And thank you RUclips for stealing all of our data and for bombarding every single video with multiple targeted ads. Sorry, I thought we were praying to the church of Google? Don't thank RUclips, MIT is the real hero here.
Thank you MIT. Please consider uploading more lecture series in abstract mathematics (e.g. real analysis, complex analysis, abstract algebra, differential geometry, topology, ...) It will be of great help to us!
I got nothing but gratitude for these amazing instructors, they are so talented and enthusiasts! For a second they made me want to come back to uni haha
What does "Stabilize the function" mean in the first proof? What exactly about the function is being stabilized? Is he just picking the last number for each n and assigning its color to the number n in the subsequence? Is this a correct paraphrasing of the argument: Create a a coloring of all of the integers by looking at the color assigned to n by our function for the integers 1 through n. This creates a coloring of all of the integers. By our infinitely assumption, there must be a mono coloring on all of the integers. Look at the first number where we must have a coloring. This is a coloring of r colors for a finite N, contradicting that no coloring must exist for any finite number.
The professor's statement that the Finatary version implies the Infinatary version is "obvious" at 6:45 is a bit inappropriate; nothing should ever be assumed to be obvious when you are teaching students. In fact, humble yourself. Nothing is ever obvious. And then his subsequent explanation seems off to me... "If you give me a finite coloring of the positive integers, well I just have to look far enough, up to this N, and I get the conclusion I want." No, that's backwards for how'd you prove Fin ==> Inf. You'd have to start with the set N mapped to from some r, and then infer that adding the next integer after the end of N to N _still solves the problem_ , and then you can continue to _extend that set_ to the set of natural numbers (since N has to be a _subset of the set of natural numbers_ from its definition), et voilà. You have the Inf thm. I believe that's the way you'd prove it. And once again, that isn't necessarily "obvious", and certainly not how the professor hand-waved it away.
The course is meant for students to whom it is obvious, if he spent time explaining every little thing, he would not get through his syllabus. Take it as an exercise an move on.
Nevermind, I get the existence, since the image is a finite set there exists a converging subsequence pointwise, and you can build these subsequences iteratively, hence a diagonal argument
Hi @@bermanmaxim I also have the same doubt. I tried to understand your explanation. If I understood correctly do you mean "any subsequence of the convergent mapping will repeat itself probably at some large value of N". I still do not get the meaning of the diagonal argument. It would be great if you (or anyone else) can shed more light on it.
@@japneetsingh5015 Think of a partial order on the set of intervals [1,N] covered by r colors where we say one interval is contained in the other if it is N
This is a graduate course. Prerequisites: No specific classes are required, but the course presupposes mathematical maturity at the level of a first-year math graduate student. See ocw.mit.edu/18-217F19 for more details. Best wishes on your studies!
Jeff Ahn your guess is better than mine. I shouldn't have assumed every coupling of English letters would automatically be reference to anything with which I or any other English-speaking American mechanic was already familiar. After all, one look at the phenotype of the subject referenced by the OP, and bam, I'm automatically disqualified from talking.
Jeff Ahn I just noticed your avatar color matches the one used by the OP. Pretty amazing, I mean what are the chances of that ever happening twice in a lifetime!
Can someone explain colors to me in this context i don't follow. Are they discrete or continuous ? (As like integer or any number on number line)Followup: does it matter?
You're just *choosing* any finite set of colours. They could all be grayscale, RGB, a subset of the reds, etc. As long as they're different. It's like saying choose any 3 numbers: 1,2,3 or pi, i, e. Under the conditions, it's all the same.
It is literally "coloring". Take numbers a1,a2,a_3,..., and then assign a color for each number. More formally, you can think of it as a function with the set of colors as the image. For example, you can color 1 with blue, 2 with green, and 3 with red. So, if an equation x+y=z has monochromatic solutions, it means that you can find x_0,y_0,z_0 with the same color such that x_0+y_0=z_0.
Yuzuru A when you can’t teach something in your own words or without looking at a book or a piece of paper then you’re a bad teacher because you don’t know enough about what you’re teaching that you still need supplementary assistance to teach them.
![BLACK BAG - Official Trailer [HD] - Only in Theaters March 14](http://i.ytimg.com/vi/Du0Xp8WX_7I/mqdefault.jpg)
Watching this as someone without a clue as to what any of this means, and I'm still fascinated. No wonder MIT is so sought after.
Thank you, MIT for making this publicly available and for free.
Thanks, RUclips for storing and serving the content.
And thank you RUclips for stealing all of our data and for bombarding every single video with multiple targeted ads.
Sorry, I thought we were praying to the church of Google?
Don't thank RUclips, MIT is the real hero here.
@@thetedmang You're trying to learn such complex math yet are unaware of something so elegant as adblockers lmao
Thank you MIT. Please consider uploading more lecture series in abstract mathematics (e.g. real analysis, complex analysis, abstract algebra, differential geometry, topology, ...) It will be of great help to us!
Exactly! I have been waiting for a good Real Analysis course for years!
Really?
@@enisten Nazis!!!
@@enisten to build a nuclear weapon Nazis!
@@enisten Sean Fujiwara
I got nothing but gratitude for these amazing instructors, they are so talented and enthusiasts!
For a second they made me want to come back to uni haha
Don't fall for it. It's a trap
@@vishwash6093 why tho? i want to come back to uni
Such video lecture series are tremendously helpful. Please keep on uploading more videos on graduate courses in mathematics.
Yufei Zhang is a legend
Fine lecture, amazing offering, PLUS, in the scuffs, dusts, and blears of the chalkboard panels, a poetry in black I haven't seen since Ad Reinhardt!
where can I read about the diagonalization technique he mentioned to prove the equivalence between infinite and finite forms
why the blackboard seems a little dirty?
This video is gold, not everyone gets the fact.
Manas Singh good but is it important?
Rýán Túçk yes
you mean, "no one gets the fact"?
I don't even understand the title of the video
What does "Stabilize the function" mean in the first proof? What exactly about the function is being stabilized? Is he just picking the last number for each n and assigning its color to the number n in the subsequence?
Is this a correct paraphrasing of the argument:
Create a a coloring of all of the integers by looking at the color assigned to n by our function for the integers 1 through n. This creates a coloring of all of the integers. By our infinitely assumption, there must be a mono coloring on all of the integers. Look at the first number where we must have a coloring. This is a coloring of r colors for a finite N, contradicting that no coloring must exist for any finite number.
Thanks MIT, winning hearts and minds.
The professor's statement that the Finatary version implies the Infinatary version is "obvious" at 6:45 is a bit inappropriate; nothing should ever be assumed to be obvious when you are teaching students. In fact, humble yourself. Nothing is ever obvious.
And then his subsequent explanation seems off to me... "If you give me a finite coloring of the positive integers, well I just have to look far enough, up to this N, and I get the conclusion I want." No, that's backwards for how'd you prove Fin ==> Inf. You'd have to start with the set N mapped to from some r, and then infer that adding the next integer after the end of N to N _still solves the problem_ , and then you can continue to _extend that set_ to the set of natural numbers (since N has to be a _subset of the set of natural numbers_ from its definition), et voilà. You have the Inf thm. I believe that's the way you'd prove it. And once again, that isn't necessarily "obvious", and certainly not how the professor hand-waved it away.
The course is meant for students to whom it is obvious, if he spent time explaining every little thing, he would not get through his syllabus. Take it as an exercise an move on.
these are wonderful lectures, please upload the lectures on Probabilistic Method from Prof. Yufei Zhao. THANK YOU!
this is pure gold
Just to check, so at about 27:38 the phi(k - j) = z? not y?
yeah should be y, z is written twice, a typo
Grazie per il contributo educativo.
OMG I love dz!!! Thank you 4 posting this treasure
This course only makes sense if you already know the content ahead of the time.
is this useful for GATE exam ??
I didn't get the diagonalization trick for the finitary Schur theorem, do someone know what "phi_N(k) stabilizes along the subsequence" mean?
Nevermind, I get the existence, since the image is a finite set there exists a converging subsequence pointwise, and you can build these subsequences iteratively, hence a diagonal argument
Hi @@bermanmaxim I also have the same doubt. I tried to understand your explanation. If I understood correctly do you mean "any subsequence of the convergent mapping will repeat itself probably at some large value of N". I still do not get the meaning of the diagonal argument. It would be great if you (or anyone else) can shed more light on it.
@@japneetsingh5015 Think of a partial order on the set of intervals [1,N] covered by r colors where we say one interval is contained in the other if it is N
Is this an undergraduate mathematics course? If so what year would students typically take this course?
This is a graduate course. Prerequisites: No specific classes are required, but the course presupposes mathematical maturity at the level of a first-year math graduate student. See ocw.mit.edu/18-217F19 for more details. Best wishes on your studies!
@@mitocw Thank you! I was curious because I'd never heard of this course before, it looks super interesting 😃
I am just here to be amazed of some math stuff.
Wierd algoritm. Suddenly I finally get what I actually subscribe. 👍🤷♀️
Gracias. Saludos
Watching this after learning my times 6 times table 😂
I love what you doing
What’s going on here?
The instructor is a genius. Look at his CV.
What is the CV?
몸빼 Constant Velocity
@@몸빼-j2s curriculum vitae. It's like a resume for academia.
Jeff Ahn your guess is better than mine. I shouldn't have assumed every coupling of English letters would automatically be reference to anything with which I or any other English-speaking American mechanic was already familiar. After all, one look at the phenotype of the subject referenced by the OP, and bam, I'm automatically disqualified from talking.
Jeff Ahn I just noticed your avatar color matches the one used by the OP. Pretty amazing, I mean what are the chances of that ever happening twice in a lifetime!
Can someone explain colors to me in this context i don't follow. Are they discrete or continuous ? (As like integer or any number on number line)Followup: does it matter?
You're just *choosing* any finite set of colours. They could all be grayscale, RGB, a subset of the reds, etc. As long as they're different. It's like saying choose any 3 numbers: 1,2,3 or pi, i, e. Under the conditions, it's all the same.
Take every natural number, assign a random color to it. Think coloring a map.
It is literally "coloring". Take numbers a1,a2,a_3,..., and then assign a color for each number. More formally, you can think of it as a function with the set of colors as the image.
For example, you can color 1 with blue, 2 with green, and 3 with red. So, if an equation x+y=z has monochromatic solutions, it means that you can find x_0,y_0,z_0 with the same color such that x_0+y_0=z_0.
He gives the vibes of mr. Mackay from south park. "Mkay? "😛
Haha, he is an amazing prof. Though. ❤️
Dude Im so gonna fail this course 😭
mathematicians have a talent to make any subject harder and duller than it really is
Yuzuru A when you can’t teach something in your own words or without looking at a book or a piece of paper then you’re a bad teacher because you don’t know enough about what you’re teaching that you still need supplementary assistance to teach them.
Lol
@@YuzuruA things non mathematicians say to make themselves feel better about themselves. Haha.
21:00
great
"tHerIs nO pRerrEquIsItiS fIr dIz cUrsE"
- MIT
I can't believe...As I'm unable to get anything except title.. 😂😂😂
"mathematical maturity"
Nice
Yufei Fucking Zhao? Omg this guy is a god.
EZ
Epic
No thank you.I'm okay with Calculus
Jor jor se bolke logo ko scheme bata de
oops wrong video
Se vale lavar el pizarrón.
++
That is a FILTHY black board......seriously.....you need better cleaning materials.
I think he is a chinese teacher
why should anybody care about this?
It's used in computer science.
@@seancashin1826 Sure it is...I do webdev and I will never need to know this crap!
@@captspeedy1899 webdev isn't computer science. It's monkey coding
@@oussematrabelsi9429 agree