: A PL topologist would call these stellar subdivisions, the operation of picking a point in a simplex of the polyhedron, and forcefully subdividing it by coning over the boundary of its neighborhood. And their inverses.
An algebraic geometer knows these operations as blowups and blowdowns.
Unfortunately, the result is quite messy. One has to go up and down a lot, alternating between both operations.
For this reason, two of the pioneers of the respective areas, Tadao Oda and James Alexander, asked independently whether one can simplify the zig and zag of these operations: Is it true that we can put all stellar subdivisions/blowups first, and only then do the inverses/blowdowns.
In other words, is there a common stellar refinement of both? To me, this question always held fascination due to its simplicity. More than that, however, it was introduced to me by a dear friend, Frank Lutz, who passed away recently. I will write a bit more on the subject in the next days. In the meantime, you can grab the preprint here.