Differential topologists working in and around dimension 4
enjoy using "diagrams" and some form of "diagrammatic calculus", the
prototype being Kirby diagrams and Kirby calculus. Mathematicians
working in other fields sometimes seem a little suspicious, or at
least feel that it's "not quite fair" that we get to prove things by
"just drawing pretty pictures". In this talk I will attempt to think
carefully about what diagrams really mean, explain some standard and
not-so-standard examples, and pose some questions about just how much
mileage we can expect to get using purely diagrammatic methods. In
particular, the usual diagram is some 2-dimensional combinatorial data
that describes a smooth 4-dimensional object up to a natural smooth
isomorphism - but what if we want to understand the isomorphisms
themselves? Isomorphisms up to what? Can diagrams help us understand
not just objects but also morphisms? Morphisms between morphisms? The
main goal is to open a can of worms and leave the audience to clean up
the mess!