3:45PM: Codepth, complete intersections, and quasicyclic modules. (Joint work with Claudia Polini, in progress)Robin HartshorneCodepth is the dual notion to depth, being the greatest length of a coregular sequence for a module, meaning the first element maps the module surjectively, the second is subjective on the kernel of the first, and so on. For a curve in P3, let M be the local cohomology module of the graded coordinate ring with supports in the ideal of the curve. Then the theorem of Hellus says that C is a set theoretic complete intersection if and only if M has codepth 2. This criterion is not directly applicable, so we define the notion of a quasicyclic module, which is an increasing limit of cyclic modules. In this talk I will recall the still open problem of whether every irreducible nonsingular curve in P3 is a set theoretic complete intersection, and derive a number of consequences using the concepts introduced above.

5:00PM: Chow rings of matroids, ring of matroid quotients, and beyondChris EurWe introduce a certain nef generating set for the Chow ring of the wonderful compactification of a hyperplane arrangement complement. This presentation yields a monomial basis of the Chow ring that admits a geometric and combinatorial interpretation with several applications. Geometrically, one can recover Poincare duality, compute the volume polynomial and verify its logconcavity, and identify a portion of a polyhedral boundary of the nef cone. Combinatorially, one can generalize Postnikov's result on volumes of generalized permutohedra, prove Mason's conjecture on the logconcavity of independent sets for certain matroids, and define a new valuative invariant of a matroid that measures its closeness to uniform matroids. This is an ongoing joint work with Spencer Backman and Connor Simpson.
