• Assistant Professor
    • Relinde Jurrius
    • Nederlandse Defensie Academie
    • Coding Theory

q-Analogues in combinatorics

Many objects in combinatorics consider subsets of a finite set that have certain properties. Take for example a finite projective plane: this is a finite set, the ‘points’, with some of its subsets, the ‘lines’, that satisfy the properties that every two points are contained in exactly one line and every two lines intersect in exactly one point. Other examples are block designs, matroids, and graphs. Many of these notions are related to each other. In the last decade the notion of a ‘q-analogue’ has gained a lot of research attention in combinatorics. A q-analogue is, roughly speaking, a generalisation from finite sets to finite spaces. So, instead of considering a finite set and some of iets subsets, we consider a vector space over a finite field and some of its subspaces. Taking q-analogues is not always as easy as it seems. The main reason is that vector spaces over finite fields lack some nice properties that finite sets have. Also, intuition based on linear algebra over the real numbers will often fail: did you ever realise that a vector over a finite field can be orthogonal to itself? This talk will survey some straightforward and less straightforward properties of vector spaces over finite fields. (Basic knowledge of linear algebra is desired, but no deep understanding of finite fields is necessary.) The goal of this overview is to shine a light on the exciting world of q-analogues, where many open problems await the curious mathematician.