English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study high dimensional expansion beyond simplicial complexes (posets) andfocus on $q$-complexes which are complexes whose basic building blocks arelinear spaces. We show that the complete $q$-complex (consists of all subspacesof a given linear space) may have non-trivial homology groups and thereforesome techniques for simplicial complexes fail. We develop new techniques towork bypass this. In particular: (i) We describe a new construction of conesand use it to determine when the homology of the complete $q$-complex istrivial. We use this construction to prove the "projective support dimensionconjecture" conjectured by Mnukhin and Siemons. (ii) We define topological highdimensional expansion for posets, and show that the complete $q$-complex haslinear (in the number of lines) coboundary expansion. (iii) We define the$q$-Linial-Meshulam model of random $q$-complexes and prove a sharp thresholdfor the connectivity of random $q$-complexes.

Topological expansion for posets and the homological $k$-connectivity of random $q$-complexes