| Abstract: | 熱帶幾何是近年來備受重視的數學領域。很粗略的說, 熱帶幾何可以將古典幾何及代數幾何的問題, 轉為組合數學的問題, 在許多情況把問題大大簡化。而實際的應用上, 熱帶幾何在短短數年間, 就有不少重要的結果。我們這個計畫以熱帶幾何的角度, 去研究萊夫謝茨束及流形退化的問題。這裡我們會需要熱帶相交理論, 而我們主要的方法是研究熱帶循環, 並把非交換餘調熱帶化。然而, 熱帶相交理論尚在起步的階段。我們準備在這個計畫中, 研究目前有的幾種熱帶相交理論, 證明其等價或找出其中的關係, 或補上不足我們使用的地方。進一步我們會使用本計畫創新的將非交換餘調熱帶化的手法, 先計算簡單非交換餘調熱帶化的問題, 再於對熱帶循環有更深刻的認識後, 能夠推廣到更一般的情況。更重要的是, 我們希望能以這些工具, 研究萊夫謝茨束及問題, 尤其是單值作用在熱帶幾何中正確的建構方式。最後, 個計畫中, 我們準備建構一個幾何社群的網站, 給不單是對熱帶幾何, 乃至所有幾何相關問題有興趣的學者, 能有交流學習的空間。
Tropical geometry is a new area in Mathematics but has several important results already. Roughly speaking, tropical geometry turns problems in classical geometry and algebraic geometry into combinatorics ones. In this project, we will use tropical geometry to study Lefschetz pencils and manifold degenerations. In order to do this, we need tropical intersection theory and notion of tropical cycles. Unfortunately, tropical intersection theory is still under development. Thus in the two-year project, we will thoroughly study several tropical intersection theories, show the relations.We also plan to tropicalizew non-abelian cohomology: we will calculate simple cases first, and then move to more general ones after we learn more about tropical cycles. Our goal is to study Lefschetz pencils and manifold degenerations, especially tropical version of monodromy theory. Finally, we plan to make a website for all scholars, and students who are interested in geometry (not only in tropical geometry), to communicate with other people their thoughts, share what they have learned, and ask any questions they might have. |
