We describe several problems in symplectic geometry which new results and methods, appearing in both Mathematics and Physics, may provide new avenues for. The geometry of the Lagrangian subvarieties of a symplectic manifold has been the subject of a great deal of current research. One version of this is a construction by Weinstein of the symplectic ``category'' whose objects are symplectic manifolds and whose morphisms are Lagrangian subvarieties. A recent monograph by Guillemin and Sternberg shows that Weinstein's category and variants of it are powerful tools in semi-classical analysis. We show hints that this category reveals symmetries in semiclassical systems which are not apparent in the underlying symplectic manifold. This may be a semiclassical version of a principle advocated by Witten, that a quantum system may be have much more symmetry than the underlying symplectic manifold. Another area we propose investigating is the relation between symplectic geometry and quantum manifold invariants. Many topological quantum field theories may be interpreted mathematically as counts of points in moduli spaces. One major exception seemed to be Chern-Simons gauge theory; in Physics language, this theory is not supersymmetric. Recent work in Physics (due to Beasley-Witten, Kapustin-Willett-Yaakov, Kallen, and others) has shows that Chern Simons Gauge theory has supersymmetric avatars. We use this insight to conjecture formulas for quantum manifold invariants which may give more insight into their topological nature. We also speculate on other possibilities raised by these constructions.
The interplay of ideas from Mathematics and Theoretical Physics has been a productive one for both fields. This project develops new perspectives on the relations between these two areas. The planned project would give both concrete mathematical results and, hopefully, new avenues of interaction between the two fields. More broadly, problems arising from Physics, and the Mathematics arising from grappling with these problems, have been at the core of Mathematics for centuries. Quantum Field Theory and String Theory show every promise of playing a similar role in the future, and stimulating new discoveries in Mathematics, which invariably lead to applications far removed from their area of origin.
