Minicourse 1: Combinatorial limits and their applications in extremal combinatorics
Lecturer: Daniel Kráľ (Masaryk University, Czech Republik and University of Warwick, UK)
Abstract: The theory of combinatorial limits provide analytic tools to represent and
analyze large discrete objects. Such tools have found important applications
in various areas of computer science and mathematics. Combinatorial limits
are also closely related to the flag algebra method, which led to solving
several long-standing open problems in extremal combinatorics.
The course will be focused on limits of dense graphs and permutations.
We will explore the links to the regularity method and present a brief
tutorial on the flag algebra method, which will be demonstrated on several
problems from extremal combinatorics.
The tentative syllabus of the tutorial is the following:
1. Introduction - dense graph convergence, graph limits
2. Graph and permutation quasirandomness via limits
3. Flag algebra method and its relation to graph limits
4. Applications of flag algebra method in combinatorics
5. Computer assisted use of flag algebras via SDP
Minicourse 2: Coxeter groups
Lecturer: Alice Devillers, The University of Western Australia
Abstract: Coxeter groups are finitely presented groups such that all generators are involutions and all relations have a specific form involving pairs of generators only. They are the building blocks for Tits buildings (indeed they are exactly the thin buildings) and hence are very useful, but also they are very interesting objects in and of themselves, from an algebraic, geometric and combinatorial point of view.
The relations of a Coxeter group can be encoded into a Coxeter diagram. Group elements can be expressed as words in terms of the generators and so we have a length function on the group.
Finite reflection groups, such as the automorphism group of a regular polygon or regular polyhedron, are examples of Coxeter groups. Finite reflection groups have the interesting property that they have a unique longest element. They have been classified and are related to the famous and ubiquitous Dynkin diagrams (plus a couple more). The proof is really beautiful.
Affine reflection groups, that is infinite groups generated by affine reflections in Euclidean space, are also examples of Coxeter groups.
Even general abstract Coxeter groups have some nice properties, such as a solution to the word problem.