## 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 polynomial maps, lattice-ordered groups represent the algebraic counterpart of the polyhedral cones and piece-wise linear homogenous 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

- 5/5/2023: Introduction to the course. Motivations and applications of the theory of abelian lattice ordered groups. Main examples. Crash course on Galois connections and categorical adjunctions.
- 11/5/2023: Overview of the main results and techniques in the study of l-groups: The integers and Weinberg’s theorem. Archimedeanicity and Hölder’s theorem. Semisimplicity and Yosida’s representation.
- 12/5/2023: Strong unit and MV-algebras. The free abelian l-groups as algebras of functions. Lattice ordered groups and piecewise linear geometry.
- 18/5/2023: The general adjunction and its fixed points: Baker&Beynon duality and polyhedral geometry.

## 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

## Term and schedule

Lecturer: Luca Spada

Course duration: 10 hours.

Course calendar: 5, 11, 12 and 18 of May, from 10:00 to 12:45. All lectures are in room P19 (last floor, building F3).

### 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 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 30 minutes.
- Solving some exercises at home.

Tags: Baker, Beynon, duality, l-group, piece-wise linear maps, Polyhedral geometry, Riez spaces, strong order unit, Vector lattices