Last updated Fri 5/11/07 5:00 PM
Abstract: Convex rank tests are coarsenings of the hyperplane
arrangement
associated to the symmetric group. Semigraphoids are combinatorial
structures that correspond to squares and hexagons on a polytope known as
the permutohedron.
These objects - convex rank tests and semigraphoids -
are equivalent, and this result allows us to answer a question posed by
Postnikov, Reiner, and Williams. No prior knowledge of hyperplane
arrangements or permutohedra will be assumed.
This work originated in collaboration with the Pourquié lab at the
Stowers Institute in
Kansas City. We describe the experiments conducted to identify the
molecular components of biological clocks, and explain how the convex
rank test called the cyclohedron test contributes to time course
microarray data analysis.
Abstract: Ideals generated by minors in rectangular matrices filled with variables are classical, particularly when the minors are all of those having some fixed size. I'll talk about a more general class of determinantal ideals, where the sizes of the minors in certain subrectangles can vary. These ideals are naturally indexed by permutations in S_n, and we'll see just how richly the combinatorics and algebra interact. The algebraic results concern primality, Gröbner bases, Hilbert series, multidegrees, and Cohen-Macaulayness; the combinatorics inlcudes antidiagonals and planar diagrams of permutations, as well as Schubert and Grothendieck polynomials. I will assume that you've seen things like Gröbner bases, (multi)graded rings, and Hilbert series, but little else; the rest will be introduced from scratch.
Last updated Fri 5/11/07 5:00 PM