What is algebraic geometry?
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- Опубліковано 16 жов 2023
- Algebraic geometry is often presented as the study of zeroes of polynomial equations. But it's really about something much deeper: the duality between abstract algebra and geometry.
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A HUGE HUGE thank you to Faisal Al-Faisal for working with me on the script and storyboard for this video!
And another thank you to Davide Radaelli for helpful conversations when making this video.
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CORRECTIONS:
At 4:26, I mistakenly wrote that g(1,1)=-2. This is a typo! The corrected version is g(1,-1)=-2.
SOURCES and REFERENCES for Further Reading!
(a) “A guide to plane algebraic curves” by Keith Kendig. It’s written in a very elementary style and has lots of really captivating diagrams throughout. If you look at the table of contents, it starts off with lots of examples that only require elementary algebra. And by the end, it actually gets to some pretty deep theorems in algebraic geometry.
(b) "Ideals, Varieties, and Algorithms” by Cox, Little, O’ Shea. This book does not assume any knowledge of abstract algebra and teaches everything from the ground up. It is a very nice book with plenty of computational examples and exercises.
(c) “Algebraic Geometry and Arithmetic Curves” by Qing Liu. This books is all about schemes and Spec. It's a rather terse theorem-proof style book, but it is beautifully written and has lots of exercises.
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MUSIC CREDITS:
The song is “Taking Flight”, by Vince Rubinetti.
www.vincentrubinetti.com/
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What is algebraic geometry?: (0:00)
Coordinate Ring: (3:04)
How algebra detects reducibility: (3:54)
How algebra detects a node: (5:15)
Schemes!: (8:00)
Please please please make more algebraic geometry or commutative algebra videos. These are really great!
your wish is my command :) more coming up real soon!
please please please commutative algebra video pretty please.
@@Aleph0 Topos Theory and Schemes/Sheaves/Stalks please.
@@Aleph0 Please please please marry my daughter.
Lookup Hodge Conjecture (David Metzler is the uploader). You're welcome 😊
Please make more videos on algebraic geometry, please. These videos are treasures.
hey thanks! more AG videos are coming up real soon :)
Best advanced math education channel on UA-cam. I struggled immensely with algebraic geometry in college. The definitions and concepts weren’t properly motivated. So I learned in a painfully mechanical way.
An excellent introductory video. I should have watched it before I took algebraic geometry or read Gathmann's.
The main fault I see with this video is that doesn’t motivate AG with purely-AG big problems but had to mention FLT or Weil conjectures (which are arithmetic geometry), making AG look like a tool for other math branches. Regardless of that, I hope this series complements well the long video series of Borcherds
@@zy9662It's hard to explain the minimal model programme or the Hodge conjecture to a wide audience. FLT and the Riemann hypothesis over finite fields is far easier to grasp to a layperson. The simplest open problem in algebraic geometry is, by far, the Jacobian conjecture. Everything else is beyond the reach of even advanced PhD students.
@@goldjoinery thanks for your comment. To your point, he didn’t explain the Weil conjectures either so he could have mentioned those and also Hodge or Riemann Roch
Wow, I don't think there is a better introduction to ideals in algebraic geometry.
Individuals that have spare money, if I were one of you, I would consider donating to this man. He has the most simple yet beautiful way of sharing knowledge I've seen since I discovered 3b1b. Give this man a chance to make more videos like this one more frequently. ❤
I am totally with you!
It's really cool that you talked about schemes! For such an advanced topic it's really nice to see a video even mentioning it
4:25
There is a typo I think: should be g(1,-1)=-2.
Aside from that, congrats for the really nice explanation.
but f(1,-1) would then be 0. I can't think of an example that works where the product is zero but the individual functions aren't.
@@psd993 yeah, i feel like he's not explaining it fully. even for the next example, y^2-x^2*(x+1)=0, the curve is fully plotted by the two curves y-x*sqrt(x+1)=0 and y+x*sqrt(x+1)=0. maybe the point is they're not always zero where the graph is? i don't get it
@@psd993 if a function is not zero then that doesn't mean it can't return zero. for a function to be considered zero it has to return zero _everywhere_ in its domain
Yeah he just wants to show any non-zero element to show it's not identically zero while its multiple with the other is identically zero (due to the constraint, or being in the quotient ring, whatever you want to call it).
@@pozatat he's talking about polynomials on reals in that part. He explains later on with the power series rings
As an algebraic geometer/commutative algebraist, this video describes exactly how we think about shapes and their corresponding rings. Great job!
(for any graduate students reading this: Read Hartshorne's Algebraic Geometry book. It is, IMHO, the end all be all reference for introductory algebraic geometry.)
Shafarevich, Gathmann and Vakil and Eisenbud (Geometry of Schemes) are also good books
For the exercises maybe but to learn from I would not recommend. Qing Liu is much easier to learn schemes from. For cohomology though, Hartshorne is pretty decent
Gathmann notes are great as well!
Ravi Vakil's Rising Sea is my favorite, it has a nice modern approach
Not a graduate student just an ethusiast just began learning math currently reading linear algebra done right by sheldon axler (Really good book imho) would you recommend this for me?
This is a really nice appetizer, it's so rare for a video on algebraic geometry to actually go far enough to talk about schemes
Such a fascinating video! Your videos tend to ignite a spark of curiousity everytime i watch them, thanks so much!
You really made my day ❤️,
Please make a series out of it, the world will remember you
Appreciable work. Keep on providing introductory videos (+ additional resources) of Advanced Math Courses. As a highly motivated undergrad, it really helped me to study these advanced topics with good intuition and a good introductory recourse (that book you mentioned). Anyway, Thanks and keep on guiding us.☺
Small error but at 4:27 I believe it should say g(1,-1)=-2 instead of g(1,1)=2
I literally noticed the same thing like 12 hours ago and thought I was losing my mind so thank you.
Not sure about that. Author is trying to show that function F(x, y) = f(x,y) * g(x,y) is 0 on (1,1) arguments while `f` and `g` are both non-zero on these, but that's not the case. g(x,y) = y - x is 0 on (1,1).
@@bydlobydlo nope, he’s trying to show they’re not identically zero (while their product is), so any non-zero element illustrates the point.
@@gi99hf60 but he didn't say that any non-zero element illustrates the point, he clearly says both are non-zero.
Lol it is fucked up because he is trying to prove that the product of both functions f(y,x) and g(y,x) with y=1 and x=1 is equal to 0, while each f(1,1) and g(1,1) are not equal to zero, which is clearly not true as g(1,1) is equal to zero. Furthermore if you have a*b = 0 how can you claim that neither a nor b are equal to 0. Are we nuts?
I was always too intimidated to begin studying this topic I’ve laways been intrested in, but this video has definitely done a good job at helping me croos that threshold. So thanks!
Great stuff, as usual
The best layman presentation of algebraic geometry I have ever seen. Great!
Was in the ER this morning, but a new aleph 0 video has made this a good day.
Damn I hope that was nothing too serious
Qing Liu’s book is great. I also really like Eisenbud and Harris “The Geometry of Schemes”, and Mumford’s “Red Book” is just a rare jewel, so beautiful, with all those drawings of schemes (some also reproduced in Eisenbud-Harris)
He teaches me commutative algebras, most of his lectures are improvised because it's too easy for him
Who?@@oportbis
@@lucianonotarfrancesco4443 Qing Liu
@@oportbis oh, Qing Liu! Awesome, you’re very lucky!
wow!! It's so amazing. You are very good at explaining everything! Well done!!!!
will you make a video about special points in algebraic geometry, such as node, biflecnode, tacnode and so on... ?
As a mathematical physicist, the immediate question that popped into my brain is, "How does this relate to differential geometry?" For example, the curve having a self intersection in one of the examples, which corresponds to the ring not being an integral domain, manifests itself in differential geometry as the curve not being a manifold-ie no diffeomorphism with R around the intersection point.
Many of the modern definitions for geometric properties in algebraic geometry come from differential geometry. For instance, the definition of the cotangent bundle of a space comes from a translation of the differential geometry construction into ring theory.
There are also many connections between the study of sheaf theory in both areas. de Rham cohomology and the usual cohomology theories in algebraic geometry agree in the study of common geometric object and can be used as tools to understand each other, for example.
Algebraic geometry also has some deep roots in the study of string theory, if you're into that :-)
There are algebraic varieties that are not manifolds (you found an example) and viceversa (e.g. the graph of e^x)
Great video! I got a my Math PhD but never explored algebra beyond my quals. I’ll give some of these books a shot sometime!
Awesome as always. For the curious and passionate, I suggest Hartshorne book on Algebraic Geometry which is the best. We used it as a basic introduction.
Great video. So nice to see a new video from you. Thank you
excellent quality of explanation. Please more videos on this topic.
Great introduction! Very thoughtful and wise presentation.
Great video as always ! I'm an applied maths guy and I'm always so puzzled when I hear people talk about algebraic geometry, it sounds to me like a bunch of cryptic, abstract nonsense. At least now I have an idea of what's going on :)
So good to have you back!
7:40 While yes, it is possible to compute many geometric properties using the algebraic description, it should be noted that doing so can be very hard, especially if the dimension of the components gets big. (in particular, most algorithms make heavy use of groebner basis which might have a size double exponential in the input. But they still work reasonably well most of the time)
It would still be a lot harder using just geometric arguments, isn't?
Brother, Ill let you know that I'm inspired!!! It's so well-done. Thanks😻😻
Thanks for suggesting the books in the end. Might take a look into the subject soon enough
hey this was a really well done video! the level of abstraction seemed just right, and that's a difficult needle to thread
AMAZING!!! Thank you so much!!!
Thank you so much for this video. I am an engineering student, interested in pure Math. I am not good by any stretch of the imagination, but I feel comfortable recommending "Lectures on Curves, Surfaces and Projective Varieties" by Beltrametti. It is a classical approach to algebraic geometry with minimal prerequisites, including basic undergraduate math and projective geometry. It predates Grothendiek and his revolution, but its extremely lucid and does not feel impenetrable at all, unlike Hartshorne for example
After so long, nice seeing you, great video
Aleph is back ! Good see you
Thanks for the lesson
This was so cool. You motivated me to keep my self studying....thank you
Ideals, Varieties and Algorithms is an outstanding book. But, you're gonna need to know how to use a computer to compute Groebner bases. You're going to struggle to learn the big ideas if you can't use MatLab or Mathematica.
I'd also add as a suggestion, the Red Book of Varieties and Schemes as a pretty good text. Hartshorne of course, but that one is really tough.
I really have to hand it to you, the style of the video, the explanation and especially the beautiful music in the background make every video of yours feel like I'm gaining +10 IQ points every time I watch them! 😊 Really great work, such beautiful explanations.
I don't even come here to learn. I love listening to these math vids where a nice person shows me something cool with a calm voice. The best
Oh, you should checkout @mathcuratorzanachan35742
Decent video. The final part about any ring (here Z) being thought of as functions on it's prime spectrum was also very mind blowing for me when I first saw it
It’s a very inspiring video, thank you for making it!
More videos on Algebraic Geometry please!
May I ask a clarification? At 4:25, you say that g(x,y) = y - x and so g(1,1) is -2. Shouldn't it 0 since g(1,1) = 1 - 1 = 0? Or am I missing something?
Exactly what I need answer for
Thanks for the correction! This is indeed a typo - I meant to write g(1,-1)=-2. I've added a correction to the description.
4:25 should be g(1,-1) or any other non-zero yielding (x,y)
Your last example reminds me of topology. Like, Z^2 counts the ways you can wrap a stretchy oriented circle around a stretchy oriented torus.
I guess that's groups and not rings though.
Super cool, I've been waiting for this
Thank-you for sharing these wonderful insights.
What a great educator and math experience!
There is also real algebraic geometry, which focuses on differential geometric techniques like morse theory/critical points of functions rather than focusing on purely algebraic techniques that come from complex number and finite field considerations.
It applies to ordinary manifolds/real geometries in a unique and different way:
1952 - John Nash proved that every closed smooth manifold is diffeomorphic to a nonsingular component of a real algebraic set
(shamelessly taken from the Wikipedia page on the history of real algebraic geometry)
Excellent video! 🎉😊
Really nice. Actually carries you across the threshold of the the two are related (even 'married' together;-).
I've had the feeling that zero and one should also be trivially prime, when staring at the empty set, because the higher number don't exist yet, so we get the somewhat trivial zero, one, two, three, before we get a (the first) repeated addition value for checking (i.e. "four", oh, that's 2+2..). [copyright: silly ideas from the internet;-) ]
If you want to learn Algebraic Geometry at the graduate level, but Liu is feeling a bit too terse and impenetrable for you, I'd also suggest "Algebraic Geometry I" by Görtz and Wedhorn. Such a lovely book.
I used both and they complement each other beautifully IMO
the algebraic geometry notes by Ravi Vakil are great too and freely available on the internet
Having basically only had Hartshorne through my university courses, a few recommendations on the lighter side is always welcome.
I enjoyed the book recommendations in conjuction eith the video thanks.
Fantastic work
Great video. I don't do marker pens, I find them messy and the noise goes through me, but you did a neat job with yours!
Wow this is really an amazing introduction of AG.
I am very happy to see many people in comment section who know about AG.
I am an undergraduate student (just started 3rd year, math major), I am also interested in AG, but unfortunately I don't know very much about it.
Currently I am studying group theory (using : Gallian's book, farilegh's book, A book of abstract algebra and D&F), real analysis (Abbott), proof writing (velleman). I will appreciate if any advice for studying mathematics towards Algebraic Geometry.
Moreover, is it necessary to study all undergraduate math subjects for better understanding (specially for AG)? Because I am less focusing on applied ones like numerical analysis, dynamics, mechanics, ODEs etc. On the other hand I am focusing on pure subjects like abstract algebra, analysis, topology, etc
Thank you.
For algebraic geometry you need commutative algebra(study of commutative rings with unity), and the more topology you know the better
ahhh you are back!!
this is so insanely good
This video isamazing!!! Very clever!
04:25 wrong calculation
g(x, y) = y - x
Okay, but below:
g(1, 1) = -2
is wrong
g(1, 1) = 1 - 1 = 0
not -2
Yeah that kind of invalidate all he said about algebra detecting irreducible curves
@@zy9662 it doesn't, because you can still input 1,-1 (which is on the curve) and it doesn't return 0
@@kingarthur4088 that makes sense lmao god damn
@@gabitheancient7664 I think this does not make sense... the important part (that would make R weird) is that y+x AND y-x != 0 for some point (x,y), yet (y+x)(y-x)=0, at this same point (x,y)
@@Blackmuhahah no that's not the important part, the important part is that the functions are not *identically* 0, it'd be literally impossible for the two factors to be different than 0 for every point but multiplying to 0
though he said that it's weird to factor 0 into non-zero things, that's just a vibe, there's nothing wrong with an identically 0 function to factor into two non-identically 0 functions, tho it does mean something in this context
Very nice explanation
I would suggest before tackling algebraic geometry to first master basic commutative algebra (like Miles Reid Undergraduate Commutative Algebra)
At 4:25, g(x,y) = y-x evaluates to 0 in (x,y) = (1,1), yet you mentioned it equals to -2. Am I completely oblivious to some mistake I made here, or did you make a mistake? You used this result to establish that a product of two nonzero elements in the quotient ring can still equal to zero. But this isn’t a good example as one of the factors ís indeed zero.
Can anyone help me out here?
A video about the weil conjectures would be great 😊😊🙏
Super interesting, thanks
One word: beautiful!
I feel like if I had been presented algebraic geometry like this when it was my master's research area, I might have finished my PhD in mathematics.
awesome video, hoping you do some homological algebra soon
Thanks for this nice video.
Happy to start learning Algebraic Geometry from you 😊
Very beautiful!
Gracias por compartir
When the world needs him, he comes !!!!!
Now I'll take this course next semester
To be symmetric, how about a video on geometric algebra?
Thank you!
There is a mistake at 4:25 that states g(1,1) = -2 while it should be 0 as 1-1=0. Was that supposed to be -y-x or I don't understand something?
highlight of my day 🥰
At 4:27, how is g(1,1) = -2? Should'nt it be 0? Or am I understanding something wrong?
4:28 should this be g(-1,1)?
This idea of algebra functions being on a geometric object sounds similar to the idea of Morse theory where smooth functions on a manifold determine the topology of that manifold.
I couldn't understand 4:26. (1, 1) is a point in the curve (y-x)*(y+x) = 0, since 0*2 = 0. So since the domain of g(x,y) = y-x is the curve we can evaluate g in the point (1,1). In fact, g(1, 1) is equal to 1 - 1 = 0 != -2. I would be grateful if someone appointed what I am missing here. Thanks.
This is fantastic video! Thank you very much.
I would be grateful if you could answer a question:
In y^2 = x^2(x+1) example, you say that the node at (0,0) can be detected by fact that you can find zero divisors in the quotient ring R[[x,y]] / (...). Does the factorisation tell the location of the node or only that the node exists?
The ring actually knows about the point (0,0). What is going on here is that we’re localizing at (0,0) (i.e. at the maximal ideal m=(x-0, y-0)) and then taking the m-adic completion of the resulting local ring. This is the construction that produces R[[x,y]]/(…). The point (0,0) is baked into the process.
If we were to apply this localization-completion process at any another point, then the resulting ring won’t have any zero divisors! (In fact it will be isomorphic to R[[x]].)
FASCINATING!
The goat has returned
If possible, could you do a video on what is differential geometry?
This is lit.
By the way, what's the Background music?
This was such a good video! I’d love to get some more algebraic geometry content at some point.
Can you explain ramification of morphisms. I know the map from unit circle to y axis has two such point, since two of them have a single preimage, rest have two preimages. In general how to think of these?
I know more topology than geometry, and this isn’t a complete answer to your question. But, you might be interested in the notion of covering maps. A covering map is a special type of map from one topological space onto another.
Consider a covering map q: X -> Y. One important property is that the number of points of X in the fiber of any point of Y is constant. Another important fact is that the fundamental group of X is mapped injectively into the fundamental group of Y. This will let you know, for example, that a circle cannot cover a line, because the circle has an infinite cyclic fundamental group while the line has a trivial fundamental group.
However, a line can cover a circle: start with the real number line, and map each integer to a base point of the circle, letting the interval between two consecutive integers wrap around the circle. In this covering map, the fiber of each point of the circle contains exactly as many points as the set of integers.
@@TheoremsAndDreams I know what covering maps are. What I'm referring to are the "almost" covering maps. Finitely many points have smaller preimage than the rest. Those are ramified with ramification number >1.
Isnt there an error at 4:26? g(x,y) = y - x and g(1,1) = 1 - 1 = 0. Or am i missing sth?
4:27 If g(x,y) = y-x, and y=1, x=1, then isn't g(1,1) = 1-1 = 0, not -2? It looks so simple but now I'm doubting myself lol. And then what are the implications because his whole point was that "non-zero" factors multiplied together give you zero, but g(1,1) = 0
What is the book displayed at the beginning?
LOOK WHOS BACK?????❤❤❤❤
Can someone explain why in 4:26 g(1,1) = -2 ?
At least someone explaining what Grothendiek worked on
Wait, what. About 4:28 it is stated that g(x,y) = y - x and g(1,1) = -2. Clearly g(1,1) = 0.
Yes, I don’t get that either
Do one on mochizuki’s impenetrable work on Teichmuller theory.
PERFECT!!!!
At 8:46, shouldn't (0) in SpecZ be a generic point? It looks like a closed point in your picture.
Yeah, I guess that depends on how you like to plot generic points. Remember a wiggle or cloud is just an useful convention, lol.