The octahedral axiom in triangulated categories (which are everywhere) is an axiom whose presence we acknowledge, but whose friendliness we often do not acknowledge. In the first half of the talk, I will make the point, by way of examples, that this axiom is a great "organisational tool" that illuminates details we may not easily notice otherwise.  The second half of the talk will make the same point for spectral sequences.

Finding Geometry in Minimal Resolutions
Let X be a closed subvariety of P^n, and let S_X be the homogeneous coordinate ring of X.  By considering the minimal free resolution of S_X we can recover many interesting geometric invariants of X.  Some of this information, like the Hilbert polynomial of X, is relatively easy to recover from the free resolution.  Other invariants, like the gonality of a curve, appear in much more surprising locations.  In this talk, we will illustrate the process of finding geometric information in a free resolution by walking through several examples.

de Rham cohomology and the infinitesimal site.

The constructions of $p$-adic cohomology (e.g. rigid and crystalline
cohomology) for smooth varieties are motivated by the construction of
de Rham cohomology of arbitrary varieties in characteristic 0. I will
explain the characteristic 0 story -- from the basics of de Rham
cohomology to Grothendieck's crystals, a differential free way to talk
about de Rham theory.
 

(Berkeley)