Aaron Abrams, (Washington and Lee University): Building a square out of triangles

If you want to build a square out of triangles, what restrictions are there on the areas
of the triangles?  For example, the triangle areas must sum to the area of the square.
Are there other restrictions?

It turns out that for each combinatorial type of tiling, there's exactly one more condition,
which is another polynomial that must be satisfied by the areas of the triangles in any 
tiling of the given type.  This polynomial invariant is full of mysteries!  My collaborator 
Jamie Pommersheim (from Reed College) and I are currently in Budapest trying to crack 
some of these mysteries.  In the talk I will describe some of our discoveries.