Algebraic Geometry Inspired by Theoretical Physics

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This project will study several situations in algebraic geometry where, for reasons connected with mirror symmetry, spaces defined in terms of a Lie group are expected to have enumerative or cohomological invariants which are related, or equal, to those where the group is replaced by its Langlands dual. The aim is to compute these invariants and verify the predictions of mirror symmetry. Among the principal examples to be studied are: (1) the stringy Hodge polynomials of the spaces of flat connections on a Riemann surface with structure group a reductive Lie group; (2) the stringy Hodge polynomials of the moduli spaces of solutions to Nahm's equations of magnetic monopoles; (3) the quantum cohomology of a loop group.This is research in algebraic geometry, one of the most classical parts of mathematics, concerned with finding solutions of polynomial equations. But it is aimed at corroborating some very recent hypotheses about mirror symmetry, an exciting idea which has lately emerged from theoretical physics. Mirror symmetry proposes that certain string theories about the fundamental structure of the universe can be formulated mathematically in two seemingly different but equivalent ways. This project will explore the evidence supporting one of the recent suggestions for how to construct one of these formulations in terms of the other.