CAREER: Test Ideals and the Geometry of Projective Varieties in Positive Characteristic

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The Principal Investigator will explore questions in characteristic p 0 algebraic geometry and commutative algebra. Explicitly, he proposes to apply methods of commutative algebra to study adjoint line bundles in characteristic p 0, especially to produce global sections. Such line bundles are crucial when classifying the central objects of algebraic geometry. Over the complex numbers, the standard techniques to study adjoint line bundles include Kodaira-type vanishing theorems. These theorems are unavailable in characteristic p 0, thus methods of the Frobenius morphism and test ideals from commutative algebra will be employed instead. The PI also proposes to use methods of algebraic geometry to study questions related to singularities in commutative algebra. The Principal Investigator will also run several mathematics camps for high school students. He will employ students to help develop software related to test ideals and singularities in characteristic p 0. Finally he will continue to organize conferences.Algebraic geometry is a very active field of mathematics. Explicitly, it is the study of geometric objects (called algebraic varieties) made up of the solutions to polynomial equations (for example, the parabola is the solution to y = x^2). Characteristic p 0 algebraic geometry is the study of these equations and their solutions when the underlying rules of arithmetic have changed. For example, instead of simply 4 + 4 = 8, in characteristic p = 5, once we get to 5 we start counting at 1 again so that 4 + 4 = 3 (like counting hours of a clock). This type of arithmetic and related algebra and geometry is heavily employed in modern cryptography and coding theory. The running of camps, conferences and the development of software will also help train the next generation of researchers.