Moduli theory concerns itself with understanding all possible shapes the space can have. For example, if one takes a rectangle and identifies opposite edges, one obtains a cylinder, and further identifying the remaining edges gives what mathematicians call a torus - the surface of a donut. Starting with rectangles of different shapes one ends up with different tori. Furthermore, one can also do similar identifications starting with parallelograms, to get more tori. It can be proven that the such tori are intrinsically different - if one lives on the surface of a torus and cannot see the whole surface from far away, one can still distinguish the kind of parallelogram that was used to build the torus. The moduli space of tori is then the set of all possible shapes of tori that are intrinsically different. Amazingly enough this "set of shapes" itself is a nice geometric object. This project deals with posing problems of similar type: of classifying some geometric objects, together with some further information on them, and of studying the resulting moduli spaces.
The project aims to understand better the geometry of various compact moduli spaces. Together with various collaborators, the PI will construct a modular compactification of the moduli space of complex curves together with a differential with prescribed multiplicities of zeroes, compactifying the so-called strata - phase spaces in Teichmuller dynamics. Real-normalized meromorphic differentials will be applied to bound the number of cusps of plane curves of a given degree. The project also will investigate the birational geometry and homology of various compactifications of the moduli spaces of cubic threefolds. Lastly the project will continue the investigations on homology and higher dimensional cycles on moduli spaces of abelian varieties.
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.
