## EVENT DETAILS AND ABSTRACT

**Algebra Seminar**
**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.

**Date:** Tuesday, November 18, 1997

**Time:** 4:30pm

**Where:** Lunt 104

**Contact Person:** Prof. Kevin Knudson

**Contact email:** knudson@math.nwu.edu

**Contact Phone:** 847-491-5574

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Department of Mathematics, Northwestern University.