Deficiencies of groups

Deficiency is a measure of how complicated the presentations of a particular group need to be; it is defined as the maximum of the number of generators minus the number of relators (over all finite presentations of the group). This talk will introduce the basics of deficiency, give a deft example of Swan which illustrates why our understanding of deficiency is deficient, and conclude with some new examples that defy this defeatism: finite $p$-groups can have any deficiency you could (reasonably) wish for.