Applied Math: Computational Topology

This is a series of 6 lectures on Computational Topology : Applied Math using computer in Algebraic Topology. Computer tool languages used can be Phyton  (this lecture) or Common Lisp (the preferred Functional Programming like Lisp for its rich mathematical background in Lambda  Calculus — new main feature in next Java 8).

As explained by this professor in Lecture 1, Computational Toplogy begins with Algebraic Topology aided by the arrival of computers in 1950s. The role of Algebraic Topology is to study Topology (Geometric Spaces “Manifolds” (流形) with continuous functions) using algebra (mainly Advanced Linear Algebra).

Just like Google Search revolutionises the world in 2000s, using only the classical Linear Algebra; Algebraic Topology will revolutionise Big Data Analytics using the Advanced Linear Algebra — the next wave in Mobile Age.

In layman’s term, it means using this tool to analyse Big Data in a geometric picture form (Topology), to visualise a pattern (diseases, epidemics, stocks, consumer purchasing behaviour, genetics…), which is otherwise ineffective by the current Statistical approach, in coping with the ocean of data from mobile phones / clouds / emails / Internet.

Topology was invented by the 19th century Grand Math & Science Master Henri Poincaré (France, 1854 – 1912).

Algebraic Topology Grand Master in the 20th century (post WW2) was the “Math Hermit” (数学隐士) Alexander Gothendieck (France, 1928 – 2014, 中国数学家 Wu Wenjun 吴文俊 (1919 – ) 在法国 Université de Strasbourg 教书时指导的博士学生).

Lecture 2:

continued…
◇ Linear Algebra
Quiver Representation = Barcoding
$0 \xrightarrow { } V_0 \xrightarrow {d_0} V_1 \xrightarrow {d_1} V_2 \xrightarrow { } 0$
Chain Complex:
$d_1{(d_0 (V_1)) } = 0$
Exact Sequence:
$Img ({ d_j} )= Ker ({ d_{j+1}})$

Prove:
do = Injective (Linear map)
d1 = Surjective (Linear map)
$\displaystyle \boxed { {V_1 \cong V_0\oplus V_2 } }$
Quotient Space :
$\displaystyle \boxed { V_1 / d_0(V_0) \cong V_2 }$
Revise:
coset isomorphism quotient

Lecture 3 Homology

Lecture 4

One thought on “Applied Math: Computational Topology”

1. Reblogged this on Singapore Maths Tuition and commented:

Quite easy to follow this “Mr.Bean” lecturer in Computational Topology. Pre-requisite is undergraduate elementary Abstract Algebra (Linear Algebra, Surjectivity, Injectivity, Isomorphism, Quotient Vector Space, Matrices, ….). He used Python to compute rather than Lisp.