Dimension Without Coordinates

This page is for the keynote presentation by Dr. Peter Dukes.

Abstract

How might you capture the notion of “dimension” of a space without any reference to coordinates or directions? This talk will explore this idea through the lens of finite geometries.

We’ll define a linear space as a set of points \( X \) equipped with a family \( \mathcal{L} \) of subsets of \( X \) called lines. A subspace is just what you’d expect, namely a linear space \( (X^\prime, \mathcal{L}^\prime) \) where \( X^\prime \subseteq X \) and \( \mathcal{L}^\prime \subseteq \mathcal{L} \). The dimension of a linear space is the maximum integer \( d \) such that any set of \( d \) points is contained in a proper subspace.

We’ll see several examples of finite linear spaces and examine their dimension. We’ll also see an application of these ideas to a certain problem in extremal combinatorics.

This is based on joint work with Alan Ling and former student co-authors Nick Benson and Joanna Niezen.