Groebner Fans in Combinatorics, Representation Theory and Commutative Algebra: Theory and Computation

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Abstract for award DMS-0401047 of Thomas: Groebner basis theory is today a central theoretical and computationaltool in algebraic geometry and commutative algebra with applicationsin myriad fields such as computer science, optimization, robotics,statistics, and computer modeling. The Groebner fan of an ideal ormodule is the decomposition of the space of weight vectors that induceGroebner base, such that each cone in the fan indexes a distinctGroebner basis. This project describes four problems fromcombinatorics, optimization, representation theory and algebraicgeometry that revolve around Groebner fans. The first projectinvestigates the complexity, geometry and homological properties ofinitial ideals of toric ideals. Answers to these questions haveimplications in integer programming and on the computation of toricGroebner bases which themselves have several applications. The next examines the variation of weight vectors in Kalai's theory ofalgebraic shifting for simplicial complexes. The third projectstudies the polyhedral geometry of moduli of resentations of the McKayquiver which (partially) resolve singularities that arise ingeneralized McKay correspondence. The last goal is to develop asoftware package for computing Groebner fans of arbitrary polynomialideals. Applications include the computation of tropical varieties andthe study of orbit closures and degenerations of representations offinite dimensional algebras.The proposed work will employ tools from combinatorics, discretegeometry and optimization to solve problems from commutative algebra,representation theory and algebraic geometry. All projects have astrong computational component that aims at effective algorithms andserves as a laboratory for uncovering new structure theorems. Thecollaborations include three graduate students and three current orrecent postdocs.