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Why are so many algebraists nowadays interested in cluster algebras? (This is a rewording of one half of the closed question Cluster algebras and teichmuller theory.)

Let $P$ be an $n$-element poset and $J(P)$ the distributive lattice of its order ideals (i.e., the downwards-closed sets). For each $I\in J(P)$ let $x_I \in \mathbb{R}^{n}$ be the indicator function ...

On affine toric varieties there is a classical theorem of Danilov which gives some combinatorial ways to describe the global sections of an appropriate sheaf of Kahler differentials as a vector space. ...