UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

September 11, 2007

939 Evans Hall


3:45PM: Some monomial ideals associated to simplicial complexes

Mauricio Velasco

The study of the minimal free resolutions of monomial ideals is important for both geometry and combinatorics (via Grobner bases and the Stanley-Reisner correspondence respectively). There are very few classes of monomial ideals whose minimal free resolutions are explicitly known. In this talk we introduce the class of "Nearly scarf ideals of simplicial complexes" and characterize their minimal free resolutions completely, showing that these ideals are a natural generalization of the Scarf ideals introduced by Bayer,Peeva and Sturmfels. Nearly Scarf ideals allow us to "turn the tables around" and use topological methods to construct minimal monomial free resolutions with prescribed properties. As an application of this idea we will construct the first example of a minimal monomial free resolution which cannot be supported by a CW-complex. Some of the results in this talk are joint work with I. Peeva

5:00PM: Toric Dynamical Systems

Anne Shiu

Abstract: In a chemical reaction network, the concentrations of chemical species evolve in time, governed by the differential equations of mass-action kinetics. This talk provides an introduction to the algebraic study of chemical reaction network theory. Chemical reactions can be represented by directed edges of a graph. A basic question is whether such a network has a steady state. The locus of all steady states is defined by the steady state ideal. We introduce the moduli space of toric dynamical systems of a digraph representing a chemical network. The nicest chemical reaction networks are the toric dynamical systems: their steady state loci and moduli spaces are toric varieties. In chemistry, they are the systems whose steady states are a special kind, called complex balancing steady states. This is joint work with Gheorge Craciun, Alicia Dickenstein, and Bernd Sturmfels.

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