## Seminar

## The LCM-lattice: a crossroads for combinatorics, topology and commutative algebra

## Dr. Timothy Clark

### Loyola University Maryland

### February 6, 2012

3:30-4:30

### B01 Knott Hall

Abstract:
Let N be an ideal in a polynomial ring which is generated by monomials. Under the relation of monomial divisibility, the set of least common multiples of the generators of N forms a finite atomic lattice (LCM-lattice) which encodes a tremendous amount of information about N. In this talk, we describe a method for pruning an LCM-lattice so that only elements relevant to homological analysis of the ideal remain. We then turn our attention to constructing the minimal free resolution of N using the information in the pruned poset. This technique proves to be particularly useful in the study of minimal resolutions of rigid monomial ideals and generalizes related work of Miller, and Bayer, Peeva and Sturmfels. The talk describes joint (and ongoing) work with Sonja Mapes of the University of Notre Dame. Contextual motivation for the study of free resolutions will be provided.

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