Foundations and Applications of Tropical Geometry

Tropical geometry is geometry over the tropical semiring, which is the usual real numbers with addition replaced by minimum, and multiplication replaced by addition. While the tropical semiring has had applications in optimization and computer science for several decades, the connection to algebraic geometry was first made only at the start of this century. The subject has expanded rapidly in the past decade.

Classically algebraic geometry studies the geometry of the sets of solutions to polynomial equations, called varieties. The fundamentals of algebraic geometry changed dramatically in the 60s with Grothendieck's introduction of schemes. In the first decade of tropical geometry, however, only tropical versions of varieties were considered. This has changed in the past few years, beginning with the work of the Giansiracusas, and of the PI with Rincon, which together introduced a scheme theory into tropical geometry. This allows much more of the power of modern algebraic geometry to be used in tropical geometry.

The primary aim of this project is to further develop the theory of tropical schemes, and apply this technology to problems in algebraic geometry.

The first goal is to develop more of the basic commutative algebra and algebraic geometry of the new theory of tropical schemes. This will then be used to construct tropical versions of important moduli spaces in algebraic geometry, starting with the Hilbert scheme, and use this to address fundamental open questions about the Hilbert scheme. It will also be used to address realizability questions in tropical geometry, which have applications to birational geometry.

The fact that tropical geometry reduces algebraic geometric objects to simpler combinatorial ones allows techniques from other areas to be used, and also more direct application of ideas from algebraic geometry to outside mathematics.

One striking example of this latter point is the use of tropical geometry by Paul Klemperer (Oxford) to design a new type of auction (a "product-mix auction") that was used by the Bank of England during the financial crisis to arrange loan terms. The key idea here is that each "bundle" of goods (such as collections of debts of varying credit risk) determines a tropical hyperplane, and the optimal pricing is determined by the region of the associated hyperplane arrangement. This used ideas that were significantly less than a decade old at the time, which is highly unusual for this style of pure mathematics, which usually has a much longer time to impact.

A slightly older example of the impact of tropical mathematics comes from optimization. Optimization over the tropical semiring was used around 15 years ago to optimize the entire Dutch intercity train network.

These examples indicate the possibility of true impact of pure mathematics in this area. To maximize the chance of this happening, the PI will participate in conferences attracting a mixture of mathematicians and people interested in applying mathematics, such as the SIAM applied algebraic geometry meeting, and the LMS Tropical Network. She will continue to be in contact with Klemperer's group. She will also focus on implementing algorithms that arise from this project, and incorporate them into Macaulay2 software packages, organising a Macaulay2 workshop to help ensure that this happens successfully.