The Basic Notions Seminar

Organized by William Stein

Mondays 3-4pm in Science Center 507 at Harvard University

List of Talks Last Year's Seminar

STACKS -- La Joie des Champs

Liber Accusationis, caput I

Michael McQuillen (Harvard)

May 2, 2005 (note change!)

The very definition of Riemann surfaces, differential manifolds, analytic varieties, schemes, or whatever else your favourite geometric object may be has to be considered as wholly non-functorial with respect to the ideas. The appropriate, and basic notion, is stack, or rather better in Grothendieck's original French, `champs'.

It's essential to the understanding of the talk that the audience should know at least one example of a non-euclidean geometry, e.g., even spherical trigonometry would be fine. Knowing what Hausdorff/separated/T2 topological space means would also be very useful (even close to essential). As in all things the more you know, the more you're likely to understand, e.g., if you're familiar with so called `non-commutative geometry' or you think that a `scheme' which isn't quasi-projective is anything other than a pathology of commutative algebra you should definitely come to have your misconceptions corrected.

An Anecdote: Stacks as a basic notion reminds me of an incident at a party: the visitor (a psychiatrist) was questioned by a mathematician for a bit as to what she did. Subsequently, she asked him what he did, and not unreasonably he asked, `Do you know what a Mobius strip is'. Good try by the mathematician, but he got a negative answer, so he followed up with `Do you know what an etale pricinple homogeneous space is'.