本計畫將從兩方面拓展計畫主持人新近有關單調拉格朗日環面單值群及合痕性之研究. 其一是將研究結果推廣至高維辛空間中的單調拉格朗日子流形, 研究其單值群及合痕性之關聯, 以及單值群不變量之高維推廣. 其二是以單調拉格朗日環面之漢彌爾頓單值群之對稱性為出發點, 尋求單調拉格朗日環面餘集上辛形式合痕類之代數刻劃, 並希望能獲致對應的手術程序, 進而提供一個具有一般性的研究方法, 以利於一般辛流形上辛形式合痕性之研究. 對拉格朗日子流形及辛形式之合痕性質之理解向來為辛拓樸領域之重要課題, 然吾人對其之掌握仍相當有限, 本計畫的結果將對此二議題作出新的貢獻. ; This project consists of two directions related the PI’s recent research on monodromy groups of Lagrangian tori in R4. One is to extend the study of monodromy groups and their invariants to Lagrangian submanifolds in higher dimensional symplectic spaces. The other one, based on the PI’s characterization of Hamiltonian monodromy groups of monotone Lagrangian tori and Chekanove tori in R4, seeks to obtain an algebraic invariant and even a surgical procedure to characterize the two associated relatively non-isotopic symplectic forms on the complement of a monotone Lagrangian torus. Isotopy properties of Lagrangian submanifolds and symplectic forms are important topics in the field of symplectic topology. And one would like to have a complete understanding of these properties for symplectic manifolds as simple as R2n. Results of the first part of this project will improve our understanding of Lagrangian submanifolds in dimensions greater than four, whilst results of the second part of this project will pave the road to a more systematic study of isotopy problems of symplectic forms in dimension four. ; 研究期間 9808 ~ 9907
