Mathematical Sciences: Tight Closure and Absolute Integral Closures, Briancon-Skoda Theorems and Graded Rings

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9413870 Aberbach This award supports research in commutative algebra. The principal investigator will explore the structure of the absolute integral closure of excellent local rings. He will also study the Cohen-Macaulayness of graded rings for ideals of higher analytic deviation. This research is concerned with a number of questions in commutative algebra and algebraic geometry. Algebraic geometry studies solutions of families of polynomial equations. One can either study the geometry of the solution set or approach problems algebraically by investigating certain functions on the solution set that form what is called a commutative ring. This dual perspective creates a close connection between commutative algebra and algebraic geometry.