Mathematics Colloquium
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Speaker: Arindam Banerjee, IIT Kharagpur
Date: 30 November, 2022 at 2.30 pm
Venue: Ramanujan Hall
Title: Castelnuovo-Mumford regularity of edge Ideals of graphs and their powers
Abstract: Regularity of edge ideals of graphs and their powers have been a very popular area of research in commutative algebra for the last one decade.
Edge ideals are one of the rare classes of ideals where linear resolution of the ideal implies linear resolution for higher powers i.e minimum possible regularity
for an edge ideal implies minimum possible regularity for all its higher powers(proved by Herzog, Hibi and Zheng). It was also characterised (by Froberg)
that an edge ideal has linear resolution (or minimum possible regularity) if and only if the underlying graph is chordal.This motivated people to
take up various projects for finding sharp upper bounds for regularity for various powers of edge ideals. Two questions largely guided this research:
1. The Nevo-Peeva Question: Is it true that all higher powers (greater than equal to 2) have linear resolutions for edge ideals with
a. linear presentation (that is edge ideals whose first differential matrix of minimal free resolution has linear entries) and
b. regularity less than equal to $3$?
2. The sharp upper bound Conjecture for regularity of powers in terms of the regularity of the ideal: It is believed that
regularity of an edge ideal is r implies for the s th power the regularity is bounded above by 2s+r-1.
There has been much progress towards both of these but both remain open so far. Recently the second question has been
solved for all bipartite graphs. Also for all graphs the bound has been proven for the second power (r+2).
There has been some effort to study how regularity behaves "on an average for all graphs" using some probabilistic methods.
In this talk we plan to discuss the history and current state of research in this area.