## Oliver Club

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!