For someone at about the upper math undergraduate level, the topics starting from localization here are interesting, because he gives a very crisp and clear introduction.
+baseballpro 42 cool that's ok. I guess more than a panic attack he is prolly bored & restless having to explain to laymen abt math. He gets impatient with fellow mathematicians as well during seminars.
So basically the reason to abstract things is so that we can go back and answer/re-answer questions about the things that were used to create those abstractions.
People are so intelligent where they are.. and that doesn't allow them to understand you Jacub ! You are so intelligent wherever you want to be .. But they are not . You are just so awesome ! . .
If commutative rings are an exotic homotopy theory (22:38 slide), then topology is to commutative rings as integers are to what? (What are the "commutative rings of commutative rings"?)
Great question! There are several ways to answer that. One of them is the following: we start with the observation that commutative rings are abelian groups equipped with an additional structure, namely a commutative multiplication that is compatible with the additive structure. This can be formalized by saying that commutative rings are commutative algebra objects in the category of abelian groups, or CRing = CAlg(Ab) for short. Now, if you try to infuse homotopy theory into the notion of abelian group in a coherent way, what you get is the category of spectra Sp. From any spectrum we can naturally extract an ``underlying abelian group" by taking ``connected components". To get commutative rings in these settings, we imitate the classical construction of CRing from Ab and consider CAlg(Sp): this is known as the category of ``commutative ring spectra" - whose theory is very well developed, in part thanks to Jacob Lurie's contributions, and requires the language of ``infinity categories" to be set up properly. I should note that the simplicial commutative rings mentioned in Lurie's lecture agree with the notion of (connective) commutative ring spectrum in characteristic zero, but not in general. Inside CAlg(Ab), the ring of integers Z can be singled out as the commutative ring with the property that for any other commutative ring R, there is a unique ring homomorphism from Z to R (1 must go to the unit in R, and the rest is uniquely determined by definition). This property can be spelled out in CAlg(Sp), and it turns out that the commutative ring spectrum satisfying it is the sphere spectrum S, which you can think of as obtained by ``strapping spheres of every dimension together". Nicely enough, if we take the connected components of S, we recover precisely Z. This reflects the fact from algebraic topology that \pi_n(S^n) = Z for every n > 0. Thus, one possible interpretation of your analogy request is the following: Classical commutative rings : Integers :: Commutative rings in homotopy theory : Sphere spectrum
Wow he tried to jump off the ceiling of advanced math to the ground where we mortals could understand. As a great teacher does he takes historical approach. Abstraction is natural tool for any intellectual activity . even programmers abstract or generilizes using programming structures, to functions(modules) to objects to processes etc.
@@mlevy2429 Unless you are specifically studying homotopy groups of spheres or whatever, as I understand spectral sequences and all that can essentially be absorbed into infinity category framework in a natural way.
I am missing something at 6:30 (Fermat Theorem). Isn't 21 a prime number of the form 4n+1 (for n=5)? It cannot be written as a sum of 2 squares though. What am I thinking wrong?
I like to add one more property of number, to Jacob Lurie's list: Complex number i is defined by Tristan Needham (page 217 of Visual Complex Analysis) as the ratio of change with y of the image f to the rate of change with x of the image f of the complex number z=x+iy when mapped to the w-plane, implying change in y due to change in x, or the ratio of change in effect y due to change in cause x. Empiricists toiled for hundreds of years to find what relates cause and effect. Failing to find any physical relation, they settled that they were not related, or they were independent. Little did they know that a number related cause and effect.
I guess the speaker was out of the world genius and clear so the audience either didn't understand well to ask a question or understood too well not to ask a question
If I may ask a question, is there any particular reason for which this talk is so elementary? I mean, I can understand this is a colloquium and there is no need to go throw technical details but I guess that the general public of this events are, at least, at a high undergraduate level.
Usually when these big events happen where people from multiple fields are given prizes they go watch talks from other people. So in the audience there are medical/life scientists and phycisists who most likely have never heard of abstract algebra
hmm that makes sense. Then I see this talk as a good opportunity to motivate some concepts and give a rapid taste of how mathematicians develop their field as Jacob did.
Colloquia are always very low level. It's anticipated that there will be lots of applied mathematicians in the audience who don't even know what the fundamental group is.
@@MK-13337physicists use abstract algebra. For example in the creation of gauge theories, manifolds, and representation theory, etc. Not sure where you’re getting this notion from.
@@hambonesmithsonian8085 Yes, some physicists use abstract algebra, I would say that some physicists use almost all kinds of math. If a physicist is not specialized in QFT and hasn't worked with group theory to define symmetries then they don't need abstract algebra.
05:34: Can someone explain his "simple observation" that the left must be odd and the right must be even for: X2+X+1=y3 - y Do you Maths guys just "see this" as obvious? Hahah I can never get in even at the ground floor of maths ☹️
Sure. Odd numbers have the form 2n + 1 for some integer n, and even numbers have the form 2n. On the left hand side, try plugging both of those in for x and you will find that you get something of the form 2( ) + 1 for each case so it must be odd. Similarly, plug both in on the right and you will get something of the form 2( ) for each case so it must be even. Also, terms like "clearly" and "obvious" are sometimes abused a bit in math, even in textbooks, and people in the math community often joke about it and how it can be a bit frustrating at times when a textbook or professor says that.
I forgot to mention that, as Lurie said, you could instead work in mod 2 by plugging in 0 and 1 on both sides. In mod 2, even numbers are equivalent to 0 and odd numbers are equivalent to 1. On the left, plugging in 0 gives you 1, and plugging in 1 gives you 3 which is equivalent to 1 in mod 2. So the left will always be odd. On the right, you will get 0 in both cases so it must be even. Notice that doing this is more simple than the algebraic method I mentioned before, and this is a good example of why abstract tools can be helpful.
@@michaelmerkle297 haha thanks! I follow your first post. But how do you become convinced that 2n+1 = odd? I mean, I am convinced. I see it. But maths is supposed to be about proof and logical watertightness right? Is there more than an induction on n=1, n=2 and so on for a few examples here?? Is induction acceptable in maths? Here my question is in the spirit of: I know I am wrong, but I don't know why!!! Hahah
@@edwardjones2202 If you really want a proof, I would just go by contradiction. Assume n is a whole number and 2n + 1 is even. Then when we divide by 2, we should get a whole number. But dividing 2n + 1 by 2 gives us n + 1/2 which is not a whole number.
Great lecture. Of course, it is important to glue the edge of one Mobius strip to the middle of another, otherwise you can land with a Klein bottle or RP^2.
@12:34 I'd call Sphere & Torus 3D and Circle 2D, rather than 2D & 1D. I understand the motivation in wanting to emphasize the contour over the content, but ultimately you need a minimum of THREE spatial dimensions to even discuss the Sphere and Torus, while you only need a minimum of TWO for the circle. So the content supersedes the contour.
I'm pretty sure he's referring to their dimension as manifolds, which is built into the definition of a maniold (each point has a neighbourhood homeomorphic to a disc in R^(their dimension)). This is very standard in mathematics - you might say that the point is that the Torus and Sphere should live independant of their embedding - they exist on their own, without reference to putting them in a larger space.
Yes, Every mathematicians are different and their interests is also different and their skills & techniques are very different from one to other, and their research topics are very different.
Shame for the mathematics committees in America, especially for neglecting my solution. They and the rest of the world's mathematicians were defeated by solving the Collatz Sequence. These actions towards me are an indication that humanity is just empty talk and lies.
Never got to Harvard, but I've always been good at "your math". So you run into a wall without a solution, then create it ? You can substitute any random "variable" into any chosen equation, given it it is psychologically finite. All the math that you have learned have been from books and history...none from your own intuition that I view from your video. This is what Harvard and MIT breeds? to shun inventors and appraise "scholars" lol so called. Addition is an X and Subtraction is Y, Z is always imaginary when dividing a "proof" or things that are not there, visibly. I would like to see more of your videos, because you touch on many ideas that I ponder on.
![Felix "Unfair" | [Stray Kids : SKZ-PLAYER]](http://i.ytimg.com/vi/Oswujxm2Ag0/mqdefault.jpg)
This talk demonstrates perfectly why algebraic topologists must be deeply admired
For someone at about the upper math undergraduate level, the topics starting from localization here are interesting, because he gives a very crisp and clear introduction.
I just fell in love with abstract algebra.....again ! :)
No I think he has a disability
+baseballpro 42 That makes no sense ! disability in what? Mathematics??
sorry i was trying a reply to a different comment sorry
+Nishant Sawant because somone said he having a painic attack
+baseballpro 42 cool that's ok. I guess more than a panic attack he is prolly bored & restless having to explain to laymen abt math. He gets impatient with fellow mathematicians as well during seminars.
So basically the reason to abstract things is so that we can go back and answer/re-answer questions about the things that were used to create those abstractions.
It's very interesting to hear someone very smart giving a lecture. I mean a genius one.
+Jamnian Nantadilok ;) Very smart isn't quite enough.
this guy would need to drop 50 iq points to get smart.
Simon wut was his iq?
Nayr it’s a meme.ruclips.net/video/6RHlggRGy28/видео.html
Why hasn't he got a Fields yet?
People are so intelligent where they are.. and that doesn't allow them to understand you Jacub !
You are so intelligent wherever you want to be .. But they are not .
You are just so awesome ! . .
I'm five years late, but this seems like a REALLY passive-aggressive comment, lol.
If commutative rings are an exotic homotopy theory (22:38 slide), then topology is to commutative rings as integers are to what? (What are the "commutative rings of commutative rings"?)
Great question! There are several ways to answer that. One of them is the following: we start with the observation that commutative rings are abelian groups equipped with an additional structure, namely a commutative multiplication that is compatible with the additive structure. This can be formalized by saying that commutative rings are commutative algebra objects in the category of abelian groups, or CRing = CAlg(Ab) for short. Now, if you try to infuse homotopy theory into the notion of abelian group in a coherent way, what you get is the category of spectra Sp. From any spectrum we can naturally extract an ``underlying abelian group" by taking ``connected components". To get commutative rings in these settings, we imitate the classical construction of CRing from Ab and consider CAlg(Sp): this is known as the category of ``commutative ring spectra" - whose theory is very well developed, in part thanks to Jacob Lurie's contributions, and requires the language of ``infinity categories" to be set up properly. I should note that the simplicial commutative rings mentioned in Lurie's lecture agree with the notion of (connective) commutative ring spectrum in characteristic zero, but not in general.
Inside CAlg(Ab), the ring of integers Z can be singled out as the commutative ring with the property that for any other commutative ring R, there is a unique ring homomorphism from Z to R (1 must go to the unit in R, and the rest is uniquely determined by definition). This property can be spelled out in CAlg(Sp), and it turns out that the commutative ring spectrum satisfying it is the sphere spectrum S, which you can think of as obtained by ``strapping spheres of every dimension together". Nicely enough, if we take the connected components of S, we recover precisely Z. This reflects the fact from algebraic topology that \pi_n(S^n) = Z for every n > 0.
Thus, one possible interpretation of your analogy request is the following:
Classical commutative rings : Integers :: Commutative rings in homotopy theory : Sphere spectrum
@14:32 _'and sew on and sew forth'_
(I wonder if Lurie intended this pun)
Probably.
I'm surprised he chose the topic of homotopy theories rather than something more related to his work in infinity categories and higher topoi.
+Christian LaPointe Yeah that's the first thing that popped in my mind.
I wouldnt understand wht he is talking about, this is palatable
Wow he tried to jump off the ceiling of advanced math to the ground where we mortals could understand. As a great teacher does he takes historical approach. Abstraction is natural tool for any intellectual activity . even programmers abstract or generilizes using programming structures, to functions(modules) to objects to processes etc.
Infinity categories and homotopy theory and almost exactly the same thing. That is the point of the infinity-categorical perspective... duh
@@mlevy2429 Unless you are specifically studying homotopy groups of spheres or whatever, as I understand spectral sequences and all that can essentially be absorbed into infinity category framework in a natural way.
I am missing something at 6:30 (Fermat Theorem). Isn't 21 a prime number of the form 4n+1 (for n=5)? It cannot be written as a sum of 2 squares though. What am I thinking wrong?
***** You are absolutely right of course... Sorry for spending your time with my silly question. I got stuck.
21 is not a prime 🤣
I like to add one more property of number, to Jacob Lurie's list: Complex number i is defined by Tristan Needham (page 217 of Visual Complex Analysis) as the ratio of change with y of the image f to the rate of change with x of the image f of the complex number z=x+iy when mapped to the w-plane, implying change in y due to change in x, or the ratio of change in effect y due to change in cause x.
Empiricists toiled for hundreds of years to find what relates cause and effect. Failing to find any physical relation, they settled that they were not related, or they were independent. Little did they know that a number related cause and effect.
How in the world were there no questions from the audience?
I guess the speaker was out of the world genius and clear so the audience either didn't understand well to ask a question or understood too well not to ask a question
If I may ask a question, is there any particular reason for which this talk is so elementary? I mean, I can understand this is a colloquium and there is no need to go throw technical details but I guess that the general public of this events are, at least, at a high undergraduate level.
Usually when these big events happen where people from multiple fields are given prizes they go watch talks from other people. So in the audience there are medical/life scientists and phycisists who most likely have never heard of abstract algebra
hmm that makes sense. Then I see this talk as a good opportunity to motivate some concepts and give a rapid taste of how mathematicians develop their field as Jacob did.
Colloquia are always very low level. It's anticipated that there will be lots of applied mathematicians in the audience who don't even know what the fundamental group is.
@@MK-13337physicists use abstract algebra. For example in the creation of gauge theories, manifolds, and representation theory, etc. Not sure where you’re getting this notion from.
@@hambonesmithsonian8085 Yes, some physicists use abstract algebra, I would say that some physicists use almost all kinds of math. If a physicist is not specialized in QFT and hasn't worked with group theory to define symmetries then they don't need abstract algebra.
Absolutely lighting!
pure mathematician anthony thompson
陳永霖
For such a contribution, I expected a lot more questions. Just one?! ...
That's pretty normal after a mathematics talk.
05:34:
Can someone explain his "simple observation" that the left must be odd and the right must be even for:
X2+X+1=y3 - y
Do you Maths guys just "see this" as obvious? Hahah I can never get in even at the ground floor of maths ☹️
Sure. Odd numbers have the form 2n + 1 for some integer n, and even numbers have the form 2n. On the left hand side, try plugging both of those in for x and you will find that you get something of the form 2( ) + 1 for each case so it must be odd. Similarly, plug both in on the right and you will get something of the form 2( ) for each case so it must be even. Also, terms like "clearly" and "obvious" are sometimes abused a bit in math, even in textbooks, and people in the math community often joke about it and how it can be a bit frustrating at times when a textbook or professor says that.
I forgot to mention that, as Lurie said, you could instead work in mod 2 by plugging in 0 and 1 on both sides. In mod 2, even numbers are equivalent to 0 and odd numbers are equivalent to 1. On the left, plugging in 0 gives you 1, and plugging in 1 gives you 3 which is equivalent to 1 in mod 2. So the left will always be odd. On the right, you will get 0 in both cases so it must be even. Notice that doing this is more simple than the algebraic method I mentioned before, and this is a good example of why abstract tools can be helpful.
@@michaelmerkle297 haha thanks! I follow your first post.
But how do you become convinced that 2n+1 = odd? I mean, I am convinced. I see it. But maths is supposed to be about proof and logical watertightness right? Is there more than an induction on n=1, n=2 and so on for a few examples here?? Is induction acceptable in maths?
Here my question is in the spirit of: I know I am wrong, but I don't know why!!! Hahah
@@edwardjones2202 If you really want a proof, I would just go by contradiction. Assume n is a whole number and 2n + 1 is even. Then when we divide by 2, we should get a whole number. But dividing 2n + 1 by 2 gives us n + 1/2 which is not a whole number.
Great lecture
Very talented person and I see on him Abbel and Galois like revolutionary capacities.
Great lecture. Of course, it is important to glue the edge of one Mobius strip to the middle of another, otherwise you can land with a Klein bottle or RP^2.
FYI pi is being critisized severely.
Please go back to preschool
Sounds like yap
Ben nefes nefese kaldım be hocam!
hocam
@12:34 I'd call Sphere & Torus 3D and Circle 2D, rather than 2D & 1D.
I understand the motivation in wanting to emphasize the contour over the content, but ultimately you need a minimum of THREE spatial dimensions to even discuss the Sphere and Torus, while you only need a minimum of TWO for the circle. So the content supersedes the contour.
I'm pretty sure he's referring to their dimension as manifolds, which is built into the definition of a maniold (each point has a neighbourhood homeomorphic to a disc in R^(their dimension)). This is very standard in mathematics - you might say that the point is that the Torus and Sphere should live independant of their embedding - they exist on their own, without reference to putting them in a larger space.
A great lecture, but not quite understandable for non-mathematicians. I guess.
Is this guy smarter than Terry Tao?
They're all at very high level. There area of research are different. He work is in higher topoi theory.
Yes, Every mathematicians are different and their interests is also different and their skills & techniques are very different from one to other, and their research topics are very different.
Completely different research areas, they're both extremely talented.
Shame for the mathematics committees in America, especially for neglecting my solution. They and the rest of the world's mathematicians were defeated by solving the Collatz Sequence. These actions towards me are an indication that humanity is just empty talk and lies.
What was your solution?
Sounds like yap
This guy's nose is too high. If one eye is blind, the other eye will be blocked by the nose.
Thanks for this very valuable and useful comment.
Never got to Harvard, but I've always been good at "your math". So you run into a wall without a solution, then create it ? You can substitute any random "variable" into any chosen equation, given it it is psychologically finite. All the math that you have learned have been from books and history...none from your own intuition that I view from your video. This is what Harvard and MIT breeds? to shun inventors and appraise "scholars" lol so called. Addition is an X and Subtraction is Y, Z is always imaginary when dividing a "proof" or things that are not there, visibly. I would like to see more of your videos, because you touch on many ideas that I ponder on.
Flutang What the heck are you on about?
@@vlix123 Schizophrenia
Sounds like someone who’s coping and seething with their yap.