AI with Advanced Math helps in discovering new drugs

https://theconversation.com/i-build-mathematical-programs-that-could-discover-the-drugs-of-the-future-110689?from=timeline Advanced Mathematical Methods with AI is a powerful tool:

• Algebraic Topology (Persistent Homology)
• Differential Geometry
• Graph Theory

https://sinews.siam.org/Details-Page/mathematical-molecular-bioscience-and-biophysics-1 代 数拓扑 Algebraic Topology (Part 1/3)

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 Topology from Euler Characteristic (V -E + R = 2)

Motivation of Algebraic Topology : Find Invariants of various topological spaces (in higher dimension). 求拓扑空间的“不变量” eg.

• Vector Space (to + – , × ÷ by multiplier Field scalars);
• Ring (to + x) in co-homology
• 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).
Note:  Analogy of an”Invariant” in Population: eg. “Age” is an invariant can be added in the “Population Space” as the average age of the citizens.

Side Reading (Very Clear) : Invariant and the Fundamental Group Primer

Note : Homology 同调 = same “tune”.

“谁谓古今殊，异代可同调

(希腊 homo = 同, -logy = 知识 / 调)

– “Reading an ancient text  allows us to think “in tune” (or resonant) with the ancient author.”

[温习] Category Theory Foundation – 3 important concepts:

• Categories
• Functors
• Natural Transformation

[Skip if you are familiar with Category Theory Basics: Video 16:30 mins to 66:00 mins.]

[主题] Singular Homology Groups 奇异同调群  (See excellent writeup in Wikipedia) (Video 66:20 mins to end)

1. Singular Simplices 奇异 单纯
2. Singular Chain Groups 奇异 链 群
3. Boundary Operation 边界
4. Singular Chain Complex 奇异 单纯复形
5.  Part 1/3 Video (Whole) :

Simplicial Homology 单纯同调

Continued from Computational Topology (1 ~4):

MATH 496/696 2016/02/10 Lecture

Homology from another angle: $\displaystyle H_{k}$ = coKernel $(m)$ ○ kernel $(d_{k}) \:$ ${(C_{k})}$ Simplicial Homology:

1. Define Chain Space C•(X)
2. Define Boundary Map d•
3. Define Simplicial Chain Complex (C•(X), d•) Homotopy 同伦 & Fundamental Groups

NJ Wildberger AlgTop24: The fundamental group Homotopy 同伦: When playing skip rope, the 2 ends of the rope are held by 2 persons while a 3rd person jumping over the “swings of rope” – these swings at any instant are  homotopic.  Fundamental Group of Surface $\pi (M; \alpha)$

Fundamental Group of Torus $\pi (T) = Z.Z$

Fundamental Group of Projective Plane (Torsion ) $\pi (P) =Z_{2}$

Darcy Lecture 5 ~ 9: Applied Algebraic Topology

Lecture 5: 4/9/2013 (三)  Clustering Via Persistent Homology

Lecture 7: 6/9/2013 (五) Calculating Homology using matrix

Lecture 8: Column Space and Null Space of a matrix

Lecture 9: 9/9/2013 (一)  Create your own Homology: (Important lecture in Applied Algebraic Topology)