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. The general adjunction and its fixed points: Baker&Beynon duality.
  • 18/5/2023:

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.