Periodic Reporting for period 1 - TROPDIFFGEO (Tropical Differential Geometry) Période du rapport: 2018-06-01 au 2020-05-31 Résumé du contexte et des objectifs généraux du projet "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." Travail effectué depuis le début du projet jusqu’à la fin de la période considérée dans le rapport et principaux résultats atteints jusqu’à présent "The work carried out on this project was typical for research in pure mathematics. We had in mind several extensions of the Fundamental Theorem which would allow it to be applied in more general contexts. By studying the properties of certain examples we were able to show that some of the extensions we hoped for were not in fact true. For example, if the set of numbers that we allow to appear in our solution is too small -- say, just fractions, instead of any complex numbers -- then there might be ""fake"" tropical solutions which can't be turned back into genuine solutions.In other cases the work consisted of coming up with mathematical proofs that our extensions were true. For example, we were able to do this for the case of ""partial differential equations"", where the equations can contain derivatives with respect to more than one variable. Partial differential equations are also very common in mathematical models of the world: for example, the heat equation, that describes how heat diffuses through an object which is not all at the same temperature, refers to change not only in time but also in each direction in space.We have disseminated our results by writing papers which we have submitted for publication in scientific journals and made freely available on the arXiv, the mathematics preprint server: see . We have also given seminar talks on the work in several countries and online: the fellow has given seven talks, and the experienced researcher one more.We also held a workshop in December 2019 at Queen Mary University of London . In addition to a targeted dissemination opportunity, bringing together a collection of experts in nearby fields and scheduling some focussed time to work in groups with them has kickstarted more research in the area." Progrès au-delà de l’état des connaissances et impact potentiel prévu (y compris l’impact socio-économique et les conséquences sociétales plus larges du projet jusqu’à présent) The project has now ended. Each of the results described in the previous passage constitutes an extension of the state of the art, compared to 2.5 years ago when this grant started. Still, we intend to continue work on some outstanding projects. One of these is aimed at another generalisation of the Fundamental Theorem, to handle solutions involving only real numbers. The statement we currently have uses complex numbers, but in real-world differential equations usually only real numbers are wanted. We will also be seeking out particular problems in applied mathematics where the tools we have developed may provide insights; to this end we're currently in conversation with groups of applied researchers.As a basic research project in pure mathematics, likely impacts to economy and the society are not immediately identifiable. As innovations are made in applied mathematics or physical or social sciences, generating new mathematical problems to be solved, any one of them could turn out to be solved by the theory we have developed, and enable technological or social improvements from there. A differential operator acting on a tropical polynomial.
