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.
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