Rees algebras, Monomial Subrings and Linear Optimization Problems

TL;DR

This thesis explores the algebraic properties of monomial algebras linked to combinatorial structures and optimization problems, establishing connections between algebra, combinatorics, and optimization.

Contribution

It investigates the normality, Gorenstein property, and other algebraic features of Rees algebras and subrings related to linear optimization and combinatorial structures.

Findings

Characterization of normality and Gorenstein properties of Rees algebras

Analysis of algebraic properties of edge ideals with the max-flow min-cut property

Study of symbolic Rees algebras and Stanley-Reisner rings

Abstract

In this thesis we are interested in studying algebraic properties of monomial algebras, that can be linked to combinatorial structures, such as graphs and clutters, and to optimization problems. A goal here is to establish bridges between commutative algebra, combinatorics and optimization. We study the normality and the Gorenstein property-as well as the canonical module and the a-invariant-of Rees algebras and subrings arising from linear optimization problems. In particular, we study algebraic properties of edge ideals and algebras associated to uniform clutters with the max-flow min-cut property or the packing property. We also study algebraic properties of symbolic Rees algebras of edge ideals of graphs, edge ideals of clique clutters of comparability graphs, and Stanley-Reisner rings.