The 2 mathematicians showed that: even very complicated polynomials have just a

finite number of solutions.

http://www.livescience.com/61103-2017-breakthrough-mathematics-awarded.html

The 2 mathematicians showed that: even very complicated polynomials have just a

finite number of solutions.

http://www.livescience.com/61103-2017-breakthrough-mathematics-awarded.html

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[**Continued** from previous BM Category Theory …]

Intuition:[Artificial Intelligence] You teach the computer like to a Primary 6 kid, thatAlgebrais atypeof expression(f) which, after evaluation, returns a value.

**If a = i (initial) **

**Intuition**: *Fix-point because, the Initial “i”, after evaluating the expression f, returns itself “i”.*

**Lambek’s Lemma**

**Note**: Endo-functor is a **functor** (equivalent to *function *in Set Theory) within the same Category (Endo = Self = 自)

Video 8.1 F-Algebras & Lambek’s Lemma

Video 8.2 Catamorphism & Anamorphism

foldr ~ catamorphism (浅层变质) of a Fix-point endo-functor on a List.

**Examples**: Fibonacci, Sum_List

Remark: **Cool Math**! the more advanced concept it is, the more closer to Nature (eg.Geology, Biology) : Catamorphism 浅层(风化)变质, or *“thin-layer change in nature” *(in Functional Programming languages: foldr or map) eg : **add1** to a list (1 5 3 8…)

= (2 6 4 9 …)

Intuition:Reverse of Algebra, given a value, Coalgebra returns an expression (f).

**Anamorphism (合成变质) ~ unfoldr**

**Example**: Prime numbers

**Remark**: Anamorphism (合成变质) or “*synthesised change in nature*“: eg. Start from a “**seed**” prime number “2” generates all other infinite prime numbers (3 5 7 9 11 13 17 …)

**Note: In Haskell, no difference** between Initial and Terminal Fix-points. However, since Fix-point is not unique, in **Category Theory **there is the **Least** Fix-point (Initial) and **Greatest** Fix-point (Terminal).

Ref:

Reading “**Understanding F-Algebra** ” by BM: https://bartoszmilewski.com/2013/06/10/understanding-f-algebras/

Catamorphism (下态) : https://www.zhihu.com/question/52180621/answer/129582557

Anamorphism : https://zhuanlan.zhihu.com/cofree/21354189

F-Algebra & F-coalgebra: http://stackoverflow.com/a/16022059/5822795

This French Classic (8 series lectures) on “Introduction to Modern Maths” made in 1969 is still **valid** today for Modern Maths students who need to be “**initialized**” for abstract, rigorous math, different from high-school maths.

Partie 1 : ensembles et parties

View the other 7 series below:

Initiation à l’algèbre (TV 1969): http://www.youtube.com/playlist?list=PLGwb7STcll74lv38PsnYmB4sr5fpWTEsq

Mathematics is divided into 2 major branches:

1. **Analysis** (Continuity, Calculus)

2. **Algebra** (Set, Discrete numbers, Structure)

In between the two branches, **Poincaré** invented in 1900s the **Topology (拓扑学)** – which studies the ‘holes’ (disconnected) in-between, or ‘neighborhood’.

Topology specialised in

– ‘local knowledge’ = **Point-Set Topology**.

– ”global knowledge’ = **Algebraic Topology**.

__Example:__

The local data of consumer behavior uses ‘Point-Set Topology’; the global one is ‘BIG Data’ using Algebraic Topology.

The beauty I see in algebra: Margot Gerritsen at TEDxSTANFORD:

From equations to matrices… to Google search, MRI body scan, …

**Modern Algebra:** Based on the 1931 influential book “Modern Algebra” written by Van de Waerden (the student of E. Noether). Pioneered by the 20th century german Göttingen school of mathematicians, it deals with Mathematics in an abstract, axiomatic approach of mathematical structures such as **Group, Ring, Vector Space, Module and Linear Algebra**. It differs from the computational Algebra in 19th century dealing with Matrices and Polynomial equations.

This phase of Modern Algebra emphasises on the algebraization of *Number Theory: * {N, Z, Q, R, C}

**Post-Modern Algebra:** The axiomatic, abstract treatment of Algebra is viewed as boring and difficult. There is a renewed interest in explicite computation, reviving the 19th century invariant theory. Also the structural coverage (Group, Ring, Fields, etc) in Modern Algebra is too narrow. There is emphasis on other structures beyond Number Theory, such as **Ordered Set, Monoid, Quasigroup, Category, ** etc.

Example:

The non-abelian Group S3 (order 6) has structures appeared in many branches of maths and sciences:

**[Reference]** :

“*Post-Modern Algebra*” by

Jonathan D.H. Smith and Anna B. Romanowska

1999, John Wiley & Sons