Algebra and Algebraic Geometry Seminar – Kieran Bhaskara and Aniketh Sivakumar – Regularity and projective dimension of toric ideals of bipartite graphs
Nov 9, 2023
9:30AM to 10:30AM
Date(s) - 09/11/2023
9:30 am - 10:30 am
Location: HH 207
Speakers: Kieran Bhaskara and Aniketh Sivakumar (McMaster University)
Title: Regularity and projective dimension of toric ideals of bipartite graphs
Abstract: The regularity and projective dimension of combinatorially-defined ideals are frequently studied invariants in combinatorial commutative algebra. In particular, much work has been done towards understanding the values these invariants can achieve for toric ideals I_G associated to a graph G. In this talk, we fully describe the possible values of these invariants for I_G as G ranges over all bipartite graphs on a fixed number of vertices. As a corollary, we show that any pair of positive integers can be realized as the regularity and projective dimension of a toric ideal of a bipartite graph. Finally, we demonstrate how our main result allows us to determine the values all five major invariants studied in the literature for this family of graphs.
Title: An overview of binomial edge ideals
Abstract: In this talk we will introduce an ideal associated to a graph, called a binomial edge ideal. These ideals were first introduced in 2010, with an application to conditional independence statements. Since then a lot of work has been done studying various homological invariants of these ideals. Here, we will define these ideals and give some results on their minimal primes and grobner basis. If time permits, we will talk about some results on the Betti numbers and regularity of these ideals.