Algebraic Combinatorics

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Fomin 9700927 The research project is devoted to exploring new relations between combinatorics and algebra, in particular representation theory and algebraic geometry. An approach to total positivity based on generalized pseudo-line arrangements suggests new developments and applications ranging from combinatorics of canonical bases in quantum groups to numerical and symbolic algorithms of linear algebra. The combinatorial problems arising in the theory of quantum cohomology provide another promising direction of research. The proposal outlines a plan for a detailed study of quantum Schubert polynomials. This research concerns the interplay between combinatorics and algebraic geometry. One of the goals of combinatorics is to find efficient methods of studying how discrete collections of objects can be arranged. The behavior of discrete systems is extremely important to modern communications. For example, the design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis, topology, and combinatorics, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.