Invariant Theory and Algebraic Combinatorics

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In Invariant Theory, one studies algebraic expressions which remain the same under spatial symmetries. Such algebraic expressions are called invariants. For example, the distance to the origin does not change under the rotation of the plane around the origin. The distance to the origin is an invariant. The PI will study a new approach to classical invariant theory. In collaboration with mathematicians and engineers at UIUC, the PI will study applications of theory of subspace arrangements to computer vision and image compression.The PI will work on various other topics in commutative algebra, algebraic combinatorics, number theory. Cluster algebras were introduced by Fomin and Zelevinsky. These algebras have deep connections with the theory of quiver representations. The PI will explore this further in collaboration with Weyman and Zelevinsky. To a matroid or polymatroid one can associate various quasi-symmetric functions. The PI will study universal properties of such invariants.The PI will collaborate with Masser to find an effective version of the Mordell-Lang problem in positive characteristic.