Combinatorics in geometry and representation theory

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The proposed project studies combinatorial problems arising from representation theory and algebraic geometry. A particular focus is on problems arising from the study of loop groups, or affine Lie algebras. The PI with collaborators will study the Schubert calculus of flag varieties of affine Lie groups, and the relations to quantum cohomology. This includes understanding the geometry of Schubert varieties and the combinatorics of the polynomials representing Schubert varieties in (co)homology. Together with collaborators, the PI also plans to develop a theory of total positivity for loop groups, aiming to extend work of Lusztig for finite-dimensional reductive groups. Connections with the theory of electrical networks, and to graphs embedded in surfaces will be investigated. Another direction of research is the study of infinite reduced words in infinite Coxeter groups. The PI will study partial orders on infinite reduced words, and explore probabilistic aspects of random infinite reduced words.Combinatorics is the study of discrete structures. Such structures come up in varied contexts throughout science. One such structure that arises in the proposed work is an electrical network consisting of resistors. A basic problem is to understand the relationship between the electrical properties of the network and the connectivity properties of the network. These problems have applications to electrical impedance tomography, a technique in medical imaging. Another type of discrete structure studied in the proposed work are structures with a large number of symmetries, including for example tilings of high-dimensional space in a regular pattern. A basic problem with these structures is to classify them and analyze the large scale behavior. Such structures are important for example in coding theory.