POWRE: Excellent Local Rings

PI将研究优秀局部环与其完井之间的关系。人们早就知道，环的形式纤维编码有关环的重要信息。因此，理解局部积分域的零理想（称为一般形式光纤）上的形式光纤是至关重要的。PI将致力于探索环与其一般形式纤维之间的关系。她将通过检查R和R[X]的一般形式纤维来研究环R和R[X]之间的关系（其中X是不确定的）。她将研究环满足完全同态象性质时的刻画问题，并将研究紧闭的局部化问题。此外，她将探讨几个刚性问题。该项目将于1999-2000学年在密歇根州立大学进行。PI将花一年的时间与世界顶尖的优秀戒指专家之一Christel Rotthaus合作。PI将参加交换代数研讨会，并与密歇根州立大学的常驻代数学者和访问代数学者讨论重叠的研究兴趣。这将有助于PI拓宽她的研究重点和学习新的研究技能。此外，PI将与这些代数学者进行头脑风暴，以发现该领域的新问题，这些问题将适合她的本科生研究。来自这次合作的想法将在未来几年推动本科生的研究项目。该项目的PI目前是她在威廉姆斯学院担任助理教授的第三年。她曾指导过四名本科生的研究，目前正在指导另一名本科生的研究。她致力于通过继续和拓宽自己的研究路径，并为本科生的原始研究提供建议，为本科数学研究的进步做出贡献。该项目的主要部分将用于拓宽她的研究重点，以及她的本科生可以研究的问题的重点。在这个关键阶段，一次性的资金投入将极大地促进她作为一名数学家在一所高度选拔的文理学院的职业生涯。

The PI will study the relationship between excellent local rings and their completions. It has long been known that the formal fibers of a ring encode important information about the ring. Because of this, understanding the formal fiber over the zero ideal (called the generic formal fiber) of a local integral domain is critical. The PI will work to explore the relationship between a ring and its generic formal fiber. She will study the relationship between a ring R and R[X] (where X is an indeterminate) by examining the generic formal fiberof R and R[X]. She will work on the problem of characterizing when a ring satisfies the complete homomorphic images property, and will also study a question relating to the localization problem for tight closure. In addition, she will explore several rigidity questions.The project will take place at Michigan State University during the 1999-2000 academic year. The PI will spend the year collaborating with Christel Rotthaus, one of the world's leading experts in excellent rings. The PI will attend commutative algebra seminars and discuss overlapping research interests with the resident algebraists and visiting algebraists at Michigan State. This will aid the PI in broadening her research focus and learning new research skills. In addition, the PI will brainstorm with these algebraists to discover new problems in the field that will be suitable for her undergraduate students to study. The ideas coming fromthis collaboration will fuel undergraduate research projects for years to come.The PI for this project is currently in her third year as an Assistant Professor at Williams College. She has previously advised the research of four undergraduate students and is currently advising the research of another. She is dedicated to contributing to the advancement of undergraduate research in mathematics by continuing along and broadening her own research path and advising the original research of undergraduates.The PI will devote a major portion of this project to broadening the focus of her research and the focus of the problems on which her undergraduate students can work. A one-time input of funds at this critical stage will serve to greatly advance her career as a mathematician at a highly selective liberal arts college.