A lozenge tiling of a dilation of a triangle is a tiling with triangular and rhomboidal tiles. Ardila and
Billey proved that the set of positions which can be occupied by triangular tiles in a lozenge tiling have
a simple geometric characterization, and they called such configurations of triangles spread out.
They then proposed the spread out simplices conjecture, which states that the same characterization
holds in higher dimensions, where lozenge tilings are generalized as fine mixed subdivisions of a
dilated simplex. Part of the interest in the conjecture comes from the fact that spread out
configurations (in arbitrary dimension) are the bases of a matroid.
In this talk, we will introduce the necessary notions to formulate the spread out simplices conjecture,
and show how it can be written as a statement about certain cells in triangulations of a product of two
simplices. We will then mention some special cases where the conjecture is proven, and touch upon
the connection between this conjecture and other topics in discrete mathematics.