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PhD course: Abstract polytopes, groups and incidence geometries, 5 ECTS

Guest Lecturer: Prof. Dimitri Leemans

Dimitri Leemans is currently a professor at the Université Libre de Bruxelles, Belgium. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will be a Guest Professor at the Department of Mathematics and Mathematical Statistics, Umeå University.

Content

In this PhD course we will introduce abstract polytopes and study two families of abstract polytopes with high level of symmetry: the regular and the chiral ones. We will show how to characterise these geometric objects using their group of automorphisms. We will also show that they are part of a much broader category of objects, namely incidence geometries. Finally we will talk about recent developments in the field of hypertopes which are a generalisation of abstract polytopes.

Format

The course consists of 16 hours of lectures given during 4 weeks in March 2022. The lectures will be given in hybrid format, making it possible to take the course online only. 

Prerequisites

Besides general mathematical maturity (master in mathematics or equivalent) a solid background in group theory and graph theory is helpful.

Registration

Registration is free and is done by sending an email to klara.stokes@umu.se. The number of places for examined participants is limited. 

Examination

Examination (hand-in assignments) and course certificates are provided to participants that so require.

Schedule

Room schedule for in real life lectures: TimeEdit

Zoom link: https://umu.zoom.us/j/62193384699

13:15-15:00 Wednesday March 2. Chapter 1: Incidence geometry and coset geometry.

10:15-12:00 Friday March 4. Chapter 2: Abstract polytopes.

13:15-15:00 Wednesday March 9. Chapter 3: Regularity and string C-groups.

10:15-12:00 Friday March 11. Chapter 4: String C-group representations of symmetric groups.

13:15-15:00 Wednesday March 16. Chapter 4: String C-group representations of symmetric groups (cont).

10:15-12:00 Friday March 18. Chapter 5: String C-group representations of PSL(2,q) groups.

13:15-15:00 Wednesday March 23. Chapter 6: Chirality and string C^+-groups.

10:15-12:00 Friday March 25. Chapter 7: Hypertopes.

Background

Polytopes have fascinated mathematicians for millenia. The most famous ones are probably the five platonic solids, namely, the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron.

The study of polytopes has also led to important interdisciplinary discoveries in the 20th century such as fullerenes and nanotubes in chemistry, or the efficient frontier of an investment portfolio using Markovitz’ method in economy. 

The work of Donald Coxeter on regular polytopes and regular maps, the work of Jacques Tits on buildings, which he earlier referred to as polyhedral geometries, and the later work of Egon Schulte on regular incidence complexes, led Schulte and Peter McMullen to develop the theory of abstract regular polytopes. These combinatorial objects, sometimes linked to classical geometry, allow/enable us to better understand groups when these groups are seen as automorphism groups of polytopes. They also include most of the regular maps, namely the non-degenerate ones, which are tesselations of closed 2-dimensional manifolds. They are a special class of diagram geometries and also appear as apartments in buildings.

The study of abstract regular and abstract chiral polytopes associated to families of almost simple groups allows us to gain a better geometric understanding of these groups, to find new sets of generators for them and to better grasp their subgroup structure., and also opens the door to studying representation theory for these groups as every layer of a polytope gives a possible representation of the group.

In order to study abstract polytopes, establishing/developing bridges to other branches of mathematics like group theory, Coxeter groups, graph theory, computational algebra, combinatorics, geometry and even number theory really helps. This course will try to show some of these connections.

In this course Dimitri will introduce abstract polytopes and study two families of abstract polytopes with high level of symmetry: the regular and the chiral ones.

We will see how to characterise these geometric objects using their group of automorphisms. We will also see that they are part of a much broader category of objects, namely incidence geometries. Finally we will talk about recent developments in the field of hypertopes which are a generalisation of abstract polytopes.

Literature

Buekenhout, Francis ; Cohen, Arjeh M. Diagram geometry.Related to classical groups and buildings.Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 57. Springer, Heidelberg, 2013. xiv+592 pp. ISBN: 978-3-642-34452-7; 978-3-642-34453-4 


McMullen, Peter ;  Schulte, Egon . Abstract regular polytopes. Encyclopedia of Mathematics and its Applications, 92. Cambridge University Press, Cambridge,  2002. xiv+551 pp. ISBN: 0-521-81496-0

Contact

Klara Stokes
Associate professor
E-mail
Email