CAREER: A unified study of singularities

职业：奇点的统一研究

• 批准号: 0449465
• 负责人: Mattias Jonsson
• 金额: --
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0449465&HistoricalAwards=false
• 关键词: CAREER; unified; study; singularities

0449465 JonssonAbstractThe investigator will use valuations to undertake a broad and unified, yet detailed study of singularities arising in different mathematical fields, including algebraic geometry, complex analysis and dynamical systems. A cornerstone in the analysis is to encode singularities in terms of functions and measures on suitable valuation spaces, that is, sets of valuations on an appropriate ring. In the case of singularities at the origin in the affine plane, the valuation space has the structure of a tree with infinite and dense branching. In more general cases, it is expected to be a building, or projective limit of simplicial complexes. Being fundamental objects in complex analysis, positive closed currents may be approximated by varieties. As an analogue of Hironaka's theorem, the investigator will study whether every positive closed current admits an approximate resolution of singularities. He will also investigate the corresponding question for dynamical systems defined by holomorphic fixed point germs. Other problems to be addressed are semicontinuity of multiplier ideals and degree growths of iterates of rational maps of projective space. The investigator will design an undergraduate course and continue his development of a masters course to suit the needs at the University of Michigan (UM). Additionally he will engage middle and high school students in mathematical activities, in particular through a summer camp at the UM.Singularities play a prominent role throughout mathematics, even when the primary objects of study are regular (smooth) objects. An example of a singularity is that of a curve in the plane that does not look like a line on a small scale, but rather as a cross or cusp, or even more complicated.It is known that plane singular curves can be viewed as "shadows" of nonsingular curves in space. This provides a way of understanding complicated object thorugh simpler ones. Other examples of singularities appear in dynamical systems, when studying the speed at which iterative algorithms converge. Working with students at the graduate and undergraduate levels, the investigator will undertake a unified study of singularities in different branches of mathematics. In addition he will develop a new undergraduate course and continue the redesigning of a graduate course. Finally he will work with K-12 students, in particular by continuing the development of a course in the framework of the Michigan Mathematics and Science Scholars summer program for high school students.

0449465 JonssonAbstractThe调查员将使用估值进行广泛的和统一的，但在不同的数学领域，包括代数几何，复杂的分析和动力系统中产生的奇点详细的研究。在分析的基石是编码奇点的功能和措施，在适当的估值空间，也就是说，一个适当的环上的估值集。在仿射平面中原点处的奇点的情况下，赋值空间具有树的结构，该树具有无限且稠密的分支。在更一般的情况下，它被期望是单纯复形的建筑物或投影极限。正闭合电流是复分析中的基本对象，可以用各种形式来近似。作为Hironaka定理的类似物，研究者将研究是否每个正的闭合电流都允许奇点的近似解。他还将调查相应的问题，动力系统定义的全纯不动点芽。其他要解决的问题是乘子理想的不连续性和射影空间的有理映射迭代的度增长。 调查员将设计一个本科课程，并继续他的硕士课程的发展，以适应密歇根大学（UM）的需求。此外，他将从事初中和高中学生的数学活动，特别是通过在UM夏令营。奇点在整个数学中发挥着突出的作用，即使当研究的主要对象是定期（光滑）对象。奇异性的一个例子是平面上的一条曲线，它在小尺度上看起来不像一条线，而是一个十字或尖点，甚至更复杂。众所周知，平面奇异曲线可以被看作空间中非奇异曲线的“阴影”。这就提供了一种通过简单对象来理解复杂对象的方法。在研究迭代算法收敛的速度时，奇点的其他例子出现在动力系统中。与学生在研究生和本科水平的工作，调查员将进行统一的研究奇点在不同的数学分支。此外，他将开发一个新的本科课程，并继续研究生课程的重新设计。最后，他将与K-12学生合作，特别是继续在密歇根州数学和科学学者暑期项目的框架内为高中生开发一门课程。