The combinatorial transverse intersection algebra
Daniel An, Ruth Lawrence, Dennis Sullivan

TL;DR
This paper develops a combinatorial graded intersection algebra on the three torus, addressing complex obstacles to ensure algebraic properties and applicability to 3D fluid motion modeling.
Contribution
It introduces a novel combinatorial intersection algebra with weighted transversality and a subcomplex for fluid dynamics computations, preserving key algebraic properties.
Findings
Constructed a commutative, associative intersection algebra on the three torus.
Defined a subcomplex with star bijection useful for fluid motion simulations.
Ensured algebraic properties are maintained under chain crumbling mappings.
Abstract
This paper constructs (with challenging obstacles) on the three torus with its cubical decomposition: Firstly, a combinatorial graded intersection algebra (graded by the codimension) which is commutative and associative defined by transversality on the usual chains which are in general position. This, (with extra elements added) on the entire -cubulated three torus whose differential satisfies the product rule and which agrees with the set theoretic intersection product appropriately weighted. The construction is characterized given these properties (see Comprehensive Theorem below). The challenge is to minimally adjoin infinitesimal elements when the geometric elements have glancing but transversal intersections weighted in such a way that the associativity (and commutativity) is not destroyed and the Leibniz product rule for the boundary operator is restored. Secondly, there is…
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Taxonomy
TopicsAdvanced Algebra and Logic · Data Management and Algorithms · graph theory and CDMA systems
