Minimal Model Program in Birational Geometry

主要研究人员建议继续研究出生几何中射影变种的分类。最小模型程序由Mori在20世纪70年代前后由S发起，旨在将射影曲面的分类推广到高维变体。这一方案是在1980年成功地进行了S的投射三折。主要研究人员计划在更高的维度上执行最小模型程序，目的是完成复杂射影变种的分类。尽管这个项目涉及的技术非常不同，但他希望获得与经典最小模型项目一样强大的结果。抛开自己的兴趣不谈，希望这项研究将对完成复杂射影变种的分类大有裨益。最后，首席研究员打算通过将Mori的工作翻译成一种分析语言，在广泛的投影种类上继续他对Kahler-Ricci流的研究。数学工具和概念已广泛应用于物理学、工程学和经济学等广泛的科学领域。特别是，双曲几何在理论物理中被证明是一个非常有用的工具，特别是在弦理论中。卡西尼最近的工作已经启发了几个重要的会议。特别是，数学科学研究所组织了一个为期一周的题为《热点问题：代数几何中的最小和典型模型》的研讨会，讨论卡西尼及其合作者所取得的上述成果。与此同时，在这个国家的许多数学系都组织了关于同一主题的研讨会。

The principal investigator proposes to continue the study of the classification of Projective Varieties in Birational Geometry. The Minimal Model Program, started by Mori around the 1970's, aims to generalize the classification of projective surfaces to higher dimensional varieties. This Program was successfully carried out in the 1980's for projective three-folds. The principal investigator plans to carry out the Minimal Model Program in higher dimension, aiming to complete the classification of complex projective varieties.Moroever, the principal investigator plans to extend the Minimal Model Program to a broader class of varieties defined in positive characteristic. Although the techniques involved in this program are very different, he expect to obtain results that are as strong as in the classicalMinimal Model Program. Quite apart from its own interest, it is hoped that this study will be very useful in completing the classification of complex projective varieties. Finally, the principal investigator intends to continue his study of the Kahler-Ricci flow on a wide range of projective varieties, by translating Mori's work into an analytic language.Mathematical tools and concepts have been extensively applied in a wide range of sciences such as physics, engineering and economics. In particular, birational geometry has proven to be a very useful tool in theoretical physics, especially in string theory. Cascini's recent work has already inspired several important conferences. In particular, the Mathematical Sciences Research Institute organized a one-week workshop entitled "Hot Topics:Minimal and Canonical Models in Algebraic Geometry" to discuss the aforementioned results obtained by Cascini and his collaborators. At the same time, seminars on the same topics were organized in many departments of Mathematics in this country.