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# 数学代写|微分拓扑作业代写differential topology代考|Higher Dimensions

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## 数学代写|微分拓扑作业代写differential topology代考|The Poincaré Conjecture and Thurston’s Geometrization Conjecture

In $1904 \mathrm{H}$. Poincaré ${ }^{6}$ conjectured that any simply connected compact and closed 3manifold is homeomorphic to the 3-sphere. This problem remained open for almost 100 years, although the corresponding problem was resolved in higher dimensions by S.

In the academic year $2002 / 2003 \mathrm{G}$. Perelman ${ }^{9}$ published a series of papers building on previous work by R. Hamilton ${ }^{10}$, which by now have come to be widely regarded as the core of a proof of the Poincaré conjecture. The proof relies on an analysis of the “Ricci flow” deforming the curvature of a manifold in a manner somehow analogous to the heat equation, smoothing out irregularities. Our encounter with flows will be much more elementary, but will still prove essential in the proof of Ehresmann’s fibration theorem, 8.5.10.

Perelman was offered the Fields Medal for his work in 2006, but spectacularly refused it. In this way he created much more publicity for the problem, mathematics and himself than would have otherwise been thinkable. In 2010 Perelman was also awarded the USD1M Millennium Prize from the Clay Mathematics Institute ${ }^{11}$. Again he turned down the prize, saying that Hamilton’s contribution in proving the Poincaré conjecture was “no less than mine”

Of far greater consequence is Thurston’s geometrization conjecture. This conjecture was proposed by W. Thurston ${ }^{13}$ in 1982 . Any 3 -manifold can be decomposed into prime manifolds, and the conjecture says that any prime manifold can be cut along tori, so that the interior of each of the resulting manifolds has one of eight geometric structures with finite volume. See, e.g., the Wikipedia page ${ }^{14}$ for further discussion and references to manuscripts with details of the proof filling in Perelman’s sketch.

## 数学代写|微分拓扑作业代写differential topology代考|The History of Manifolds

Although it is a fairly young branch of mathematics, the history behind the theory of manifolds is rich and fascinating. The reader should take the opportunity to check out some of the biographies at The MacTutor History of Mathematics archive $^{15}$ or the Wikipedia entries of the mathematicians mentioned by name in the text I have occasionally provided direct links.

There is also a page called History Topics: Geometry and Topology Index ${ }^{16}$ which is worthwhile spending some time with. Of printed books, I have found Jean Dieudonné’s book 6 especially helpful although it is mainly concerned with topics beyond the scope of this book.

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MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。