# Defence of doctoral thesis in the field of mathematics, MSc Milo Orlich

Title of the doctoral thesis is "Homology and combinatorics of monomial ideals"

This thesis is in abstract algebra, more specifically in the theory of monomial ideals. The main algebraic objects considered here are polynomials, and all polynomials can be built by summing together monomials, which are products of variables. A monomial ideal is the set of polynomials that one can obtain starting from some fixed monomials. The key idea in this subject is to associate to any monomial ideal a combinatorial object, for instance a graph (also called a network), consisting of nodes and edges. Such a graph is a finite structure that can be easily "counted", and many algebraic properties of the monomial ideal are encapsulated in this graph. Vice versa, starting from a graph one can suitably define a monomial ideal, and tackle graph-theoretic problems from an algebraic point of view.

Some of the main concepts in this subject were defined in the 70's and 90's. However, many fundamental problems remain open. In order to tell graphs apart and understand their behaviour, one can associate numbers to them, like the number of nodes, the number of edges, the number of connected components, etc. There is in fact an infinite family of such numbers, called Betti numbers, that one can associate to a given graph. Starting from some specific basic information about the graph, there are algorithms to determine all the Betti numbers, but one does not know what to expect a priori. One of the main open problems is to give closed formulas that describe the Betti numbers in essentially one line, without going through a long algorithm. The main results of this thesis are in this direction.

In the first paper of the thesis we determine exactly this kind of formulas for a specific class of graphs. In the second paper we generalize this to a larger class of monomial ideals, not necessarily associated to graphs. The third paper introduces a new approach to this subject, using critical graphs, which allow to control the Betti numbers asymptotically.

Opponent: Doctor Emil Sköldberg, National University of Ireland, Ireland

Custos: Professor Alexander Engström, Aalto University School of Science, Department of Mathematics and Systems Analysis

Contact details of the doctoral student: [email protected]

The public defence will be organised on campus and via Zoom. Link to the event

The doctoral thesis is publicly displayed 10 days before the defence in the publication archive Aaltodoc of Aalto University.

Electronic thesis

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