Principal Investigator: Kevin J. Costello
The moduli spaces of Riemann surfaces play a central role in many areas of mathematics and theoretical physics, including geometric and algebraic topology, symplectic topology, string theory, and number theory. This project explores another manifestation of the moduli spaces of Riemann surfaces, which is maybe not so widely known. This is their appearance in homological algebra, and more precisely in the study of cyclic A-infinity algebras (a kind of "Frobenius algebra up to homotopy"). The philosophy underlying much of this project is that everything one can say about the homotopy types of the various moduli spaces can be expressed in terms of the homotopy theory of cyclic A-infinity algebras. One precise manifestation of this philosophy (which will be proved in this project) is that the moduli space of surfaces arises as certain "homology operations" for cyclic A-infinity algebras. This theoretical result will be used to investigate some concrete higher-genus aspects of the mirror symmetry conjecture, which plays a prominent role in current work on algebraic and symplectic geometry. In particular, this project will attempt to compute the conjectural mirror partner of the higher-genus Gromov-Witten invariants in some examples.
The space of all possible two-dimensional shapes -- known as the moduli space of surfaces -- has long been a fundamental object of study in many areas of mathematics, from geometry to number theory. This space also plays an important role in string theory, the putative "theory of everything". This project is concerned with setting up a correspondence between two-dimensional geometry (the moduli space of surfaces) and a kind of abstract algebra. This correspondence will be used to test certain mathematical conjectures coming from string theory. String theorists have predicted that two different simplified models of string theory-- known as the A model and the B model -- are mathematically equivalent. This prediction has stimulated a great deal of mathematical work in the last 15 years. The PI will tackle some computations in the B model which have been heretofore out of reach. The results of these computations will then be compared with known computations in the A model, hopefully leading to further verification of the predictions of string theory.
