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Everything about Simplicial complex
Type of mathematical set
In mathematics, a simplicial complex is a structured set of simplices such that all the faces and intersections of the elements are also included in the set. Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called a geometric simplicial complex. Wikipedia →
🧭 Start here: a 6-step reading path
- Read the overview: Simplicial complexA 5-minute orientation
- Skim a recent paper: Higher-order interactions shape collective dynamics differently in hypergraphs and simplicial complexesInfluential work since 2022
- Pick up a classic book: Organized CollapseThe most-reprinted book on the subject
- Read the landmark paper: Simplicial complexes and complex systemsCited 152 times
- Go sideways: Triangulation (topology)A closely related topic
- Go sideways: Barycentric subdivisionA closely related topic
📖 Definitions
Noun A collection of simplices of various dimensions, connected to one another according to certain rules.
🏆 Landmark papers (most cited)
Simplicial complexes and complex systems
The Geometric Realization of a Semi-Simplicial Complex
The facet ideal of a simplicial complex
Simplicial complex augmentation framework for bijective maps
Topology-adaptive interface tracking using the deformable simplicial complex
Competing spreading dynamics in simplicial complex
Simplicial Complex Neural Networks
A non-partitionable Cohen–Macaulay simplicial complex
🆕 Influential since 2022
Higher-order interactions shape collective dynamics differently in hypergraphs and simplicial complexes
Markovian approach to tackle competing pathogens in simplicial complex
Cooperative epidemic spreading in simplicial complex
Capturing complex interactions in disease ecology with simplicial sets
Two competing simplicial irreversible epidemics on simplicial complex
📚 Books worth reading
- 📕Organized CollapseDmitry N. Kozlov · 2020
- 📕Persistence theorySteve Y. Oudot · 2015
- 📕Toric topologyV. M. Buchstaber · 2015
Algebraic And Combinatorial Aspects Of Tropical Geometry Ciem Workshop On Tropical Geometry December 1216 2011 International Center For Mathematical Meetings Castro Urdiales SpainCIEM Workshop · 2013- 📕Stacks and catetories in geometry, topology, and algebraTony Pantev · 2015
- 📕Combinatorial Reciprocity TheoremsMatthias Beck, Raman Sanyal · 2019
- 📕Homotopy TheoryDaniel G. Davis, Hans-Werner Henn · 2019
- 📕Mathematical Legacy of Richard P. StanleyPatricia Hersh, Thomas Lam · 2016
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
🔗 Related topics
- Triangulation (topology) In mathematics, triangulation describes the replacement of topological spaces with simplicial complexes by the choice of an appropriate homeomorphism.…
- Barycentric subdivision In mathematics, the barycentric subdivision is a standard way to subdivide a given simplex into smaller ones. Its extension to simplicial complexes is…
- Simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex…
- Abstract simplicial complex In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking…
- Shelling (topology) In mathematics, a shelling of a simplicial complex is a way of gluing it together from its maximal simplices (simplices that are not a face of another…
- Delta set In mathematics, a Δ-set, often called a Δ-complex or a semi-simplicial set, is a combinatorial object that is useful in the construction and triangulation…
- Simplicial homology In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of…
- Simplicial set In mathematics, a simplicial set is a sequence of sets with internal order structure (abstract simplices) and maps between them. Simplicial sets are higher-dime…