Research at the Interface of Algebraic Geometry and String Theory

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The award supports the principal investigator's research at the interface of algebraic geometry and string theory. Algebraic geometry is the mathematical study of spaces described by arbitrary algebraic equations. Fundamental investigation of such diverse scientific disciplines as high energy physics, cryptography, phylogenetics, robotics, or control theory, often reveals that key concepts of the discipline can be encoded in terms of such geometric spaces. In some cases, such interactions suggest deep new problems in algebraic geometry whose solution is necessary for further progress. In other instances, the scientific intuition actually suggests new methods for solving old problems in algebraic geometry that were otherwise inaccessible. String theory and quantum field theory (QFT) explore physics at the smallest length scales, or correspondingly at the highest energy levels. Exploration of the interactions of these physical theories with algebraic geometry has been extremely productive for both math and physics, and the power of this combination of tools and approaches only seems to strengthen with time.The goal of this project is to explore and push forward the interface of algebraic geometry with string theory. This will be done by focusing on a number specific research directions, each representing a major open problem in math and/or in physics, whose solution will make a major contribution to the field, and is likely to benefit from the application of techniques of the opposite discipline. Specifically, the principal investigator proposes to explore the extension of the classical theory of curves and their moduli to super Riemann surfaces, with a view towards establishing the foundations of perturbative superstring theories and studying the superstring measure; to prove the geometric Langlands conjecture via non abelian Hodge theory, and explore its relation to QFT and to mirror symmetry; to extend his construction of Calabi-Yau integrable systems realizing Hitchin's system to meromorphic and parabolic versions, and explore the physical applications; to use his new parametrization of the moduli space of 6 dimensional principally polarized abelian varieties to analyze this space and determine its Kodaira dimension; and to explore further aspects of F theory and attempt to establish its mathematical foundations.