PhD project offering: tropical scheme theory

Tropical geometry has burgeoned in the decade and a half since its coalescence as a field of study. It is a combinatorialization of algebraic geometry, associating to each algebraic variety a so-called tropical variety, a polyhedral complex that is its combinatorial “shadow”. In the simplest cases tropical geometry can be seen as the algebraic geometry over the tropical semiring T = (R ∪ {∞}, min, +). Tropical methods have been quite effective for certain classes of problems in algebraic geometry, particularly in enumeration and in intersection theory: one concrete example has been finding explicit recursions for a class of Gromov-Witten invariants in the plane [1]; others include applications to the Brill-Noether theorem and mirror symmetry. Outside algebraic geometry, applications to fields like optimisation and mathematical biology are also numerous. However, tropical varieties, as standardly defined, behave more like algebraic cycles than varieties per se. For instance, the operation of tropicalization factors through the map from a variety to the associated algebraic cycle [5]. Tropical intersection theory has been defined, and one finds that it is essentially a limit of algebraic intersection cohomology of toric varieties [2]. Recent work [3, 4] has introduced the definition of a scheme into tropical geometry, and with it the opportunity to replicate within tropical geometry such scheme-theoretic notions as non-reduced structure, tangent bundles, and syzygies. The aim of the proposed research would be, on one hand, to take part in laying the foundations of tropical commutative algebra that will underlie the scheme theory, and on the other, to develop applications of this theory, for which many potential directions exist, including intersection theory, lifting problems, deformation theory, or classification of curves. In particular, I will have a postdoctoral researcher working on problems related to tropical Chow varieties and Hilbert schemes, with whom collaborations could be envisaged.