Mathematical Sciences: Number of Defining Equations, Local Cohomology

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This project is concerned with the study of some interrelated questions of local cohomology, etale cohomology and the number of defining equations of algebraic varieties. Some progress has been made on these problems recently, but much remains to be done. It is still unknown whether every non-singular subvariety of affine space is a set-theoretic complete intersection. The structure of local cohomology modules remains a mystery. The relationship between the cohomological dimension and the etale cohomological dimension is still unclear. Some technical questions about local etale cohomology remain open. The principal investigator will study these questions using methods that have been successful in the past as well as developing some new methods. Algebraic geometry is the study of the geometric objects arising from the sets of zeros of systems of polynomial equations. This is one of the oldest and currently one of the most active branches of mathematics. It has widespread applications in mathematics, computer science and physics.