Tropical Mathematics

By David E. Speyer, Bernd Sturmfels. 2004

169 citations recorded by the source corpus.

This article is based on the Clay Mathematics Senior Scholar Lecture that was delivered by Bernd Sturmfels in Park City, Utah, on July 22, 2004. The topic of this lecture was the tropical approach in mathematics. This approach was in its infancy at that time, but it has since matured and is now an integral part of geometric combinatorics and algebraic geometry. It has also expanded into mathematical physics, number theory, symplectic geometry, computational biology, and beyond. We offer an elementary introduction to this subject, touching upon arithmetic, polynomials, curves, phylogenetics, and linear spaces. Each section ends with a suggestion for further research. The proposed problems are particularly well suited for undergraduate students. The bibliography contains numerous references for further reading in this field. The adjective tropical was coined by French mathematicians, including JeanEric Pin [16], in honor of their Brazilian colleague Imre Simon [19], who was one of the pioneers in what could also be called min-plus algebra. There is no deeper meaning in the adjective tropical. It simply stands for the French view of Brazil.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/4ff6df58075368f0d9d66356043ebb0a335d5cc7

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Citations in the corpus

Supertropical semirings and supervaluations (2010, background): “ered monoid rather than an ordered group), which has led researchers in tropical mathematics to utilize valuation theory. In order to avoid the introduction of the negative, some researchers, such as [SS], have used the min-plus algebra instead of the max-plus algebra. There is a deeper result which”
Geometry of the Restricted Boltzmann Machine (2009, background): “For a first introduction see [30], and for further reading see [4, 8, 10, 23] and references therein.”
Degree-growth of monomial maps (2006, background): “tiplication by addition, division by subtraction, and addition by max, or min, has attracted a good deal of attention lately; it is known as “tropicalization”, and a good introduction to the topic is [SS].) It is more interesting to compare (x,y) → (y,(y2 + 1)/x) with (a,b) → (b,max(2b,0)− a). Itera”
Brane tilings and their applications (2008, background): “According to [84], the name ‘tropical’ was coined by a French mathematician Jean-Eric Pin [85], in honor of their Brazilian colleague Imere Simon [86].”
Enumerative tropical algebraic geometry in R^2 (2003, background): “Similarly we can set up the problem of counting all (not necessarily irreducible) curves.”