Math Blog

Replace Field scalar as in a Vector Space to Ring scalar in a Module.

Module is more powerful than Vector Space – because Ring has an “Ideal” (理想) which can partition it to Quotient Ring, but Field (scalar in a Vector Space) can’t.

Abstract Algebra studies all kinds ofmathematical structures(Group, Ring, Vector Space, Category, …) , and relationship between them if the structures are preserved after mapping.

[Hint]:

$latex (a + x)^{frac {1}{3}} + (a – x)^{frac {1}{3}} = 2(a)^{frac {1}{3}} + 2(x)^{frac {1}{3}} $

Terence Tao:

• Weakness in Math: Algebraic Topology
• Collaborative & “Lone Wolf”
• Not attempting the Riemann Hypothese: tools not there yet.

3 Phases of Math Training:

1. Pre-rigourous: intuition
2. Rigorous: formal prove
3. Post-rigourous: 1 + 2

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 it elegantly. Terence Tao (Australia) only got 1 mark out of 7 in this question.

Solution:

https://www.wired.com/2017/04/elusive-math-proof-found-almost-lost

$latex boxed {P (a + b) geq P (a) times P (b)}&fg=aa0000&s=3$

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 arxiv.org website – likePerelman did with the “Poincaré Conjecture”.

Answer : (scroll below)

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Answer = 3 mins

See clearer if change person to taxi car, bun to passenger.

9 taxi cars send 9 passengers will take the SAME timing as 3 taxi cars send 3 passengers.

Adjunction is the “weakeningof Equality” in Category Theory.

Equivalence of 2 Categories if:

5.2 Adjunction definition: $latex (L, R, epsilon, eta )$ such that the 2 triangle identities below ( red and blue) exist.

6.1Prove: Let C any category, D a Set.

$latex boxed {text {C(L 1, -)} simeq text {R}}&fg=aa0000&s=3$

$latex {text {Right Adjoint R in Set category is }}&fg=aa0000&s=3$ $latex {text {ALWAYS Representable}}&fg=aa0000&s=3$

1 = Singleton in Set D

From an element in the singleton Set always ‘picks’ (via the function) an image element from the other Set, hence :
$latex boxed {text{Set (1, R c) } simeq text{Rc }}&fg=0000aa&s=3$

Examples : Product & Exponential are Right Adjoints

Note: Adjoint is a more powerful concept to understand than the universal construction of Product and Exponential.

Adjoint (伴随) Functors (函子):

https://plato.stanford.edu/entries/category-theory/

[INMO 1993]

Russian mathematician Grigori Perelman proved the Poincaré Conjecture in 7 years of solitude research in his Russian apartment - same 7 years of solitude forAndrew Wiles (The Fermat’s Last Theorem)in the Cambridge attic house andZhang Yitang 张益唐 (7-Million-Gap Twin Primes) in the “Subway” sandwich kitchen.

7 is the Perfect Number. 1 week has 7 days, according to the “Book of Genesis”, God created the universe in 6 days and rested “Sabbath” on the beautiful 7th day.

People involved in his journey:

His mother who supported his early education in Math, all the way to his international fame and adulthood —Erdös Paulalso had a mother supporting her single son in the entire Mathematical life.

Prof Hamilton who “unintentionally”gave him the inspiration of the “Ricci-flow” tool – the first key to the door of The Poincaré Conjecture.

His closed Chinese friend the MIT…

Representable Functor F of C ( a, -):

$latex boxed {(-)^{a} = text {F} iff a = text {log F}}&fg=aa0000&s=3$

4.2Yoneda Lemma

Prove :

Yoneda Lemma:
$latex text {F :: C} to text {Set}$

$latex boxed {alpha text { :: [C, Set] (C (a, -),F) } simeq text {F a}}&fg=0000aa&s=3$

$latex alpha : text {Natural Transformation}$
$latex simeq : text {(Natural) Isomorphism}$

Proof: By “Diagram chasing” below, shows that

Left-side: $latex alpha text { :: [C, Set] (C (a, -),F) } $ is indeed a (co-variant) Functor.

Right-side: Functor “F a“.

Note: When talking about the natural transformations, always mention their component “x”: $latex alpha_{x}, beta_{x}$

Yoneda Embedding (Lagatta)

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 Functional Programming:

• Strong Types Safety: detect bugs at compile time.
• Data Protection thru Immutability: Share data safely in Concurrent / Parallel processing.
• Software ‘Componentisation’ ie Modularity: Each function always returns the same result, ease of software reliability testing.

Each “small” function is a Monoid.
f : a -> a (from input of type ‘a‘ , returns type ‘a’)
g: a -> a

compose h from f & g : (strong TYPING !!)
h = f。g : a -> a

[Note]: Object in Category, usually called Type in Haskell, eg.’a’ = Integer)

You already know a Monoid (or Category in general) : eg Clock

1. Objects: 1 2 3 …12 (hours)
2. Arrow

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 strongTyping.

The examples use Haskell to explain the basic category theory : product, sum, isomorphism, fusion, cancellation, functor…

10.1Monads

$latex begin{array}{|l|l|l|}
hline
Analogy & Compose & Identity
hline
Function & : : : : “.” & : : : : Id
hline
Monad & ” >> = ” (bind) & return :: “eta”
hline
end{array}$

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

$latex boxed {text {Monad} = {T , eta , mu} }&fg=aa0000&s=3$

$latex eta :: Id dotto T$
$latex mu :: T^{2} dotto T$
$latex text {Natural Transformation : } dotto $

http://adit.io/posts/2013-04-17-functors,_applicatives,_and_monads_in_pictures.html#functors

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 without modifying it (rotate, flip, move around, exchange…).

Category consists of :
1) Find pattern thru Universal Construction in Objects (Set, Group, Ring, Vector Space, anything )
2) Functor which operates on 1).
3) Natural Transformation as in 3).

$latex boxed {text {Natural Transformation}}&fg=000000&s=3$
$latex Updownarrow $

$latex boxed {text {Morphism of Functors}}&fg=aa0000&s=3$

Analogy:

Functors (F, G) :=operation inside a container
$latex boxed { F :: X to F_{X}, : F :: Y to F_{Y}}&fg=0000aa&s=3$

$latex boxed {G :: X to G_{X}, : G :: Y to G_{Y}}&fg=00aa00&s=3$

Natural Transformation ($latex {eta_{X}, eta_{Y}}&fg=ee0000&s=3$) := swap the content ( $latex F_{X} text { with } G_{X}, F_{Y} text { with } G_{Y} $) in the…

https://en.m.wikipedia.org/wiki/Andr%C3%A9_Lichnerowicz

See the 1970s FrenchBaccalaureate Math Textbooks:(for UK Cambridge GCE A-level Math students, this is totally new “New Math” to us !)

Curry-Howard-Lambek Isomorphism:

$latex boxed {text {Category Theory = High School Algebra = Logic = Lambda Calculus (IT)}}&fg=aa0000&s=3$

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 Grothendieckto the Bourbakian Mathematicians who rewrote all Math textbooks after WW2, instead of in Set Theory, should switch to Category Theory. His idea was turned down by André Weil.

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 ?

Why Facebook choosesHaskell Functional language for Spam rule engine?