Differential calculus for free algebra and constrained homology

TL;DR

This paper develops differential calculus for free algebras and applies it to constrained (co)homology, hypergraphs, and filtrations, providing new tools for analyzing complex combinatorial structures.

Contribution

It introduces a differential calculus framework for free algebras and explores its implications for constrained (co)homology and hypergraph analysis.

Findings

Realized simplicial complexes as invariant traces of hypergraphs

Constructed constrained persistent (co)homology for filtrations

Extended differential calculus to hypergraph structures

Abstract

In [Discrete differential calculus on simplicial complexes and constrained homology, Chin. Ann. Math. Ser. B 44(4), 615-640, 2023], the constrained (co)homology for simplicial complexes and independence hypergraphs is constructed via differential calculus on discrete sets. In this paper, we study the differential calculus for free algebra and subsequently study the homomorphisms of constrained (co)homology induced by inclusions of simplicial complexes and independence hypergraphs. We apply the differential calculus to hypergraphs. We realize simplicial complexes and independence hypergraphs as certain invariant traces of hypergraphs. As an application, we give the constrained persistent (co)homology for filtrations of simplicial complexes and filtrations of independence hypergraphs.