CAREER: A unified study of singularities

[0449465] jonsson[摘要]研究者将利用估值对不同数学领域中出现的奇点进行广泛、统一而又详细的研究，包括代数几何、复杂分析和动力系统。分析中的一个基石是在合适的估值空间（即合适环上的估值集合）上用函数和测度来编码奇点。在仿射平面原点存在奇点的情况下，估值空间具有具有无限密集分支的树形结构。在更一般的情况下，它预计是一个建筑物，或简单的综合体的投影限制。正闭合电流是复杂分析中的基本对象，它可以用变量来近似。作为Hironaka定理的类比，研究者将研究是否每个正闭合电流都允许奇点的近似解析。他还将研究由全纯不动点胚定义的动力系统的相应问题。其他要解决的问题是乘子理想的半连续性和投影空间的有理映射迭代的度增长。研究者将设计一个本科课程，并继续开发一个硕士课程，以适应密歇根大学（UM）的需要。此外，他还将带领初高中学生参与数学活动，特别是通过在澳大的夏令营。奇点在整个数学中扮演着重要的角色，即使当主要研究对象是规则（光滑）对象时也是如此。奇点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直线，而是像一个十字或尖，甚至更复杂。平面奇异曲线可以看作是空间中非奇异曲线的“影子”。这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，在动力系统中也会出现其他奇点的例子。与研究生和本科生合作，研究者将对数学不同分支中的奇点进行统一研究。此外，他将开发一门新的本科课程，并继续重新设计研究生课程。最后，他将与K-12学生合作，特别是继续在密歇根数学和科学学者暑期高中生项目的框架下开发课程。

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.