This work targets several problems in algebraic and geometric combinatorics. These problems have applications to many areas of pure and applied mathematics and theoretical physics, e.g., convex geometry, matroid theory, representation theory, algebraic geometry, commutative algebra, inverse boundary problems, total positivity, cluster algebras, theory of hyperplane arrangements, tropical geometry, branched polymers, theory of solitons, Lie theory, Schubert calculus, quantum cohomology, discriminants, and other areas. Many problems involve the study of special classes of convex polytopes, their combinatorial structure, valuations, and triangulations, and the study of related combinatorial objects. Combinatorics plays an increasingly important role in science. The project focuses on applications of combinatorial methods to algebraic and geometric problems. Its results will have impact in several areas of pure and applied mathematics and physics. This work will lead to development of new combinatorial techniques and constructions. Some topics from this project are suitable for undergraduate research. This project will increase public awareness of combinatorics and mathematics in general.
