Combinatorial design theory is the part of mathematics which studies systems of finite sets whose arrangement satisfy some concept of balance or symmetry. Its roots lie in the design of biological experiments but modern applications are found across a wide array of mathematical areas. Of particular interest in this seminar is the role it plays within projective geometry and finite group theory.
In this seminar we will focus on a type of combinatorial block design known as a Steiner system. Whilst these systems are simple to define, the unfortunate truth is we know relatively little about them. We will see some examples of these systems, discuss some long-standing open problems in this area, and see how a miraculous set of images can help us understand the structure of a particularly complex Steiner system. If we have time to spare, I will talk a little about how Steiner systems play a pivotal role within my own research.