EVENT DETAILS AND ABSTRACT

Title: Cohomology of Subspace Arrangements via Minimal Free Resolutions (Peeva)/ A-graded Algebras (Gasharov)
Speaker: Professors Irena Peeva and Vaselin Gasharov
Speaker Info: MIT
Brief Description:
Special Note:
Abstract:

Cohomology of subspace arrangements > via minimal free resolutions > > Irena Peeva > MIT > >Methods from Algebraic Geometry can be applied to study >the complement of an arrangement of linear subspaces in >complex space. However, very little is known for real >spaces. The classical algorithm to compute the cohomology >of the complement of a real arrangement of linear subspaces >is to construct the intersection lattice, compute the homology >of all lower intervals in this lattice, and then apply a formula >of Goresky and MacPherson. We introduce an approach of expressing the cohomology of the complement of a real diagonal arrangement of >linear subspaces by the Betti numbers of a minimal free resolution. >We apply this especially to the arrangements of r-equals, which >have been extensively studied recently. This is a joint work with >Victor Reiner and Volkmar Welker. >-------------------------------------------------------------------- > A-graded algebras > > Vesselin Gasharov > MIT > >Let A be a finite subset of N^d and J be the toric ideal >defined by A. An ideal M in a polynomial ring is called >A-graded if it has the same multigraded Hilbert function as J. >The study of A-graded ideals was initiated by Arnold, who >showed that the structure of such ideals is encoded in continued >fractions in the case when J defines a monomial curve in A^3. >Arnold, Korkina, Post and Roelofs proved that if J defines a monomial curve in A^3 then any A-graded ideal is isomorphic >to an initial ideal of the toric ideal J. Later, Sturmfels studied >arbitrary A-graded ideals and related their structure to >subdivisions of the convex envelope of A. He conjectured the >following generalization of Arnold-Korkina-Post-Roelofs' result: >if the toric ideal J has codimension 2 then any A-graded ideal >is isomorphic to an initial ideal of J. We prove this conjecture. >This is a joint work with Irena Peeva.