Tropical geometry and integrable systems

The proposal seeks funding to support an international conference on integrable systems and tropical geometry. For the UK's traditionally strong standing in the research area of integrable systems it is vital to maintain an active communication between pure and applied mathematicians and to connect with emerging new research topics.

The relatively young discipline of tropical geometry is such an emerging field within algebraic geometry having strong overlaps with combinatorics and enumerative geometry. Central are the notions of the min-plus algebra and tropical curves which independently appeared in the area of integrable systems when taking the so-called ultradiscrete limit of soliton equations. The declared aim of the proposed conference is to connect the two communities, integrable and tropical, and "jump start" a mutual awareness of developments in each field, a discussion process to find a common language and in the long term the formulation of common research interests resulting in high quality papers and grant applications transgressing traditional subject boundaries. It will contribute to the UK's strong research position in the mathematical sciences and increase its potential to harness new mathematical insight for applications in the sciences, applied and pure.

Tropical geometry has already found applications in statistical sciences and biology, more specifically phylogenetics where one tries to classify the genetic codes of animals and humans. The proposed conference is a first step to explore new avenues for applications of tropical geometry related to integrable systems. The latter appear for instance in the study of nonlinear optics and telecommunication, plasma physics, and ocean, atmospheric and planetary sciences.

"Classical" integrable systems, or soliton, theory has found practical importance in aeronautical design and in optical fibre communications systems. The modern, discrete approach will therefore find a place in comparable contexts where the discrete or particulate nature of the medium is being engineered, for instance in digital transmission systems and the modelling of smooth surfaces by polyhedra either in construction or representation problems. Indeed the area of tropical geometry or algebra alone already finds application in integer programming and optimization problems. Tropical algebras also provide examples of non-classical logics liable to be of use in the application of category theory to theoretical computing. Further applications of tropical geometry include phylogenetics and statistical modelling.

To reach these other disciplines and disseminate the results of the scientific exchange beyond the communities participating in the workshop, we will use newly set-up web-forums, websites and contacts within the recently formed College of Science of Engineering at the University of Glasgow publishing a non-technical, widely accessible account of the new research field and its possible applications.

The reduction of problems to a combinatorial level also has the advantage of making them accessible to an audience with a less technical background. This makes the subject a prime candidate for popularization and an influence on the undergraduate teaching curriculum. A wider audience more aware of the possibilities of mathematical insight is one that will more readily engage in scientific and technological innovation crucial to the economy at every level.

In addition the topic of the conference will be formative of the public perception of mathematics through its influence on the presentation of undergraduate courses in mathematics at Glasgow and other Universities, articles in the University Newsletter and Magazine, the recruitment of students through media events such as University Open Days and on the provision of Master Class events for school pupils. We will also arrange for a public lecture-type event on the island of Islay whilst the conference is in progress.