Syracuse Algebra Seminar

Schedule of Talks Spring 2025:

• Next talk (February 28): Liana Șega (University of Missouri-Kansas City), TBA

  • Abstract: TBA

• March 7: Eloísa Grifo (University of Nebraska-Lincoln)

• March 21: Keller VandeBogert (Fields Institute)

• March 28:  Rachel Diethorn (Oberlin College)

• April 4: Ryan Watson (University of Nebraska-Lincoln)

• April 11: Oana Veliche (Northeastern University)

• April 18: Prashanth Sridhar (Auburn University)

• April 25: Justin Lyle (Auburn University)

Past talks:

• January 31: Jen Biermann (Hobart and William Smith Colleges), Algebraic invariants of weighted oriented graphs

  • Abstract: Monomial ideals associated to (unweighted, unoriented) graphs have been studied extensively since the introduction of edge ideals by Villarreal in 1990.  More recently, motivated by applications in coding theory, interest has arisen in a generalization of edge ideals to weighted, oriented graphs.  In this talk I will review basic definitions and what is known about the regularity of the edge ideals of unweighted, unoriented graphs before discussing the regularity of edge ideals of weighted, oriented paths and cycles.

• February 7: Des Martin (Syracuse University), An introduction to DG ideals

  • Abstract:  Let G be a tree and I_G the edge ideal of G. In this talk we use discrete Morse theory to construct the minimal free resolution of k[x_1,.., x_n]/I_G from the Taylor resolution. We then explore when the minimal free resolution obtained from this process inherits the differential graded (dg) algebra structure from the Taylor resolution. In joint work with Hugh Geller and Henry Potts-Rubin.

• February 14: Henry Potts-Rubin (Syracuse University), DG-sensitive pruning for squarefree monomial ideals

  • Abstract: Given a differential graded (dg) algebra resolution F of k[x_1,...,x_n]/I, what kinds of processes can we apply to F to obtain a new resolution (possibly of a different module, possibly over a different ring) that also admits the structure of a dg algebra?  In this talk, we describe one such process, a "pruning process" introduced by Boocher in 2012, which we prove is "DG-sensitive" when I is a squarefree monomial ideal.  In the language of combinatorics, our result states that when the minimal free resolution of the facet ideal of a simplicial complex ∆ admits the structure of a dg algebra, then so does the minimal free resolution of the facet ideal of each facet-induced subcomplex of ∆ (over the smaller polynomial ring).  We will also discuss the existence of a dg algebra structure on the minimal free resolution of the edge ideal of a diameter-four tree, which we combine with pruning and results discussed in the previous week’s seminar to completely classify the trees and cycles with edge ideal minimally resolved by dg algebras.  This is joint work with Hugh Geller and Des Martin.

• February 21: Nawaj KC (University of Nebraska-Lincoln), Loewy lengths of modules of finite projective dimension

  • Abstract: If M is a module of finite length over a local Noetherian ring (R, m), its Loewy length is the smallest non-negative integer i such that m^iM = 0. I will sketch a proof of the following result: if R is a strict Cohen-Macaulay ring, the Loewy length of a module of finite projective dimension is at least as big as the Loewy length of a quotient by sufficiently general linear system of parameters on R. This is joint work with Josh Pollitz.