Emerging connections between homology theory and set theory

This talk will survey a family of results and questions — many of them quite recent — which lie at the interface of homology theory and set theory. All of these trace at some fundamental level to Mardešić and Prasolov's 1988 paper Strong homology is not additive; animating each of them are the connections between what may loosely be thought of as continuity properties of strong homology and the derived functors of the inverse limit, and deep questions in infinitary combinatorics. We will begin by reviewing the relevant background from each of these areas; in particular, no more than a basic awareness of ordinals, cardinals, the ZFC axioms, and the functors $\mathrm{lim}^p$ will be assumed of our audience (though we will briefly review the latter). We will then show how two prominent further assumptions, namely the Continuum Hypothesis (CH) and the Open Coloring Axiom (OCA), have opposite effects on the vanishing of a main obstruction to additivity, the first derived limit of an inverse system indexed by the functions from $\mathbb{N}$ to $\mathbb{N}$. From this we will turn to more contemporary results. A brief history of these developments appears in the introduction the speaker's joint work with Chris Lambie-Hanson Simultaneously vanishing higher derived limits, although more recent results should be mentioned as well, if time permits.

Zoom: https://mi-ras-ru.zoom.us/j/91599052030
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