Toric fiber products and graph decompositions

The toric fiber product is a general procedure for gluing two
ideals, homogeneous with respect to the same grading. Toric fiber
products generalize familiar constructions in commutative algebra like
adding monomial ideals and the Segre product. We will introduce the
construction and show how it can be used to understand properties of
ideals by a divide-and-conquer approach. This method has been applied
most successfully to families of ideals parametrized by combinatorial
objects like graphs. We will show how to use toric fiber product to
prove structural theorems about classes of ideals from algebraic
statistics.