New Directions in Homology of Moduli Spaces

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The research conducted under this grant concerns the study of moduli spaces of algebraic and geometric objects, and in particular those of manifolds. Manifolds are mathematical objects generalizing the three-dimensional space in which we live, and appear throughout mathematics and science. Locally they look like ordinary space, though possibly of a different dimension, but globally their shape may be complicated. As a consequence, they can have interesting symmetries or can be deformed in interesting ways. It is a classical problem to understand which symmetries and deformation exist, and how to distinguish these in a computationally effective manner. Moduli spaces provide a geometric method to access this information, and we will contribute to the progress of science by furthering their study.To understand of the topology of moduli spaces of algebraic and geometric interest, and in particular their homology, the Principal Investigator will apply the techniques of homotopy theory and higher algebra. Firstly, the PI shall use the existence of additional higher-algebraic structures on moduli spaces. The PI will then use higher-algebraic cellular structures to reinterpret homological stability results and obtain meta-stable results about those homology groups for which homological stability does not hold. Secondly, the PI will destabilize recent stable results in manifold theory by comparing closely related moduli spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.