Supertropical Matrices: Matrix Semigroups and Quadratic Forms

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 many 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, introduced by the PI, 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. Developing this theory requires building a solid theoretical foundation, incorporating new concepts, which includes the establishment of fundamental algebraic notions tier by tier. This process has been carried out in an extensive series of the PI's papers that deal with structure theory, polynomial algebra, matrix and linear algebra, and basic polyhedral geometry; all provide the building blocks of this theory.

The proposed research is a step forward in the evolution of supertropical algebra, enhancing the study of matrices with emphasis on matrix semigroup and quadratic forms. In this theory, matrices have a special importance as they correspond uniquely to weighted digraphs (now possibly with multiple edges) and intimately compose in graph theory. Supertropical matrices have a rich structure that permits an easy incorporation of methods from combinatorics that usually involve sophisticated computational aspects, but become transparent in the supertropical setting. These attributes makes the theory utilizable for realizations of complicated topological-combinatorial objects (e.g., quivers, matroids, or simplicaial complexes), assisting in their analysis.

A matrix semigroup can therefore be viewed as a collection of combinatorial objects, where its algebraic properties specify the behaviour of these objects. Semigroup identities provide a characterization of a collection of matrices as a whole, while quadratic forms relate to each matrix as an individual. Both provide useful insights on matrices, reflected in their combinatorial view. This proposal utilizes these perspectives, together with a systematic study of supertropical matrices, based on varied disciplines (algebra, semigroup theory, and combinatorics), to better understand families of matrices and their invariants.

In the long run, the goal of this study is to develop a theory of supertropical algebraic semigroups and supertropical K-theory, analogous to those in classical mathematics, that in addition to their algebraic significance also have a deep topological-combinatorial meaning.

The proposed research is a contribution in tropical mathematics, with a direct impact in pure mathematics, first in tropical algebra and semigroup theory and later in algebraic geometry, combinatorics, operator algebra, valuation theory, and model theory. The introduction of supertropical matrix semigroups and their utilization for semigroup representations will bring in new methods for studying semigroup endomorphisms and identities. In future, it will pave the way for developing a supertropical K-theory and a theory of algebraic semigroups, as a tropical analogous to group varieties, which will have a future impact on algebraic geometry.

Tropical algebra generalizes the familiar boolean algebra, which is at heart of computer science, and is strongly connected to automata theory and formal languages that are central subjects of study in computer science and will be directly advantaged by the outcome of this project. In the long run, beside its theoretical influence, the project may potentially have a contribution to applications in other areas of study (as tropical theory already has), including: finance, analysis of discrete event systems, control theory, combinatorial optimization, string theory, phylogenetics, and computational biology.

The project impact will be achieved through writing research papers on the results, to be available on preprint servers, publishing them in leading academic journals, and publicising the work at conferences and seminars throughout the UK and abroad to amplify the impact of our work.

Pure mathematics is an abstract subject that is pursued for its intrinsic worth, providing also computational and modeling tools for many branches of science. Many of these tools, whose usefulness is now unquestioned, were designed firstly by pure interest and only years later being utilised for real applications. Nowadays mathematics is challenged by new needs of modern science, requested to model phenomena or objects having a discrete nature. As tropical algebra has already been shown to provide good solutions for these needs, we expect that it will continue to play a significant role in future studies.