Tropical geometry is often thought of as the algebraic geometry over the tropical semiring. A good part of it is combinatorial in nature, with relations to the (geometry and combinatorics of) polyhedra and toric geometry. Recently it found applications in explaining mirror symmetry at a more fundamental level.
Tropical algebraic geometry establishes and studies some very general principles to translate algebro-geometric problems into purely combinatorial ones.
Example In algebraic geometry one often work with polynomials. In tropical geometry, these polynomials are “tropicalized” and this turns them into piecewise linear functions.
For instance: .
This tropicalizes to , and this is a piecewise linear curve.
(To see this remember that in the tropical semiring, the sum of two numbers is their minimum, and their product is their sum. , and so converts to . The 0 will remain mysterious for the moment. (If you cannot wait look at the AARMS notes listed below.))
Textbook accounts/lecture notes include
Diane Maclagan, AARMS Tropical Geometry:lecture notes from a four week graduate summer school on Tropical Geometry held at the University of New Brunswick in July/August 2008 under the auspices of the Atlantic Association for Research in the Mathematical Sciences (AARMS).
MSRI introductory workshop on tropical geometry page, Aug 24-28, 2009 (with videos of the lectures)
I. Itenberg, G. Mikhalkin, Geometry in the tropical limit, arXiv: 1108.3111
Walter Gubler, A guide to tropicalizations, arxiv/1108.6126
E. Katz: A tropical toolkit. Expo. Math. 27, No. 1, 1-36 (2009).
Oleg Viro, Hyperfields for tropical geometry I. hyperfields and dequantization, arxiv/1006.3034; Tropical geometry and hyperfields, talk at Mathematics - XXI century. PDMI 70th anniversary, video; On basic concepts of tropical geometry, Trudy Mat. Inst. Steklova 273 (2011), 271–303
Patrick Popescu-Pampu, Dmitry Stepanov, Local tropicalization, arxiv/1204.6154
W. Gubler, Tropical varieties for non-Archimedean analytic spaces, Invent. Math. 169 (2007), 321–376.
Andreas Gathmann, Tropical algebraic geometry, math.AG/0601322
Paul Johnson, Hurwitz numbers, ribbon graphs, and tropicalization, arxiv/1303.1543 (pages 55-72 in CM580)
Brugalle Erwan, Markwig Hannah, Deformation of tropical Hirzebruch surfaces and enumerative geometry, arxiv/1303.1340
Qingchun Ren, Steven V Sam, Bernd Sturmfels, Tropicalization of classical moduli spaces, arxiv/1303.1132
An alternative algebraic framework for tropical mathematics (not based on semirings), “more compatible with valuation theory” has been proposed in