the combinatorics of permutations in the symmetric group $s_n$ has deep connections to algebraic properties of the {\em coinvariant ring} (through work of artin, chevalley, lusztig-stanley, and others) and geometric properties of the {\em flag variety} whose points are complete flags in $\mathbb{c}^n$ (through work of ehresmann, borel, and others). we will discuss new generalizations of the coinvariant ring and flag variety indexed by two positive integers $k \leq n$. the algebraic and geometric properties of these objects are controlled by ordered set partitions of $[n]$ with $k$ blocks. there are connections between these objects and the delta conjecture in the theory of macdonald polynomials. joint with jim haglund, brendan pawlowski, and mark shimozono. many important maps in algebraic combinatorics (the rsk bijection, the schutzenberger involution, etc.) can be described by piecewise-linear formulas. these formulas can then be ``de-tropicalized,'' or ``lifted,'' to subtraction-free rational functions on an algebraic variety, and certain properties of the combinatorial maps become more transparent in the algebro-geometric setting. i will illustrate how this works in the case of the promotion map on semistandard tableaux of rectangular shape. i will also indicate how promotion can be viewed as the combinatorial manifestation of a symmetry coming from representation theory, and how its geometric lift fits into berenstein and kazhdan's theory of geometric crystals.