A model for statistical ranking is a family of probability distributions
whose states are orderings of a fixed finite set of items. We represent the
orderings as maximal chains in a graded poset. The most widely used ranking
models are parameterized by rational function in the model parameters, so they
define algebraic varieties. We study these varieties from the perspective of
combinatorial commutative algebra. One of our models, the Plackett-Luce model,
is non-toric.

Convex algebraic geometry concerns the interplay between optimization theory
and real algebraic geometry. Its objects of study include convex semialgebraic
sets that arise in semidefinite programming and from sums of squares. This
article compares three notions of duality that are relevant in these contexts:
duality of convex bodies, duality of projective varieties, and the
Karush-Kuhn-Tucker conditions derived from Lagrange duality.

A Mustafin variety is a degeneration of projective space induced by a point
configuration in a Bruhat-Tits building. The special fiber is reduced and
Cohen-Macaulay, and its irreducible components form interesting combinatorial
patterns. For configurations that lie in one apartment, these patterns are
regular mixed subdivisions of scaled simplices, and the Mustafin variety is a
twisted Veronese variety built from such a subdivision. This connects our study
to tropical and toric geometry.

An orbitope is the convex hull of an orbit of a compact group acting linearly
on a vector space. These highly symmetric convex bodies lie at the crossroads
of several fields, in particular convex geometry, optimization, and algebraic
geometry. We present a self-contained theory of orbitopes, with particular
emphasis on instances arising from the groups SO(n) and O(n). These include
Schur-Horn orbitopes, tautological orbitopes, Caratheodory orbitopes, Veronese
orbitopes and Grassmann orbitopes.

The diagonal in a product of projective spaces is cut out by the ideal of
2x2-minors of a matrix of unknowns. The multigraded Hilbert scheme which
classifies its degenerations has a unique Borel-fixed ideal. This Hilbert
scheme is generally reducible, and its main component is a compactification of
PGL(d)^n/PGL(d). For n=2 we recover the manifold of complete collineations. For
projective lines we obtain a space of trees that is irreducible but singular.
All ideals in our Hilbert scheme are radical. We also explore connections to
affine buildings and Deligne schemes.