Moduli Spaces of Solutions to Non-Linear PDES in (Derived) Differential Geometry

Abstract

Many moduli spaces in geometry and physics, like those appearing in symplectic topology, quantum gauge field theory and in relation to homological mirror symmetry, are constructed as parametrizing spaces of solutions to nonlinear elliptic differential operators modulo symmetries of the underlying theory. A plethora of difficulties arise in constructing such spaces; for instance, the spaces are often not smooth, with multi non-equidimensional components, and in cases where they are given as intersections of higher domensional components they exhibit singularities due to non-transverse intersections. That is why in constructing such spaces, often given by quotient of a parametrizing scheme (or stack) by a suitable group of symmetries, such as the gauge group action, one needs to enhance the space to higher stacks, and the intersection theory must be enhanced so that it takes into account a suitable geometric model for non-transverse intersection loci. This suggests that one should treat such spaces within the framework of higher and derived differential/algebraic geometry. In this talk we will present two general approaches to describing moduli spaces of solutions to non-linear PDES (not necessarily from an elliptic operator nor specific to the setting of differential geometry), discuss some relevant finiteness issues and strive to present many examples. This talk is based on a current joint work in progress with Artan Sheshmani and Shing-Tung Yau.