**RESEARCH IN PAIRS**

**Quadratic Forms Over Supertropical Semirings: Trigonometry and Convex Geometry**

*Formes quadratiques sur des semi-anneaux super-tropicaux: trigonométrie et géométrie convexe***22 October - 2 November 2018**

The theory of tropical mathematics has shown a tremendous development in recent years that both established the field as an area in its own right and unveiled its deep connections to numerous branches of pure and applied studies. This theory combines several fields of study and aims for a better understanding of the interplay among algebraic, combinatorial, and geometric features of tropical mathematics. Beside its own theoretical significant, the theory has numerous applications in diverse areas of study including computer science, physics, finance, and computational biology. It provides a natural algebraic formulation of objects which were previously not accessible, as well as a new approach to address problems such as representations of semigroups and realizations of discrete combinatorial objects. The merit of this theory is the ability to translate problems from one domain of study to another, and thus to employ mathematical methods associated with one domain of study in another domain.

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.

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.

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.

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