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 Invariantsof 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  – is to find “holes” in the Big Data in the 10,000 (e.g.) dimensional space the hidden information (co-relationship, patterns, etc).
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