Moduli Problems in Algebraic Geometry, Their Structures and Their Applications

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This project is a research in algebraic geometry, a branch of mathematical science. The Principal Investigator (PI) will study properties of certain spaces (called moduli space) of objects that can be characterized by algebraic properties. (An example of such are the roots of polynomials). These spaces describe solution spaces that are vital to research in many branches of mathematical researches and in theoretical physics. The PI will work on several research directions. He will work toward a full understanding of high genus Gromov-Witten invariants of Calabi-Yau threefolds; he will also develop an alternative theory on generalized Donaldson-Thomas invariants of Calabi-Yau threefold; develop necessarily tools to study and prove the conjecture on BPS-states of Calabi-Yau threefolds.This research project is the continuation of PI's long term research goal of broadening mathematical research by understanding new idea from theoretical physics and contributing to the development of theoretical physics by providing mathematical foundation vital to its advancement. The progress on studying high genus GW invariants and DT invariants will advance our understanding of moduli spaces in general; enrich the research in algebraic geometry. It will also strengthen the interaction between algebraic geometry and other subjects of mathematics, and with mathematical physics. This project will promote teaching, learning and training young mathematical researchers. Over all, it will contribute its share in advancing the science research in the country.