PhD course on lattice-ordered groups and polyhedral geometry (Spring 2023)
Introduction
The course is an introduction to the theory of abelian lattice-ordered groups from different perspectives. Initially, we study these structures with purely algebraic methods. We will analyse some important theorems and connections with other parts of mathematics, such as AF C * -algebras. Later we will move on to their geometric study, through the Baker-Beynon duality. It will be seen that, just as the commutative rings provide an algebraic counterpart for the study of affine manifolds with rational maps, lattice-ordered groups represent the algebraic counterpart of the polyhedral cones and piece-wise linear maps between them.
Course topics
- Abelian lattice-ordered groups: definition and examples.
- Representation results.
- Archimedeanity and strong (order) unit.
- Free and finitely presented abelian l-groups.
- Baker&Beynon duality.
- Polyhedral geometry
Lecture by lecture topics
Course material
- Bigard, A., Keimel, K., & Wolfenstein, S. (2006). Groupes et anneaux réticulés (Vol. 608). Springer.
- Anderson, M. E., & Feil, T. H. (2012). Lattice-ordered groups: an introduction (Vol. 4). Springer Science & Business Media.
- Goodearl, K. R. (2010). Partially ordered abelian groups with interpolation (No. 20). American Mathematical Soc.
- Glass, A. M. W. (1999). Partially ordered groups (Vol. 7). World Scientific.
Practical aspects
Teacher: Luca Spada
Course duration: 10 hours.
Exam
You can choose to take the final exam in one of the following ways:
- A short oral interview (about 30 minutes) in which the knowledge acquired on the basic and more advanced concepts of category theory will be evaluated.
- The presentation of a topic agreed with the teacher and not covered in the course, in the form of a short seminar also open to other doctoral students lasting about 45 minutes.
- Solving some exercises at home.
Term and schedule
To be arranged with the course participants. Presumably in April-May.
Tags: Baker, Beynon, duality, l-group, piece-wise linear maps, Polyhedral geometry, Riez spaces, strong order unit, Vector lattices
