TROPDIFFGEO · Tropical Differential Geometry

Tropical Differential Geometry

"Tropical mathematics (named in oblique tribute to Brazilian mathematician Imre Simon) is a branch of mathematics that replaces plus with taking the minimum or maximum, and times with plus. Many mathematical problems and real-world systems can be described by the solutions of a system of polynomial equations. Think for example of a circle, the solutions to x^2 + y^2 = 1. Tropical mathematics changes the shapes formed by these solutions from curved, usually smooth, shapes (like circles) to collections of flat pieces. It is easier to understand each flat piece, but there are a large number of them. So ""tropicalising"" is a way to move the complexity of a problem away from the algebra of solving the original equations, towards what's called combinatorics, i.e. keeping track of all the different possibilities for what flat pieces there are. If the combinatorics is easier overall, as in many cases it is, we've made progress on solving the problem. Here are some examples of successes of tropical approaches: - Reconstructing the tree of life. By studying the same gene in a collection of species, we can see what mutations each species has accumulated in that gene, and use that to build an evolutionary tree for that gene. But different genes can have different trees: what is the right tree for the species as a whole? It turns out that a possible evolutionary tree is the same thing as a tropical line in high-dimensional space, so the tropical geometry of these lines provides ways to compute the ""average"" tree. - Design of auctions. If many simultaneous items are to be auctioned and each bidder values different particular combinations of the items differently, it's hard to work out how the auctioneer can realise greatest profit. ""Product-mix auctions"" are a tropical solution to this problem, used by the Bank of England in the 2008 financial crisis and thereafter. This project set out to build tropical methods for working with differential equations. The equations best suited to the description of physical systems that change in time are not the simple polynomial equations above but differential equations, which can also refer to rates of change of the quantities involved. This topic was initiated by Prof. Dmitri Grigoriev, who invented algorithms for finding a solution of tropical differential equations in certain cases. The particular objectives of the project were to develop the Fellow's theory (""the Fundamental Theorem of Tropical Differential Geometry"") that gave a way to ""tropicalise"" differential equations, in the sense of the first paragraph."

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