Mathematical Sciences: Interactions of Commutative Algebras with Analysis, Geometry and Computer Science

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9625308 Smith The Career Development Plan combines a research program in commutative algebra with an educational program for graduate and undergraduate students at the University of Michigan. Most of the proposed research is part of a broader program to illuminate the connections between analytic ideas such as differential operators and $L^2$-estimates with characteristic $p$ techniques in commutative algebra such as tight closure. The goals of the proposed research are: (1) to apply characteristic $p$ techniques towards a solution of Fujita's Conjecture in algebraic geometry; (2) to understand vanishing theorems for cohomology of line bundles on complex varieties in terms of tight closure; (3) to study the ring theoretic properties of differential operators on non-smooth varieties by using reduction to characteristic $p$ methods; and (4) to study a particular instance where there appears to be a direct link between $D$-modules and modules with an action of Frobenius. Parts (I) and (II) of the Career Development Plan outline specific methods by which the research will be conducted. The main tools are closure operations on ideals, such as the tight closure and the integral closure, and the Frobenius action on local cohomology. A computer project suitable for a graduate student is proposed as a method for understanding the module structure of a ring over its ring of differential operators. Part (III) proposes a four year plan to teach both graduate and undergraduate students projective algebraic geometry and computer algebra while engaging them in state of the art research. The proposed project is a greatly expanded version of research already in progress on sparse systems of parameters and Noether Complexity. This long term research and educational project includes plans for a new graduate course at the University of Michigan stressing the use of computers in algebraic geometry, and for a junior level seminar for majors in mathematics and computer scien ce in which teams of undergraduate students led by a graduate student learn projective geometry through self-discovery. The research begun in this seminar will be expanded to a full scale research project on computer algebra. This project will culminate in a conference in which students present their findings to a broader audience. Part (IV) addresses the educational and human resources aspects of thePIs career goals. A program to bring real-life practitioners of mathematics in business and industry to the undergraduate classroom is proposed. The objectives are to simultaneously motivate the curriculum and introduce students to people like themselves who have used mathematics to create a successful career.