We study how Betti numbers of ideals in a local ring change under small perturbations. Given p ? N and given an ideal I Noetherian ( R , m ) our main result states that there exists > 0 such if J is with ? mod the same Hilbert function as then ? i / coincide for ? . Moreover, we present several cases which forced to have therefore numbers.

In the present paper, we characterize all possible Hilbert functions of graded ideals in a polynomial ring whose regularity is smaller than or equal to d, where d is a positive integer. In addition, we prove the following result which is a generalization of Bigatti, Hulett and Pardue’s result: Let p ≥ 0 and d > 0 be integers. If the base field is a field of characteristic 0 and there is a grade...

We give a leisurely account of the relationship between volume and L2-Betti numbers on closed, aspherical manifolds based on the results in [4] – albeit with a different point of view. This paper grew out of a talk presented at the first colloquium of the Courant Center in Göttingen in October 2007. 1. Review of L2-Betti numbers The L2-Betti numbers of a closed Riemannian manifold, as introduce...

We give explicit formulas for the asymptotic Betti numbers of unordered configuration spaces an arbitrary finite graph over field.

We prove a conjectured lower bound of Nagel and Reiner on Betti numbers of edge ideals of bipartite graphs.

1. Write-up Wish List 2 2. To Do List 2 3. History 3 4. Basic definitions and notation 4 5. Codimension One Finite Length [Sarah] 5 6. Codimension Two Finite Length [Sarah] 5 7. Codimension Three Finite Length 7 8. Attempts at Codimension Four 8 9. Betti Death 8 10. Alternative diagram decompostions for complete intersections 9 11. Decomposition Via Separating the Betti Diagram 13 11.1. Heurist...