GT seminar: "Conditional Convergence Revisited"

Abstract: Spectral sequences have proven to be immensely powerful tools in modern mathematics. One of the main questions when working with spectral sequences is whether the spectral sequence is computing what we want it to compute. This is the question of convergence. At the inception of the subject, dealing with this question usually involved imposing quite severe finiteness conditions, but as the subject developed it became apparent that better considerations were needed. One groundbreaking point of view is Boardman's notion of conditional convergence. The concept of conditional convergence is very useful in that it allows one to deduce strong convergence from conditions which are entirely internal to the spectral sequence in question, and which in many cases are easy to check.

From a structural point of view, convergence is often treated a bit abusely. We speak of a spectral sequence being convergent, even though technically speaking convergence is extra data, and not just a property of the spectral sequence. Moreover, the notion of conditional convergence is not even internal to spectral sequences. In other words, given a spectral sequence, with no information on how it arose, the question “Does the spectral sequence converge conditionally?” does not even
make sense. In this talk, I instead propose the two notions derived weak and derived strong convergence which recaptures Boardman's work, but are more structurally well-behaved. This is joint work in progress with Achim Krause and Thomas Nikolaus.

This is a talk in our geometry and topology seminar series.