“Topological combinatorics”的意思、由来-开放百科全书

词条

Topological combinatorics

释义

See also
References
Further reading

The discipline of combinatorial topology used combinatorial concepts in topology and in the early 20th century this turned into the field of algebraic topology.

In 1978 the situation was reversed — methods from algebraic topology were used to solve a problem in combinatorics – when László Lovász proved the Kneser conjecture, thus beginning the new study of topological combinatorics. Lovász's proof used the Borsuk–Ulam theorem and this theorem retains a prominent role in this new field. This theorem has many equivalent versions and analogs and has been used in the study of fair division problems.

In another application of homological methods to graph theory Lovász proved both the undirected and directed versions of a conjecture of András Frank: Given a k-connected graph G, k points , and k positive integers that sum up to , there exists a partition of such that , , and spans a connected subgraph.

In 1987 the necklace splitting problem was solved by Noga Alon using the Borsuk–Ulam theorem. It has also been used to study complexity problems in linear decision tree algorithms and the Aanderaa–Karp–Rosenberg conjecture. Other areas include topology of partially ordered sets and bruhat orders.

Additionally, methods from differential topology now have a combinatorial analog in discrete Morse theory.

References

{{Citation| first=Mark | last=de Longueville| coauthors=| contribution=25 years proof of the Kneser conjecture - The advent of topological combinatorics| title=EMS Newsletter| editor-first=| editor-last=| coeditors=| publisher=European Mathematical Society| place=Southampton, Hampshire | pages=16–19| year=2004| id= | contribution-url=http://www.emis.de/newsletter/current/current9.pdf| format=PDF| accessdate=2008-07-29 }}.

| last = Björner | first = Anders | author-link = Anders Björner
| editor1-last = Graham | editor1-first = Ronald L. | editor1-link = Ronald Graham| editor2-last = Grötschel | editor2-first = Martin | editor2-link = Martin Grötschel
| editor3-last = Lovász | editor3-first = László | editor3-link = László Lovász| contribution = Topological Methods
| isbn = 978-0-262-07171-0
| publisher = The MIT press
| title = Handbook of Combinatorics
| url = http://www.math.kth.se/~bjorner/files/TopMeth.pdf
| volume = 2
| year = 1995}}.

{{citation

| last = Kozlov | first = Dmitry | author-link = Dmitry Feichtner-Kozlov
| arxiv = math.AT/0507390
| title = Trends in topological combinatorics
| year = 2005}}.

{{citation

| last = Kozlov | first = Dmitry | author-link = Dmitry Feichtner-Kozlov
| isbn = 978-3-540-71961-8
| publisher = Springer
| title = Combinatorial Algebraic Topology
| year = 2007}}.

{{citation

| last = Lange | first = Carsten
| publisher = Berlin Institute of Technology
| series = Ph.D. thesis
| title = Combinatorial Curvatures, Group Actions, and Colourings: Aspects of Topological Combinatorics
| url = http://opus.kobv.de/tuberlin/volltexte/2005/753/pdf/lange_carsten.pdf
| year = 2005}}.

{{citation

| last = Matoušek | first = Jiří | author-link = Jiří Matoušek (mathematician)
| isbn = 978-3-540-00362-5
| publisher = Springer
| title = Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry
| year = 2003}}.

{{citation

| last = Barmak | first = Jonathan | author-link =
| isbn = 978-3-642-22002-9
| publisher = Springer
| title = Algebraic Topology of Finite Topological Spaces and Applications
| year = 2011}}.

{{citation

| last = de Longueville | first = Mark | author-link =
| isbn = 978-1-4419-7909-4
| publisher = Springer
| title = A Course in Topological Combinatorics
| year = 2011}}.{{DEFAULTSORT:Topological Combinatorics}}

3 : Combinatorics|Topology|Algebraic topology

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。

See also

References

Further reading