Tropical geometry: Difference between revisions

→‎Applications: improved wording
(→‎Applications: improved wording)
A tropical line appeared in [[Paul Klemperer]]'s design of [[auction]]s used by the [[Bank of England]] during the financial crisis in 2007.<ref>{{Cite web|url = https://www.economics.ox.ac.uk/news/how-geometry-came-to-the-rescue-during-th |title = How geometry came to the rescue during the banking crisis|accessdate = 24 March 2014|website = Department of Economics, University of Oxford}}</ref> Yoshinori Shiozawa defined subtropical algebra as max-times or min-times semiring (instead of max-plus and min-plus). He found that Ricardian trade theory (international trade without input trade) can be interpreted as subtropical convex algebra.<ref>{{Cite journal |doi = 10.1007/s40844-015-0012-3|title = International trade theory and exotic algebras |url=https://www.researchgate.net/publication/280646264 |journal = Evolutionary and Institutional Economics Review|volume = 12|pages = 177–212|year = 2015|last1 = Shiozawa|first1 = Yoshinori}} This is a digest of Y. Shiozawa, "[https://www.researchgate.net/publication/236020268 Subtropical Convex Geometry as the Ricardian Theory of International Trade]" draft paper.</ref>
Moreover, several optimization problems arising for instance in job scheduling, location analysis, transportation networks, decision making and discrete event dynamical systems can be formulated and solved in the framework of tropical geometry.<ref>{{cite book |last= Krivulin |first= Nikolai |arxiv=1408.0313 |chapter= Tropical optimization problems |year=2014 |title=Advances in Economics and Optimization: Collected Scientific Studies Dedicated to the Memory of L. V. Kantorovich |pages=195–214 |publisher=Nova Science Publishers |location=New York |isbn=978-1-63117-073-7 |editor1=Leon A. Petrosyan |editor2=David W. K. Yeung |editor3=Joseph V. Romanovsky}}</ref> A tropical counterpart of the [[Abel–Jacobi map]] can be applied to a crystal design.<ref>{{cite book |author-link=Toshikazu Sunada|last=Sunada |first=T. |year=2012 |title=Topological Crystallography: With a View Towards Discrete Geometric Analysis |series=Surveys and Tutorials in the Applied Mathematical Sciences |volume=6 |publisher=Springer Japan |isbn=9784431541769}}</ref> The weights in a [[weighted finite-state transducer]] are often required to be a tropical semiring. Tropical geometry showscan show [[self-organized criticality]] behaviour.<ref>{{Cite journal|last=Kalinin|first=N.|last2=Guzmán-Sáenz|first2=A.|last3=Prieto|first3=Y.|last4=Shkolnikov|first4=M.|last5=Kalinina|first5=V.|last6=Lupercio|first6=E.|date=2018-08-15|title=Self-organized criticality and pattern emergence through the lens of tropical geometry|journal=[[Proceedings of the National Academy of Sciences of the United States of America]]| volume=115|issue=35|language=en|pages=E8135–E8142|doi=10.1073/pnas.1805847115|issn=0027-8424|pmid=30111541|pmc=6126730|arxiv=1806.09153|bibcode=2018arXiv180609153K}}</ref>
 
==See also==