nLab tropical geometry

Idea

Tropical geometry is often thought of as algebraic geometry over the tropical semiring. Many central results are combinatorial in nature, with relations to the (geometry and combinatorics of) polyhedra, matroids, cluster algebras and toric geometry. Recently it found applications in explaining mirror symmetry at a more fundamental level (see e.g. Gross 11).

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: $f (x, y) = x^2 + y^2-1$.

This tropicalizes to $trop(f ) = x^2 \oplus y^2 \oplus 0 = min(2x, 2y, 0)$, 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. $x^2 = x\oplus x$, $y^2 = y\oplus y$ and so $x^2+y^2$ converts to $min(2x,2y)$. The 0 will remain mysterious for the moment. (If you cannot wait look at the AARMS notes listed below.))

Related concepts

• tropical semiring

References

Books and Lecture Notes

Textbook accounts/lecture notes include

• Diane Maclagan, Bernd Sturmfels Introduction to tropical geometry, draft book

• Mark Gross, Tropical geometry and mirror symmetry, CBMS regional conf. ser. 114 (2011), based on the CBMS course in Kansas, AMS book page, pdf

• G. Mikhalkin, Tropical geometry, book, early draft, scribd; Tropical geometry and its applications, Proc. Intern. Congr. Math., V. 2 (Madrid, 2006), Eur. Math. Soc., Zürich, 2006, 827–852 djvu pdf

• 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)

• Erwan Brugallé, Kristin Shaw, A bit of tropical geometry, arxiv/1311.2360

Collections of articles

• (CM580) Tropical geometry and integrable systems, Contemp. Math. 580, Amer. Math. Soc., Providence, RI, 2012
• Tropical and idempotent mathematics, pdf, proceedings conf. Moscow 2012

Papers and Preprints

• Dan Abramovich, Moduli of algebraic and tropical curves, arxiv/1301.0474

• Dan Abramovich, Lucia Caporaso, Sam Payne, The tropicalization of the moduli space of curves, arxiv/1212.0373

• I. Itenberg, G. Mikhalkin, Geometry in the tropical limit, arXiv: 1108.3111

• M. Einsiedler, M. Kapranov, D. Lind, Non-archimedean amoebas and tropical varieties, math.AG/0408311

• Mikhail Kapranov, Thermodynamics and the moment map, arxiv/1108.3472

• Walter Gubler, A guide to tropicalizations, arxiv/1108.6126

• E. Katz, A tropical toolkit, math.AG/0610878 (Expo. Math. 27 (2009), No. 1, 1-36)

• 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.

• David Speyer, Bernd Sturmfels, Tropical mathematics, math.CO/0408099; The tropical Grassmannian, Adv. Geom. 4 (2004) 389–411 math.AG/0304218 emis

• Andreas Gathmann, Tropical algebraic geometry, math.AG/0601322

• Diane Maclagan, Polyhedral structures on tropical varieties, arXiv:1302.5372

• 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

• Martin Ulirsch, Functorial tropicalization of logarithmic schemes: The case of constant coefficients, arxiv/1310.6269

• Luis Felipe Tabera, On real tropical bases and real tropical discriminants, arxiv/1311.2211

• Simon Hampe, Tropical linear spaces and tropical convexity, arxiv/1505.02045

• Tyler Foster, Introduction to adic tropicalization, arxiv/1506.00726

• G. Mikhalkin, Quantum indices of real plane curves and refined enumerative geometry, arxiv/1505.04338

We associate a half-integer number, called the quantum index, to algebraic curves in the real plane satisfying to certain conditions. The area encompassed by the logarithmic image of such curves is equal to $\pi^2$ times the quantum index of the curve. We use the quantum index to refine real enumerative geometry in a way consistent with the Block-G"ottsche invariants from tropical enumerative geometry.

• Diane Maclagan, Felipe Rincón, Tropical ideals, arxiv/1609.03838

• Kiumars Kaveh, Christopher Manon, Khovanskii bases, Newton-Okounkov polytopes and tropical geometry of projective varieties, arxiv/1610.00298

• Alain Connes, Caterina Consani, Geometry of the scaling site, arxiv/1603.03191

• Keyvan Yaghmayi, Geometry over the tropical dual numbers, arxiv/1611.05508

An alternative algebraic framework for tropical mathematics (not based on semirings), “more compatible with valuation theory” has been proposed in

• Zur Izhakian, Manfred Knebusch, Louis Rowen, Algebraic structures of tropical mathematics, arxiv/1305.3906

Connections to diophantine integration (involving p-adic integration):

• Eric Katz, Joseph Rabinoff, David Zureick-Brown, Diophantine and tropical geometry, and uniformity of rational points on curves, arxiv/1606.09618

Mirror symmetry for correlation functions of tropical observables:

• Andrey Losev, Vyacheslav Lysov, Tropical mirror symmetry: correlation functions, arXiv:2301.01687

Miscellaneous: MO questions, discussions, etc.

• MathOverflow : Mikhalkin’s tropical schemes versus Durov’s tropical schemes, learning-tropical-geometry

• wikipedia page

• $n$Cafe: Tight spans, Isbell completions and semi-tropical modules

category: geometry