This proposal aims to study the geometry of moduli spaces, such as their dimensions, irreducible components, birational types, and geometric invariants. The principal investigator plans to carry out the study in several directions. Firstly, he would like to study Teichmueller curves, which are rigid geodesics in the moduli space of curves, and hence can provide crucial information for the geometry of the moduli space. Secondly, he wants to explore moduli spaces parameterizing curves in an ambient space, focusing on the comparison between the moduli space of stable maps, the moduli space of semi-stable sheaves, Hilbert scheme and Chow variety. Finally, he plans to study the Jacobian variety of line bundles on a non-reduced curve, which may reveal geometric properties of smooth curves by deformation and degeneration techniques.
This project belongs to the subject of algebraic geometry, whose main objects are algebraic varieties defined by the solution sets of polynomial equations. Moduli spaces parameterize varieties of a given type. For instance, a donut and a car tire are of the same type, because they both have one hole, but a pretzel with three holes is different. An attractive aspect is that a moduli space for its objects tends itself to be a variety. Therefore, studying moduli spaces can help us understand the classification of algebraic varieties. In addition, moduli spaces have been extensively used in many other fields. The principal investigator expects that the outcome of this project can enrich the studies of other subjects, including combinatorics, dynamics, enumerative geometry and mathematical physics.
. Moduli spaces parameterize geometric objects of a given type. For example, a donut and a car tire are of the same type, because they both have one handle, but they are different from a pretzel, which has three handles. An attractive aspect is that a moduli space itself usually has a geometric structure as well. Therefore, the results obtained by the PI can help us better understand the variation of geometric structures. In addition, moduli spaces have been extensively used in many other fields, hence those results the PI obtained can develop applications to other subjects, such as combinatorics, dynamics, number theory, and physics. During the support of the NSF award, the PI has given more than 25 reserach talks, including seminars, colloquia, conferences, and summer schools worldwide, to spread his research results. He has also organized a number of seminars and workshops, both in US and internatially, with a foucs on training students and young scholars. In addition, the PI has been advising a PhD student at his home institute, who is learning and working in the field of moduli spaces.
