Minicourse 1: Combinatorial limits and their applications in extremal combinatorics
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*: Combinatorial methods in group theory (and group-theoretical methods in combinatorics)
This course will describe a range of combinatorial methods that
are helpful in group theory. Along the way it will also include some
applications of group theory to combinatorics, and especially to the study
of symmetry of discrete objects (such as graphs and maps). It will begin
with some applications of simple counting, and then methods for finding
pseudo-random elements of a group, and lead on to Cayley graphs, coset graphs
and double-coset graphs and their various applications, and finish with a
section on Möbius inversion on subgroup lattices. Lots of examples and
figures will be given to explain and illustrate these things.
The provisional syllabus is as follows:
1. Basic applications of counting (incl. to Lagrange's theorem, the class
equation and Sylow theory)
2. Methods for generating random or pseudo-random elements of a group
3. Cayley graphs [briefly]
4. Schreier coset graphs and their applications (incl. to permutation
representations, coset enumeration, finiteness of finitely-presented groups,
the Ree-Singerman theorem on transitivity of a group of permutations, and
graphical implementation of Reidemeister-Schreier theory)
5. Back-track search methods for finding all subgroups of small index in a
6. Double-coset graphs and their applications (incl. to construction of
7. Möbius inversion on lattices, and applications to groups.
*Due to health issues, the minicourse of Alice Devillers on Coxeter groups had to be cancelled.