Current Events > Really nice intro lecture on homology

(+) Yeah! (+)

https://www.youtube.com/watch?v=ShWdSNJeuOg

If you're interested in math, this guy has some really good videos. Helped me "get" homology a bit more than before. Cohomology is still a different story...

He also doesn't believe in real numbers which is an interesting perspective, but he definitely backs up his argument with videos upon videos.

guys can we please care about the same things i care about

Is homology the study of gay people
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I feel this is bait.

UnholyMudcrab posted...

Is homology the study of gay people

no its a neat way of finding n-dimensional "holes" in a topological space

try even thinking about what a 4 dimensional "hole" looks like....pretty cool we can at least get data on it

https://www.youtube.com/watch?v=uiq-EcQz_uU

Here's a construction of a neat thing called Boy's surface

Romes187 posted...

no its a neat way of finding n-dimensional "holes" in a topological space

try even thinking about what a 4 dimensional "hole" looks like....pretty cool we can at least get data on it

I'm all about looking for holes

Reis posted...

I'm all about looking for holes

if that's the case you'll love the Tits Freudenthal magic square

https://en.wikipedia.org/wiki/Freudenthal_magic_square

Link to Tits' approach

https://en.wikipedia.org/wiki/Freudenthal_magic_square#Tits'_approach

Are you like a math professor?

No but in high school nobody told me how cool math was and instead had me calculate a bunch of stuff for no reason

then i finally learned that every single thing you learn in undergrad math is a specialized instance of a really beautiful and cool generalization

Then I learned about group theory and my mind was blown by how....vast and general a mathematical theory could get. There's the series that got me going on group theory (first run through I understood maybe 10% of it)

https://www.youtube.com/watch?v=UwTQdOop-nU&list=PLwV-9DG53NDxU337smpTwm6sef4x-SCLv

Then I wanted to understand physics at the fundamental level and you really do need to do the work to understand what some of these things mean and how they relate to reality.

just to relate my last two items there

as far as we know, the forces that govern physics can be modeled by symmetry groups.

cept gravity ofc

Gonna turn this into my "cool maths" topic

Here's another one dealing with group theory (actually its foundations)

https://en.wikipedia.org/wiki/Galois_theory

Dude was like 18 when he came up with this...and was shot and killed in a duel at 20. And his stuff was so revolutionary no one could understand it for some time.

But his theory proves there is no formula for solving quintic equations in terms of coefficients (like how you can with the quadratic formula...and there's a beefy one for cubics etc but impossible for quintics)

What a great mind

This is honestly the most boring shit I've ever seen.

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Romes187 posted...

Dude was like 18 when he came up with this...and was shot and killed in a duel at 20.

Brutal

Romes187 posted...

no its a neat way of finding n-dimensional "holes" in a topological space

try even thinking about what a 4 dimensional "hole" looks like....pretty cool we can at least get data on it

Is this related to that Crisis on Infinite Earths storyline I kept seeing advertised last month?

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CRON posted...

This is honestly the most boring shit I've ever seen.

yeah its definitely not for everyone

darkphoenix181 posted...

Brutal

yeah - guy was a hardcore radical who spent a lot of time in prison formulating his theories. Love it.

Lorenzo_2003 posted...

Is this related to that Crisis on Infinite Earths storyline I kept seeing advertised last month?

No idea what that is

https://en.wikipedia.org/wiki/Hairy_ball_theorem

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DifferentialEquation posted...

https://en.wikipedia.org/wiki/Hairy_ball_theorem

Yes! But we can comb Tori which is nice :)

I love ridiculously named theorems haha

https://en.wikipedia.org/wiki/Ham_sandwich_theorem

DifferentialEquation posted...

https://en.wikipedia.org/wiki/Hairy_ball_theorem

https://en.wikipedia.org/wiki/Monster_group

This one takes a bit of 'splaining but...

A group is a set with an operation that satisfies some stuff...one is closure so if you take two elements of the set and combine them, you get another element in the set. So integers with addition = a group. Integers with division = not a group (since you can take 2/3 and it is NOT an integer)

Groups go beyond numbers. Symmetry groups are cool...take a square and find all the ways you can "rotate" or "flip" it to bring it back on itself. Using the operation of composition you can see that if you rotate, say 90 degrees and do a flip along the horizontal axis, its the same as rotating 180 degrees. So rotations and flips are closed and thus a group (im simplifying here). Put them all together and you have the dihedral group (D4 for the square). The "order" is how many elements are in the group...so rotate 90, 180, 270, 360 (0) and flip horizontal, vertical, and diagonally either way. So the order is 8.

The monster group has an order of 8x10^53

Which means there are that many "rotations" or "flips" you can do to bring the shape (whatever it is) onto itself again.

The fact this thing logically exists is so strange...wish I could visualize it (the "shape" has 196k dimensions)

Bump for night CE

will post some more tomorrow itt

We did homology yesterday so to confuse, lets do holonomy today

https://en.wikipedia.org/wiki/Holonomy

what is the difference? Well homology deals with holes in shapes, but holonomy can measure intrinsic curvature on a surface, such as a surface of a sphere

easy way to see this is if you point your finger in a particular direction and move around the room before coming back to your original spot, your finger is still pointing in the same direction. Thats because there isn't any curvature in space where you're standing (that you can measure). However if you do this move on the surface of a sphere, depending on the route you take, your finger will be pointing in a different direction when you return. That difference measures the curvature (see the pic in the wiki article).

Why is this important? Well you can extend this idea to more subtle versions of "parallel transport" (which is the act of 'moving' your finger around) to more abstract objects, but the idea of measuring curvature intrinsically is VERY important for modelling the entire universe. Space time has curvature at parts, and we have no "outside of the universe" to embed it in to measure.

It's easy to look at a shape and say "yeah it has curvature, you can just take a tangent vector at a spot and measure its rate of change" but that implies the shape is embedded into another space. We have to assume we are like an ant crawling along the sphere (in the above example) - how would that ant ever know the sphere is curved? How can we know the earth is round? Parallel transport gives us a way!

https://www.youtube.com/watch?v=CYBqIRM8GiY

Here's a really cool video on Spinors which are strange little creatures buried deep in the code

https://youtu.be/XTeJ64KD5cg

heres a really big number

UnholyMudcrab posted...

Is homology the study of gay people

lol

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https://youtu.be/mbv3T15nWq0

this guy has 3 lecture series on youtube

one on foundations of quantum mechanics
one on differential geometry for physics
one in general relativity and gravity

the lecture above dives into tensors in a pretty abstract way but thats really how it must be done to get a grip on what a tensor is.

also touches on dual spaces and concretizes with some examples.

tensors are one of those things that really made me appreciate deeper math because one (somewhat naive) way to thin about them is as the reality behind the representations And arbitrary coordinate choices.

Bump to remind me to post after work

https://www.youtube.com/watch?v=d4EgbgTm0Bg

You've heard of real numbers
You've seen complex numbers

but what else?

Quaternions! Fun fact - the guy who came up with these bad boys was so excited he carved the equations under a bridge that you can go check out even today allegedly.

These guys are super useful for 3D representations (i.e. vidya game production) and use a LOT less computation than if you were to go at it without them.

You can extend these again to the Octonions but strangely, there are no other known number systems. I can't even begin to wrap my mind around octonions...

Tag for cool math stuff to read later

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Romes187 posted...

You can extend these again to the Octonions but strangely, there are no other known number systems. I can't even begin to wrap my mind around octonions...

Strictly speaking you can also get stuff like the split complex numbers and variations on quaternions by fiddling with some of the definitions and ideas. For example the split complex numbers have i = 1 (not -1), and IIRC are a field just like the complex numbers. i = 0 gets you the dual numbers (not a field iirc).

I forget how many fields you can make over the reals - I want to say 8? I know it was covered in Jacobson's algebra texts, but I forget the details and don't have the books with me right now

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Eevee-Trainer posted...

Strictly speaking you can also get stuff like the split complex numbers and variations on quaternions by fiddling with some of the definitions and ideas. For example the split complex numbers have i = 1 (not -1), and IIRC are a field just like the complex numbers. i = 0 gets you the dual numbers (not a field iirc).

I forget how many fields you can make over the reals - I want to say 8? I know it was covered in Jacobson's algebra texts, but I forget the details and don't have the books with me right now

Haven't looked too much into split complex numbers but apparently its how to model a lorentz boost which is neat (or one way to do so)

thx!

https://youtu.be/0z1fIsUNhO4

theres a neat one I found today

gives a nice look at stereographic projection too

https://en.wikipedia.org/wiki/Involute

Involutes are neat curves (that depend on other curves)

They are also used when making involute gears (the pathway of the gear tooth follows an involute - https://en.wikipedia.org/wiki/Involute_gear)

There's also an evolute...and the evolute of an involute of a curve is the curve itself

https://en.wikipedia.org/wiki/Evolute

Curves are cool - and I don't understand why they don't teach the proper definition of a curve in high school (continuous map of an interval onto a top. space)

Romes187 posted...

guys can we please care about the same things i care about

This cry is the most relatable thing I've ever read. But Im not subjecting myself to a lecture. I didn't even put up with that shit when I was in high school.

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dolomedes posted...

ha, i came here thinking it was about biological homology. where are all the homologous structures?!

respect to math tho

hah

well i'll split the difference with you

https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life

the things you can do with that little bugger are pretty cool. there are a bunch of youtube vids showing what comes about from such a simple algorithm

Lost_All_Senses posted...

This cry is the most relatable thing I've ever read. But Im not subjecting myself to a lecture. I didn't even put up with that shit when I was in high school.

I'm not usually a fan of lectures either, but with Youtube you can throw out the boring ones and find the teachers who fit your style

and you do need to get into depth on some of this stuff to understand it...wikipedia only takes you so far

Romes187 posted...

I'm not usually a fan of lectures either, but with Youtube you can throw out the boring ones and find the teachers who fit your style

and you do need to get into depth on some of this stuff to understand it...wikipedia only takes you so far

I feel you. I feel like I only care to be lectured when the subject is really stupid, but the deeper you go into it, the more you realize even stupid stuff has depth if you look for it.

You know Whang!? He's a Youtuber that does a series called Tales From The Internet. He takes stuff that seemed really dumb from a bystander standpoint and really shows how deep they actually go. Like 2Girls1Cup and Southpark opening floodgates leading to 4Chan/Anonymous vs. Scientology. And he seems so well researched and well spoken.

This isn't really lecturing tho. So..y bad for getting off track. Ill peep the first couple minutes of your vid as compensation

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Omg, 47 minutes.

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Nope. Nu-uh. Hell no. NO

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Lost_All_Senses posted...

Omg, 47 minutes.

There are entire classes on homology

Probably not going to be interesting unless you care about topology

but the other vids i posted are shorter

Romes187 posted...

There are entire classes on homology

Probably not going to be interesting unless you care about topology

but the other vids i posted are shorter

I don't even have a high school diploma. I literally couldn't grasp one concept of what this is. I see "Higher Dimension" and I think of a dimension where people are getting higher than we do in this dimension. That's my brain.

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Lost_All_Senses posted...

I don't even have a high school diploma. I literally couldn't grasp one concept of what this is. I see "Higher Dimension" and I think of a dimension where people are getting higher than we do in this dimension. That's my brain.

No one can really visualize whats happening in higher dimensions

but you definitely don't need a high school diploma to be interested in topology

after you do enough linear algebra, higher dimensions don't really bother as much since you manipulate do 10, 20, a billion dimensional vectors quite easily.

We get caught up in conflating "dimension" with "spatial dimension"

Romes187 posted...

No one can really visualize whats happening in higher dimensions

but you definitely don't need a high school diploma to be interested in topology

after you do enough linear algebra, higher dimensions don't really bother as much since you manipulate do 10, 20, a billion dimensional vectors quite easily.

We get caught up in conflating "dimension" with "spatial dimension"

You make me want to burn books so others can't advance any further.

No but really, I think you're overestimating my ability to learn this type of stuff. I mean, I maybe could. But seeing as my brain is no longer developing like it did in my teens/20s, it would be way more of a struggle. I'm completely fine with leaving it up to people like you and the guy in the video. Go sneak into a college course and find people who are more on the ball.

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I am definitely not the one to do any research and advance knowledge on this stuff....I just like looking at it in my spare time because im married with 2 kids so what else am i gonna do hahah kill me

https://www.youtube.com/watch?v=pCpLWbHVNhk

Here's a fractal zoom that requires zero knowledge to appreciate the beauty of

https://en.wikipedia.org/wiki/Kronecker_delta

This lil guy shows up quite a bit. A version of it (called the dirac delta) is ubiquitous to Quantum mechanics

For QM it gives a big ole "spike" in the graph (towards infinity) and 0 everywhere else. This (somewhat) showcases the "collapsing" of a wave function.

I think of the kronecker delta as an on off switch...sometimes you want a function to be 0 if your indices don't match (thinking of an inner product of basis) and "on" (or 1) if they are the same. Inner products have a cosine of the angle between them, and this allows you to create a different way of describing it (i.e. if the basis are the same, angle is 0 and cosine = 1.)

I think I have that right...maybe. In any case the concept is cool (and like anything it goes much deeper)

https://youtu.be/ZwZD5zhwJQY

https://www.youtube.com/watch?v=KKr91v7yLcM

One of my favorite videos on visualizing a wave function (for QM). Helps you visualize why imaginary numbers come into play (the rotations around the imaginary component create the wave as seen from the front)

Moves into a 3d model as well. That guy makes a bunch of cool visualization videos

Romes187 posted...

He also doesn't believe in real numbers which is an interesting perspective, but he definitely backs up his argument with videos upon videos.

So I looked into this, and I somewhat see his point of view. He dislikes notions of assuming something exists axiomatically without proving it exists, and he seems to be big on construction. For instance he doesn't seem to like the axiom of infinity (assume there exists an infinite/inductive set X, the smallest of which are the naturals) - the axiom of infinity is independent from the rest of ZFC. He also doesn't like the notion of letting something exist without constructing such explicitly: it's one thing to say "there exists Y which is X", and to say "this Y is an example of X".

Thus he takes issue with the formal construction of the reals. Going from the naturals to the rationals he has no issue with, it's when you have to invoke infinities, infinite processes, whatever, in the construction of the reals that he takes issue.

Honestly I'm not sure where to stand on the matter myself. Math philosophy is weird.

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