Author Archives: tomcircle

Originally posted on Math Online Tom Circle:
“Categories for Dummies” (French) https://youtu.be/LVHoROSF3KA Example 1: Paris (P) -> Rome (R) -> Amsterdam (A) Objects: cities {P, R , A} Morphism (Arrow): railway? Identity: railway within the city Associative: (P -> R)…

Originally posted on Math Online Tom Circle:
https://sites.google.com/site/logiquecategorique/dcm 4. Category, Logic and IT https://youtu.be/Shk3TNagC-4

Originally posted on Math Online Tom Circle:
Tom Leinster N = 0 : 0-Cat? => Set, 0-morphism = function N= 1: 1-Cat? => Cat, 1-morphism = functor N= 2: 2-Cat? => 2-morphism = Natural Transformation ? ? Definition of n-Category:?…

Originally posted on Math Online Tom Circle:
Dr. Eugenia Cheng – Professor of Category Theory (Chicago University) Author of the Best Selling Category Book Book : (for readers from 7-year-old to high school and undergraduate students) “How to Bake Pi…

Originally posted on Math Online Tom Circle:
2017 Keywords:? Category? Monad = Monoid + Endofunctor Category Theory is replacing Set Theory as the foundation of Math. Nowadays, ?few Advanced Math papers are written without using Category to explain, and this…

Originally posted on Math Online Tom Circle:
Remember when you cancel a common factor at both sides of an equation, you must check if the factor is non-zero, otherwise you would miss some answers. This is about Cancellation Law, related…

Originally posted on Math Online Tom Circle:
A weak South Korean Math student?June Huh?becomes a top Mathematician in the Princeton Institute for Advanced Study by proving the Read’s Conjecture and Rota Conjecture. “A path less taken to the peak of…

Originally posted on Math Online Tom Circle:
“Algebraic Topology” can detect the Multi-dimensional neural network in our brain – by studying the Homology (同调) and co-Homology (上同调) with the help of Linear Algebra (multi-dim Matrix) & ?Computers. Homology = compute…

Originally posted on Math Online Tom Circle:
What is Math ? https://infinityplusonemath.wordpress.com/2017/06/17/what-is-math/ Mathematics?= “that which is learned“ –(Pythagoras) Math is not about calculation, it is understanding the nature, the universe, the philosophy (logic, intelligence – both “human” and “artificial”)… What…

Originally posted on Math Online Tom Circle:
Singapore scores the 2017 PISA Test World’s First in Math. Below is the “Singapore Math” shown by an USA School adopting the pedagogy. https://youtu.be/SlAEvaxMzXQ I had the opportunity to meet the inventor of…

Originally posted on Math Online Tom Circle:
Three mysterious Math Objects: Pi,? Complex Numbers,? Prime Numbers are hiding in circle. https://youtu.be/NaL_Cb42WyY

Originally posted on Math Online Tom Circle:
After the defeat in 1st match, Ke Jie said “AlphaGo is approaching God!”.? This version of AlphaGo is 10x more powerful than last year’s version which has beateb the South-korean Go player Lee.…

Originally posted on Math Online Tom Circle:
Kotlin is the “New Java” officially supported by Google from May 2017! It is less verbose (罗唆) than Java which is clumsy with boilerplates (样板), ?interoperates with Java on JVM, with modern functional…

Originally posted on Math Online Tom Circle:
Emmanuel Macron is the youngest French President (39) since Napoleon Bonaparte (40). A brilliant student since young, he impressed his secondary school Drama teacher 24 years older, finally married her. Like any genius…

Originally posted on Math Online Tom Circle:
[Continued from previous BM Category Theory …] Intuition: [Artificial Intelligence] You teach the computer, like a Primary 6 kid, that Algebra is a?type of expression (f) which, after evaluation, ?returns a value. If…

Originally posted on Math Online Tom Circle:
Replace Field scalar as in a Vector Space to Ring scalar in a Module. https://youtu.be/IvukAijXgLE Module is more powerful than Vector Space – because Ring has an “Ideal” (理想) which can partition it…

Originally posted on Math Online Tom Circle:
https://youtu.be/prstbP2cG6M

Originally posted on Math Online Tom Circle:
Abstract Algebra studies all kinds of?mathematical structures?(Group, Ring, Vector Space, Category, …) , and relationship between them if the structures are preserved after mapping.

Originally posted on Math Online Tom Circle:
https://youtu.be/xopM9BFjcNo

Originally posted on Math Online Tom Circle:
https://youtu.be/0_ZzZzvxnP0

Originally posted on Math Online Tom Circle:
https://youtu.be/L3HXZfLsuXU

Originally posted on Math Online Tom Circle:
[Hint]:? https://youtu.be/mllw55JQy7Y

Originally posted on Math Online Tom Circle:
Terence Tao:? Weakness in Math: Algebraic Topology Collaborative & “Lone Wolf” Not attempting the Riemann Hypothese: tools not there yet. https://youtu.be/MXJ-zpJeY3E 3 Phases of Math Training: Pre-rigourous: intuition Rigorous: formal prove Post-rigourous: 1…

Originally posted on Math Online Tom Circle:
This question was submitted ?by West Germany to the IMO Committee, the examiners could not solve it in 6 hours.? In the IMO (1988) only 11 contestants solved it, ?one of them proved…

Originally posted on Math Online Tom Circle:
https://youtu.be/QvvkJT8myeI

Originally posted on Math Online Tom Circle:
https://www.wired.com/2017/04/elusive-math-proof-found-almost-lost Case “=” : if (a, b) independent? Case “>” : if (a, b) dependent Thomas Royen used only high-school math (function, derivative) in his proof in 2014. He then published it in…

Originally posted on Math Online Tom Circle:
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Originally posted on Math Online Tom Circle:
Answer : (scroll below) . . . . . . . . . . . . . . . . . . . . Answer = 3 mins See clearer if change person…

Originally posted on Math Online Tom Circle:
Adjunction is the “weakening?of Equality” in Category Theory. Equivalence of 2 Categories if: 5.2 ?Adjunction definition: such that the 2 triangle identities below ( red and blue) exist. 6.1?Prove: Let C any category,…

Originally posted on Math Online Tom Circle:
https://plato.stanford.edu/entries/category-theory/

Originally posted on Math Online Tom Circle:
[INMO 1993]

Originally posted on Math Online Tom Circle:
Russian mathematician Grigori Perelman proved the Poincaré Conjecture in 7 years of solitude research in his Russian apartment ?- same 7 years of solitude for?Andrew Wiles (The Fermat’s Last Theorem)?in the Cambridge attic…

Originally posted on Math Online Tom Circle:
Representable Functor F of C ( a, -): 4.2?Yoneda Lemma Prove 😕 Yoneda Lemma?: Proof: By “Diagram chasing” below, shows that Left-side: is indeed a (co-variant) Functor. Right-side: Functor “F a“. Note: When…

Originally posted on Math Online Tom Circle:
Brian Beckman:? You can understand Monad without too much Category Theory. Functional Programming = using functions to compose from small functions to very complex software (eg. Nuclear system, driverless car software…). Advantages of…

Originally posted on Math Online Tom Circle:
Category Theory (CT) is like Design Pattern, only difference is CT is a better mathematical pattern which you can prove, also it has no “SIDE-EFFECT” and with strong?Typing. The examples use Haskell to…

Originally posted on Math Online Tom Circle:
10.1?Monads Imperative (with side effects eg. state, I/O, exception ) to Pure function by hiding or embellishment in Pure function but return “embellished” result. Monad = functor T + 2 natural transformations http://adit.io/posts/2013-04-17-functors,_applicatives,_and_monads_in_pictures.html#functors

Originally posted on Math Online Tom Circle:
In essence, in all kinds of Math, we do 3 things:? 1) Find Pattern among objects (numbers, shapes, …),? 2) Operate inside the objects (+ – × / …),? 3) Swap the object…

Originally posted on Math Online Tom Circle:
https://en.m.wikipedia.org/wiki/Andr%C3%A9_Lichnerowicz https://youtu.be/l298jeGgroA See the 1970s French?Baccalaureate Math Textbooks:?(for UK Cambridge GCE A-level ?Math students, this is totally new “New Math” to us !)

Originally posted on Math Online Tom Circle:
Curry-Howard-Lambek Isomorphism:? Below the lecturer said every aspect of Math can be folded out from Category Theory, then why not start teaching Category Theory in school. That was the idea proposed by Alexander…

Originally posted on Math Online Tom Circle:
Facebook rewrote the SPAM rule-based AI engine (“Sigma“) ?with Haskell functional programming to filter 1 million requests / second. The Myths about Haskell : Academia, Not for Production ?? https://youtu.be/mlTO510zO78 Why Facebook chooses?Haskell…

Originally posted on Math Online Tom Circle:
https://youtu.be/02_H3LjqMr8

Originally posted on Math Online Tom Circle:
Excellent lecture using Physics and IT to illustrate the 2 totally different approaches in Programming: Imperative (or Procedural) – micro-steps or Local 微观世界 Declarative (or Functional) – Macro-view or Global 大千世界 In Math:?…

Originally posted on Math Online Tom Circle:
[Continued from 1.1 to 2.2] 3.1 Monoid?M (m, m) Same meaning in Category as in Set: Only ?1 object, Associative, Identity Thin / Thick Category: “Thin” with only 1 arrow between 2 objects;?…

Originally posted on Math Online Tom Circle:
Top 5: MIT Harvard? Stanford? Oxford Cambridge 18 University of Tokyo … 20 Peking University 北京大学 … 22 Ecole Polytechnique (France) … 26 TsingHua University 清华大学 … 28 Hong Kong University … 32…

Originally posted on Math Online Tom Circle:
During the 19th century French Revolution, a young French boy?Evariste Galois?self-studied Math and invented a totally strange math called “Group Theory“, in his own saying – “A new Math?not on calculation but on…

Originally posted on Math Online Tom Circle:
Object-Oriented ?has 2 weaknesses for Concurrency and Parallel programming 😕 Hidden Mutating States;? Data Sharing. Category Theory (CT): a higher abstraction of all different Math structures : Set , Logic, Computing math, Algebra……

Originally posted on Math Online Tom Circle:
Haskell is the purest Functional Language which is based on Category Theory. eBook:? http://learnyouahaskell.com/chapters

Originally posted on Math Online Tom Circle:
http://www.math.niu.edu/%7Ebeachy/abstract_algebra/study_guide/contents.html The Study Notes on 600 problems and solutions: http://www.math.niu.edu/~beachy/abstract_algebra/guide/contents.html

Originally posted on Math Online Tom Circle:
Study Tips: http://www.youtube.com/playlist?list=PLi01XoE8jYoi-BCj5m0rRW03vqxaQrH7u

Originally posted on Math Online Tom Circle:
Suitable for Upper Secondary School and Junior College Math Students. Abstract Algebra is scary because it is abstract, and its Math Profs are mostly fierce – but not with this pretty Math lady……