Date: 19 December 2015.
Place: Class 24, Department of Mathematics, Hacettepe University, Ankara.
Program:
11:00-12.00: Mufit Sezer
On Cohen-Macaulayness and depth of ideals in invariant rings.
12.00-13.00: Lunch
13.00-14.00: Marius Vladoiu
Combinatorial classification of toric ideals and applications.
Abstracts of the talks.
(1) Mufit Sezer. On Cohen-Macaulayness and depth of ideals in invariant rings.
We investigate the presence of Cohen-Macaulay ideals in invariant
rings and show that an ideal of an invariant ring corresponding
to a modular representation of a p-group is not Cohen-Macaulay
unless the invariant ring itself is. For modular cyclic groups of prime
order, we show that the quotient of the invariant ring modulo the transfer
ideal is always Cohen-Macaulay. This is joint with M. Kohls.
(2) Marius Vladoiu. Combinatorial classification of toric ideals and applications.
We discuss a combinatorial classification of all toric ideals based on the bouquet structure
we introduce in a joint work with Sonja Petrovic and Apostolos Thoma. On one end of the spectrum of this
classification there are ideals, that we call stable, for which bouquets capture the complexity of various generating sets
and the minimal graded free resolution, while on the other end of the spectrum of this classification lie toric ideals whose
various bases (e.g. indispensable and Graver) coincide. We discuss several applications of this classification, the main ones
being the solution of an open problem in combinatorial commutative algebra, and that all positively graded toric ideals are encoded by
toric ideals of hypergraphs, i.e. toric ideals of 0-1 matrices.