Formes quadratiques sur des semi-anneaux super-tropicaux: trigonométrie et géométrie convexe
22 October – 2 November 2018
Tropical mathematics is carried out over idempotent semirings – a ”milder” structure than the structure of fields – which better suits for mathematical descriptions, incorporating a combinatorial view, of objects having a discrete nature; such objects arise frequently in modern studies and are often not accessible by classical theory. On the other hand, the cost of using semirings (as they lack subtraction) is the inapplicability of standard algebraic methods, sometimes even definitions of basic algebraic notions. Supertropical theory is established on an algebraic structure that enriches the tropical semiring, preserving all its advantages, and at the same time allows the recovering of classical algebraic concepts. The underlying algebraic structure of this theory is novel, yet a semiring which is not accessible by the classical algebraic approaches.
Objectives. The proposed research project is a step forward in the development of tropical algebra, more precisely in quadratic forms. It composes together knowledge in algebra, geometry, and combinatorics to study supertropical analogous of quadratic forms; tropical versions of these analogous are a particular case. As tropical objects appear as the so called “tropicalization” of families of classical objects, the proposed study also aims to provide a machinery for investigating these families. To reach these goals, the research program is divided into three main tasks, interacting with each other:
A. Exploring structures of quadratic forms on general semirings, including algebraic and geometric perspectives;
B. Analysis of Cauchy-Schwartz ratios that introduce an interesting type of (discrete) trigonometry.
C. Employing supertropical semigroups for studying basic convex geometry, first in ray spaces.
This research aims to provide the foundations for a theory of quadratic forms over semirings.