I think this is the newest topic of mathematics I have ever encountered. Also, I am almost blind about everything that involves it and it was (still is) a challenge for me to write about this. Thank you for those that helped me in my search and understanding of this topic. If you know more about it let me know in the comment box bellow.
Tropical geometry is a relatively new area in mathematics, which might loosely be described as a piece-wise linear or skeletonized version of algebraic geometry. Its leading ideas had appeared in different forms in the earlier works of George M. Bergman and of Robert Bieri and John Groves, but only since the late 1990s has an effort been made to consolidate the basic definitions of the theory.
It is a young subject that has undergone a rapid development since the beginning of the 21st century. While establishing itself as an area in its own right, deep connections have been made to many branches of pure and applied mathematics.
Tropical geometry is based on tropical algebra, where the sum of two numbers is their minimum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These tropical varieties retain a surprising amount of information about their classical counterparts. Tropical geometry naturally lies in the intersection of geometric combinatorics and algebraic geometry. It is basically looking at the tropical (i.e. combinatorial) picture first and then trying to transfer the results back to the original algebro-geometric setting.
There are a number of articles and surveys on tropical geometry. The study of tropical curves (tropical hypersurfaces in ℝ2) is particularly well developed. In fact, for this setting, mathematicians have established analogues of many classical theorems; e.g., Pappus’s hexagon theorem, Bézout’s theorem, the degree-genus formula, and the group law of the cubics all have tropical counterparts. The strengths of tropical methods come from the fact that the tropical objects are intrinsically combinatorial, and computations can go farther on combinatorial objects than on algebro-geometric objects.
It has a wide range of applications in enumerative geometry, mirror symmetry, computational algebra, optimization, algebraic statistics, and computational biology. Better understanding of combinatorial structures in tropical geometry has led to new algorithms and formulas in enumerative geometry and computational algebra. Moreover, tropical geometric objects have rich combinatorial structures that also arise naturally in discrete geometry and combinatorial algebra, such as graphs, subdivisions and triangulations of polytopes (a polytope is a geometric object with flat sides, and may exist in any general number of dimensions), fiber polytopes, just to name a few. Tropical methods can be used to develop algorithms and software for solving classical problems in computational commutative algebra.
Tropical geometry was used by economist Paul Klemperer to design auctions used by the Bank of England during the financial crisis in 2007. Here is a video which explains a little more about what happened then:
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. 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. A tropical counterpart of Abel-Jacobi map can be applied to a crystal design. The weights in a weighted finite state transducer are often required to be a tropical semiring.
There are a couple of more mathematical and interesting explanations for why we need tropical geometry on Mathoverflow.
For those that want to start learning about this topic more, I recommend the article Tropical Mathematics by David Speyer and Bernd Sturnmfels. The article offers 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 in the sections seems to be well suited for undergraduate students. Next on the list should be the book Tropical Geometry by Diane Maclagan and Bernd Sturmfels. Also of these are free on the internet and that is why I think they should be a good thing to check first.
Just as you are used to so far, here are some book recommendations for this topic:
There are many other sources for learning about tropical geometry and I advice you to search and give it a try. At least a small glance over it and some of its wonderful concepts will make you see the world with different eyes.
Don’t forget that maths is everywhere! Enjoy! ~LThMath~