Excellent Advanced Math Lecture Series (Part 1 to 3) by齊震宇老師

（2012.09.10) Part I:

History: 1900 H. Poincaré invented Topologyfrom Euler Characteristic (V -E + R = 2)

**Motivation of Algebraic Topology**: Find * Invariants*[1]of various topological spaces (in higher dimension). 求拓扑空间的

**“不变量”**eg.

- AbelianFundamental Group(so as to manipulate by + -);

- Vector Space (to + – , × ÷ by multiplier Field scalars);

- Ring (to + x), etc.

then apply algebra (Linear Algebra, Matrices) with computer to compute these invariants (**homology**, **co-homology, etc**).

A topological space can be formed by a “Big Data” Point Set, e.g. genes, tumors, drugs, images, graphics, etc. By finding (co)- / homology – hence the intuitive Chinese term (上) /同调 [2] – is to find “**holes**” in the Big Data in the 10,000 (e.g.) dimensional space the hidden information (co-relationship, patterns, etc).**Note**: [1]…

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