Author Archives: tomcircle

About tomcircle

Math amateur

Les Categories Pour Les Nuls

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)…

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Debate (French): Set Theory vs Category Theory

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

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An Introduction to N-Categories

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:?…

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Category Theory : “How to Make Pi”

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…

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Program = Category 

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…

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Integral Domains 整环 (Abstract Algebra)

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…

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South Korean Young Mathematician June Huh

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…

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Is There a Multi-dimensional Mathematical World Hidden in the Brain’s Computation?

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…

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What is math? 

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…

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Singapore Math for  Primary School Kids

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…

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Pi hiding in prime regularities

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

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AI “AlphaGo” beats World’s Best Go Player Ke Jie in 1st Match

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.…

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Google officially supports Kotlin from May 2017

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…

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French youngest President Emmanuel Macron and his Education 

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…

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BM Category Theory II 8: F-Algebra, Lambek’s Lemma , Catamorphism, Coalgebra, Anamorphism

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…

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What is a Module (模)? 

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…

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American Genius Series 1 1of8 Jobs vs Gates (2015)

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

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Structure Preserving

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.

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Mathematics: Beauty vs Utility – Numberphile

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

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2 minutes pour le théorème de Bézout

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

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Maths -Topologie

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

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中国初中生集体挑战美国高考数学SAT题

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

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The World’s Best Mathematician (*)

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…

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The Legend of Question Six (IMO 1988)

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…

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The Shortest Ever Papers 

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

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Unknown German Retiree Proved The “Gaussian Correlation Inequality” Conjecture

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…

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Tough Math : Diophantine Equation

Originally posted on Math Online Tom Circle:
Math Online Tom Circle View original post

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Fun Math

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

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Category Adjunctions 伴随函子

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,…

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Category Theory (Stanford Encyclopedia of Philosophy)

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

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Beautiful Math Olympiad Problem

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

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The genius who rejected Fields Medal & Clay Prize: Grigori Perelman 

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…

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The Yoneda Lemma

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…

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Don’t fear the Monad

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…

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Programming and Math

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…

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BM Category Theory 10.1: Monads

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

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Category Theory 9: Natural Transformations, BiCategories

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…

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French New Math Lichnerowicz Pedagogy 

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 !)

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Curry-Howard-Lambek Isomorphism

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…

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Fighting spam with Haskell | Engineering Blog | Facebook Code

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…

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Haskell Tutorial in One Video

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

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BM Category Theory II 1.1: Declarative vs Imperative Approach

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:?…

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BM Category Theory 3.x Monoid, Kleisli Category (Monad)… Free Monoid 

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;?…

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QS University Ranking 2017 by Math Subject

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…

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Euler’s formula with introductory group theory

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…

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Category Theory : Motivation and Philosophy

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……

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A Category Language : Haskell

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

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Online Study Guide : Abstract Algebra

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

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How to Take Great Notes – Study Tips – How to be a Great Student – Cornell Notes

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

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What is a Field, a Vector Space ? (Abstract Algebra)

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……

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