Geometric Applications of Non-Abelian Hodge Theory

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Pantev 9800790 Pantev will use techniques from abelian and non-abelian Hodge theory to approach concrete geometric questions as well as problems in mathematical physics and string theory. Four problems will be studied. The first one is to look for a construction of a linear algebraic object that is canonically associated to a projective variety X and calculate the cohomology of all local systems on X together with their Hodge structures whenever the latter makes sense. The second concerns the non-abelian Hodge conjecture on curves and provides an intrinsic geometric characterization of the locus of motivic local systems. The third discusses a construction of a singular plane curve whose complement has a non-residually finite fundamental group. The fourth project suggests a geometric strategy for understanding the appearance of enhanced gauge symmetry in string theory and proposes a method for analyzing the exotic components of F theory moduli spaces. The understanding of these questions is essential for unifying various linearization procedures in algebraic geometry and mathematical physics. On one hand, it will bring us closer to understanding the basic structure of algebraic varieties, and on the other hand it will bring a geometric perspective to the interaction of the recently discovered M and F string theories. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one that has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.