Tropical geometry

The tropical geometry is an active field of research in algebraic geometry and thus a branch of mathematics . It can be viewed as a piece-wise linearized version of algebraic geometry. Algebraic varieties thus become combinatorial objects that can be examined with discrete mathematics . Therefore, there are close connections between tropical geometry and combinatorics , enumerative geometry , computer algebra and toric geometry .

History of Tropical Geometry

As early as the 1950s, idempotent half-rings such as the Min-Plus algebra were used in discrete mathematics and computer science . Early forms of tropical geometry can be found in the 1970s with George Bergman . A few years later, concepts of tropical geometry are also used in group theory by Robert Bieri , John RJ Groves , Walter D. Neumann and Ralph Strebel . ^{1.6 Group Theory.}

Tropical geometry is receiving greater attention, especially through its successful application in enumerative geometry, for example by Grigory Mikhalkin , who used it in 2005 in the Gromov-Witten theory . ^{1.7 Curve Counting}

Still, tropical geometry is still in its infancy. For example, there have long been no analogous objects in tropical geometry for various basic objects of algebraic geometry - such as abstract varieties and their morphisms. In 2013 Jeffrey and Noah Giansiracusa introduced tropical schemes .

Concept formation

The Min-Plus-Algebra was already called a tropical half-ring by French colleagues in the 1980s in honor of the Brazilian mathematician and computer scientist Imre Simon , which is supposed to indicate the origin of Simon. The term tropical geometry should also be understood in this context . So the term has no deeper meaning, but in short:

"It simply stands for the French view of Brazil."

"It just stands for the French view of Brazil."

- Maclagan, Sturmfels : Introduction to Tropical Geometry . 2015, chap. 1 , foreword .

Main idea behind the tropical geometry

Tropical geometry is based on the idea of ​​providing simpler and new methods for studying algebraic curves . In tropical geometry, shadows of algebraic curves are considered. These shadows are combinatorial, piecewise linear objects and are called tropical curves. They clearly correspond to simple graphs or "line drawings". Because of their piecewise linear structure, the tropical curves can be studied using simpler methods than algebraic curves. Because the tropical curves are shadows of algebraic curves, you can still read some properties of the algebraic curve from the respective tropical curve.

Tropical geometry arithmetic

In tropical addition , the sum of two numbers is either their maximum (max-plus algebra) or their minimum (min-plus algebra) and there is no subtraction .

In tropical multiplication , the tropical product of two numbers is their classical sum and the tropical division is defined as a classical subtraction .

Many of the well-known axioms of arithmetic retain their validity in tropical mathematics: the associative and commutative law applies to both tropical addition and tropical multiplication. and . The distributive law also applies : ${\displaystyle x\oplus y=y\oplus x}$ ${\displaystyle x\otimes y=y\otimes x}$ ${\displaystyle x\otimes (y\oplus z)=x\otimes y\oplus x\otimes z}$

Basic concepts

Tropical half ring

The tuple , where the addition and the multiplication are defined by and is a half ring (even a half body ). It is known as a tropical half-ring or also as min-plus algebra. ${\displaystyle (\mathbb {R} \cup \{\infty \},\oplus ,\otimes )}$ ${\displaystyle \oplus }$ ${\displaystyle \otimes }$ ${\displaystyle a\oplus b=\min(a,b)}$ ${\displaystyle a\otimes b=a+b}$

While algebraic geometry usually deals with sets of zeros of polynomials over a body, tropical geometry can be understood as geometry over the tropical half-ring.

It should be noted that the tropical half-ring is idempotent , that is, it always applies . In addition, there is a neutral element with regard to addition, namely , but no inverse elements . ${\displaystyle a\oplus a=a}$ ${\displaystyle \infty }$

It is also possible to use the Max-Plus-Algebra instead of the Min-Plus-Algebra , which some authors consider to be more natural. In Max-Plus-Algebra, the addition and multiplication for the tuple are defined by and . The Max-Plus-Algebra is mainly used in economics as well as in business administration . ${\displaystyle \oplus }$ ${\displaystyle \otimes }$ ${\displaystyle (\mathbb {R} \cup \{\infty \},\oplus ,\otimes )}$ ${\displaystyle a\oplus b=\max(a,b)}$ ${\displaystyle a\otimes b=a+b}$

Tropical polynomials

If one considers Laurent polynomials over the tropical half-ring, one obtains functions of the form

${\displaystyle p\colon \mathbb {R} ^{n}\to \mathbb {R} ,(x_{1},\dotsc ,x_{n})\mapsto \bigoplus _{i=1}^{k}\left(a_{i}\otimes x_{1}^{i_{1}}\dotsb x_{n}^{i_{n}}\right)=\min _{1\leq i\leq k}\left(a_{i}+\sum _{j=1}^{n}i_{j}x_{j}\right),}$

whereby . ${\displaystyle a_{1},\dotsc ,a_{k}\in \mathbb {R} ,i_{1},\dotsc ,i_{n},k\in \mathbb {Z} }$

It turns out that these tropical polynomials in variables are precisely the continuous , piecewise-linear , concave functions with integer slopes. ^{Lemma 1.1.2.} ${\displaystyle n}$

More generally, a tropical polynomial can be assigned to each element from the ring of Laurent polynomials over an evaluated body with evaluation . To get the tropized polynomial as . ^{Formula (2.4.1).} ${\displaystyle K}$ ${\displaystyle {\text{val}}}$ ${\displaystyle f=\sum _{u\in \mathbb {Z} ^{n}}c_{u}x^{n}\in K[x_{1}^{\pm 1},\dotsc ,x_{n}^{\pm n}]}$ ${\displaystyle {\text{trop}}(f)}$ ${\displaystyle {\text{trop}}(f)=\min _{u\in \mathbb {Z} ^{n}}\left({\text{val}}(c_{u})+\sum _{i=1}^{n}u_{i}w_{i}\right)}$

Tropical hypersurfaces

A tropical hypersurface to a Laurent polynomial is the location of all points in where the piecewise linear function is not linear. ^{Definition 3.1.1.} ${\displaystyle {\text{trop}}(\mathbb {V} (f))}$ ${\displaystyle f\in K[x_{1}^{\pm 1},\dotsc ,x_{n}^{\pm n}]}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle {\text{trop}}(f)}$

Tropical variety

Let be an ideal and its variety in the algebraic torus . ${\displaystyle I\subseteq K[x_{1}^{\pm 1},\dotsc ,x_{n}^{\pm n}]}$ ${\displaystyle X=\mathbb {V} (I)\subseteq T^{n}}$ ${\displaystyle T^{n}}$

Then the tropized variety is the intersection of the tropical hypersurfaces which are defined by the polynomials from , that is ${\displaystyle {\text{trop}}(X)}$ ${\displaystyle I}$

${\displaystyle {\text{trop}}(X)=\bigcap _{f\in I}{\text{trop}}(\mathbb {V} (f))\subseteq \mathbb {R} ^{n}.}$

A tropical variety in is a subset that arises as a tropization of a sub- variety of an algebraic torus of an evaluated body. ^{Definition 3.2.1.} ${\displaystyle \mathbb {R} ^{n}}$

Central sentences

The main theorem of tropical geometry ^{theorem 3.2.5.} provides various characterizations of tropical varieties. In the formulation for hypersurfaces, he goes back to Mikhail Kasparnov . ^{Explanation before theorem 3.1.3.}

The structure theorem for tropical varieties links tropical geometry with the discrete geometry of polyhedra. ^{Theorem 3.3.6.}

Applications of tropical geometry

Applications in algebraic geometry

In 2007, Matt Baker suspected how tropical geometry could provide a new approach to the Brill and Noether theorem , originally proven in 1980 by Phillip Griffiths and Michael Harris , which makes statements about the connection between the topological gender of curves and their degree and rank. Evidence was provided by Sam Payne and colleagues. The set is named after Max Noether and Alexander von Brill , who set it up in 1874, but only Griffiths and Harris provided strict evidence. In 2017/18 David Jensen and Dhruv Ranganathan and Jensen and Sam Payne showed how to prove a tightening of Brill-Noether's theorem with tropical geometry (strong conjecture of the maximum rank).

Applications of tropical geometry in enumerative geometry

With the help of tropical geometry, Welschinger invariants (according to Jean-Yves Welschinger ) can be calculated in enumerative geometry .

Applications of tropical geometry in computer science

In computer science , tropical matrix multiplication is used in algorithms for the shortest paths in graphs and networks. The framework for such algorithms is known as dynamic programming. In computer science, the main application is the Min-Plus-Algebra.

Applications of tropical geometry in economics and business administration

In economics and business administration , the methods of tropical geometry are used for the distribution of emergency loans in the context of financial crises, the distribution of products / shopping baskets in the context of auctions and the calculation of coalitions. In economics and business administration, the focus is on Max-Plus-Algebra.

Baldwin / Klemperer's theorem of unimodularity

A competitive equilibrium then exists for every pair of concave utility functions of demand type D for all relevant bundles of supply, if D is unimodular.

A set of vectors im is unimodular if every linearly independent subset of the vectors can be expanded to a basis im with the determinant ± 1. ^{Definition 4.2.} ${\displaystyle Z^{n}}$ ${\displaystyle R^{n}}$

Applications of tropical geometry in physics

In physics , the methods of tropical geometry are used in high energy physics . For example, there are applications in the theory of supersymmetric fields. Tropical geometry is used in physics to find equilibrium positions in the gravity field of four bodies. In physics, the main application is the Min-Plus-Algebra.

Applications of tropical geometry in biology

In computational biology, many algorithms for gene prediction and sequence alignment are based on dynamic programming , these algorithms being a further development of the tropical polynomial. The interpretation of the algorithms in dynamic programming is particularly useful for drawing statistical conclusions. In computer biology, the main focus is on the Min-Plus-Algebra.

Individual evidence

1. ↑ ^{a b c d e f g h i j k l m n o} Diane Maclagan and Bernd Sturmfels: Introduction to Tropical Geometry . In: Graduate Studies in Mathematics . tape 161 . American Mathematical Society, Providence 2015, ISBN 978-0-8218-5198-2 ( PDF; 2.7 MB ). PDF; 2.7 MB ( Memento of the original from April 23, 2016 in the Internet Archive ) Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice.  
2. ↑ Grigory L. Litvinov: The Maslov dequantization, idempotent and tropical mathematics: A brief introduction. S. 2-3 , arxiv : math / 0507014 .
3. ↑ George M. Bergman: The logarithmic limit-set of an algebraic variety. In: Transactions of the American Mathematical Society . tape 157 , 1971, p. 459-469 .
4. ↑ Mikhalkin: Enumerative tropical algebraic geometry in ${\displaystyle \mathbb {R} ^{2}}$ . 2005.
5. ^ Andreas Gathmann: Tropical algebraic geometry . 2006, Conclusion.
6. ↑ Jeffrey and Noah Giansiracusa: Equation of tropical varieties , Duke Math. Journal, Volume 165, 2016, pp. 3379-3433
7. ^ Diane Maclagan, Felipe Rincon: Tropical schemes, tropical cycles, and valuated matroids, J. Europ. Math. Soc., Arxiv 2014
8. ^ Jean-Eric Pin: Tropical Semirings . ( PDF; 215 KB ).
9. ↑ J. Gunawardena: Idempotency . In: Publ. Newton Inst. Band 11 . Cambridge University Press, Bristol 1994, Introduction, pp. 50-69 .
10. ^ Simon: Recognizable sets with multiplicities in the tropical semiring. In: Mathematical Foundations of Computer Science . Carlsbad 1988, p. 107-120 ( PDF; 194 kB ).
11. ↑ Mathoverflow: What's tropical about tropical algebra? September 23, 2011, accessed September 10, 2017 . 
12. ↑ Hannah Markwig: Tropical Geometry. In: Katrin Wendland, Annette Werner (Hrsg.): Multifaceted mathematics. Vieweg + Teubner Verlag, Wiesbaden 2011, ISBN 978-3-8348-1414-2 , chapter 15, page 295
13. ↑ Grigory Mikhalkin, Johannes Rau: Tropical geometry . 2015, Remark 1.2.2. .
14. ^ Baker, Specialization of linear systems from curves to graphs , Arxiv 2007
15. ↑ Filip Cools, Jan Draisma, Sam Payne, Elina Robeva, A tropical proof of the Brill-Noether-Theorem, Advances in Mathematics, Volume 230, 2012, pp. 759-776
16. ↑ Jensen, Ranganathan, Brill-Noether theory for curves of a fixed gonality , Arxiv 2017
17. ^ Jensen, Payne, Tropical independence II: the maximal rank conjecture for quadrics, Algebra Number Theory, Volume 10, 2016, pp. 1601-1640, Arxiv
18. ^ Jensen, Payne, Effectivity of Farkas classes and the Kodaira dimensions of and ${\displaystyle M_{22}}$ ${\displaystyle M_{23}}$ , Arxiv 2018
19. ↑ Kevin Hartnett, Tinkertoy Models Produce New Geometric Insights , Quanta Magazine 2018
20. ↑ Mikhalkin, G .: Enumerative tropical geometry in R ^ 2. J. American Mathematical Society. 18, 313-377 (2005)
21. ^ ^{A b c} Elisabeth Baldwin and Paul Klemperer: Understanding Preferences: “Demand Types”, and the Existence of Equilibrium with Indivisibilities . Oxford University October 9, 2016, p. 1.22 .
22. ↑ Barak Kol: Tropical geometry and high energy physics, in: Tropical Aspects in Geometry, Topology and Physics . Ed .: Mathematical Research Institute Oberwolfach. 2015.
23. ↑ Ashoke Sen : Tropical curves, wall crossing and supersymmetric field theory . 2012.
24. ↑ Hampton, M., Moeckel, R .: Finiteness of relative equilibria of the four-body problem. Inv. Math. 163, 289-312 (2006)
25. ^ Lior Pachter and Bernd Sturmfels: Algebraic statistics for computational biology . Cambridge University Press, 2005.

literature

Textbooks:

• Diane Maclagan, Bernd Sturmfels : Introduction to tropical geometry. American Mathematical Society, Providence 2015, ISBN 978-082-185-198-2 .
• Grigory Mikhalkin , Johannes Rau: Tropical geometry. 2015, book in preparation, ( PDF; 2.9 MB ).
• Mark Gross  : Tropical geometry and mirror symmetry. American Mathematical Society, Providence 2011, ISBN 978-0-8218-5232-3 .
• Ilia Itenberg, Grigory Mikhalkin , Eugenii Shustin: Tropical algebraic geometry. Birkhäuser, Basel 2007, ISBN 978-3-7643-8309-1 .
• Matthew Baker , Sam Payne: Nonarchimedean and Tropical Geometry, Simons Symposia, Springer, 2016, ISBN 978-3319309446 .

Brief introductions:

• Erwan Brugallé, Ilia Itenberg, Grigory Mikhalkin, Kristin Shaw: Brief introduction to tropical geometry. Gokova Geometry / Topology Conference, 2015, arxiv : 1502.05950 [math] .
• Hannah Markwig : Tropical Geometry. In: Katrin Wendland , Annette Werner (Hrsg.): Multifaceted mathematics. Vieweg + Teubner Verlag, Wiesbaden 2011, ISBN 978-3-8348-1414-2 , chapter 15.
• Andreas Gathmann : Tropical algebraic geometry. Annual report of DMV 108 (2006) no. 1, 3–32, arxiv : math / 0601322 .
• David Speyer, Bernd Sturmfels: Tropical Mathematics. 2004, arxiv : math / 0408099 .
• Jürgen Richter-Gebert , Bernd Sturmfels, Thorsten Theobald: First steps in tropical geometry. 2003, arxiv : math / 0306366 .

Scripts:

• Melody Chan: Tropical Geometry, Harvard, 2013, PDF; 777 kB

More publishments:

• Antoine Chambert-Loir : Tropical Geometry. The skeleton of the amoeba . In: Spektrum der Wissenschaft 2019, no. 6, pp. 12-17
• Grigory Mikhalkin: Enumerative tropical algebraic geometry in . ${\displaystyle \mathbb {R} ^{2}}$J. Amer. Math. Soc. 18 (2005), no. 2, 313-377, arxiv : math / 0312530 .
• Grigory Mikhalkin: Tropical geometry and its applications. Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 arxiv : math / 0601041 .
• David Speyer: Tropical Geometry. Dissertation, University of California, Berkeley, 2005, ( PDF; 731 kB ).

Web links

• Tropical geometry in nLab
• Lecture series with Bernd Sturmfels
• Brugallé, Itenberg, Shaw, Viro: Tropical Geometry , Oberwolfach, Mathematical Snapshots 2018