What is algebraic geometry?

[@SoteriosXI]

Please please please make more algebraic geometry or commutative algebra videos. These are really great!

[@Aleph0]

your wish is my command :) more coming up real soon!

[@jeffreyhowarth7850]

please please please commutative algebra video pretty please.

[@Sidionian]

​@@Aleph0 Topos Theory and Schemes/Sheaves/Stalks please.

[@SoteriosXI]

@@Aleph0 Please please please marry my daughter.

[@CharlieVegas1st]

Lookup Hodge Conjecture (David Metzler is the uploader). You're welcome 😊

[@jieyuzhang7559]

Best advanced math education channel on KZbin. I struggled immensely with algebraic geometry in college. The definitions and concepts weren’t properly motivated. So I learned in a painfully mechanical way.

[@excuti300]

Please make more videos on algebraic geometry, please. These videos are treasures.

[@Aleph0]

hey thanks! more AG videos are coming up real soon :)

[@japedr]

4:25 There is a typo I think: should be g(1,-1)=-2. Aside from that, congrats for the really nice explanation.

[@psd993]

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.

[@kingarthur4088]

@@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

[@gi99hf60]

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).

[@gi99hf60]

​@@pozatat he's talking about polynomials on reals in that part. He explains later on with the power series rings

[@mahatmaniggandhi2898]

exactly

[@RyeedAglan]

An excellent introductory video. I should have watched it before I took algebraic geometry or read Gathmann's.

[@zy9662]

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

[@goldjoinery]

@@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.

[@zy9662]

@@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

[@caspermadlener4191]

Wow, I don't think there is a better introduction to ideals in algebraic geometry.

[@Tens0r1]

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.)

[@lhmsilva011]

Shafarevich, Gathmann and Vakil and Eisenbud (Geometry of Schemes) are also good books

[@theflaggeddragon9472]

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

[@rohanjain2120]

Gathmann notes are great as well!

[@vladimirbadalyan1195]

Ravi Vakil's Rising Sea is my favorite, it has a nice modern approach

[@azap12]

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?

[@0x370c2de]

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. ❤

[@marcoottina654]

I am totally with you!

[@wilderuhl3450]

Was in the ER this morning, but a new aleph 0 video has made this a good day.

[@StratosFair]

Damn I hope that was nothing too serious

[@lucastaams353]

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

[@sandropollastrini2707]

The best layman presentation of algebraic geometry I have ever seen. Great!

[@speeshers]

Such a fascinating video! Your videos tend to ignite a spark of curiousity everytime i watch them, thanks so much!

[@considerthehumbleworm]

Small error but at 4:27 I believe it should say g(1,-1)=-2 instead of g(1,1)=2

[@andrewsantopietro3526]

I literally noticed the same thing like 12 hours ago and thought I was losing my mind so thank you.

[@bydlobydlo]

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).

[@gi99hf60]

@@bydlobydlo nope, he’s trying to show they’re not identically zero (while their product is), so any non-zero element illustrates the point.

[@victorespinosa7214]

@@gi99hf60 but he didn't say that any non-zero element illustrates the point, he clearly says both are non-zero.

[@arnaujimenez2194]

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?

[@roosh2927]

Hands down the best introduction to algebraic geometry and rings I’ve seen on KZbin! 👏🏼

[@Ruktiet]

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

[@konstaConstant]

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

[@lucianonotarfrancesco4443]

Oh, you should checkout @mathcuratorzanachan35742

[@lucianonotarfrancesco4443]

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)

[@oportbis]

He teaches me commutative algebras, most of his lectures are improvised because it's too easy for him

[@lucianonotarfrancesco4443]

Who?@@oportbis

[@oportbis]

@@lucianonotarfrancesco4443 Qing Liu

[@lucianonotarfrancesco4443]

@@oportbis oh, Qing Liu! Awesome, you’re very lucky!

[@fhtagnfhtagn]

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

[@zy9662]

Yeah that kind of invalidate all he said about algebra detecting irreducible curves

[@kingarthur4088]

@@zy9662 it doesn't, because you can still input 1,-1 (which is on the curve) and it doesn't return 0

[@gabitheancient7664]

@@kingarthur4088 that makes sense lmao god damn

[@Blackmuhahah]

@@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)

[@gabitheancient7664]

@@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

[@loicdelzenne7684]

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?

[@gauravbharwan6377]

Exactly what I need answer for

[@Aleph0]

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.

[@burnytech]

​@@Aleph0 But then f(x,x) will be 0 which breaks the whole point?

[@piandinfinity9343]

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.☺

[@StratosFair]

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 :)

[@rayschram3399]

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!

[@physira7551]

You really made my day ❤️, Please make a series out of it, the world will remember you

[@ElchiKing]

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)

[@zy9662]

It would still be a lot harder using just geometric arguments, isn't?

[@jarahfluxman20]

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.

[@GNeulaender]

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 :-)

[@TheKeyboardistVG]

There are algebraic varieties that are not manifolds (you found an example) and viceversa (e.g. the graph of e^x)

[@lookupverazhou8599]

Is no one concerned that no one can know which comment is AI and which isnt?

[@rouvey]

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

[@RepTheoAndFriends]

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

[@consumeentertainment9310]

Brother, Ill let you know that I'm inspired!!! It's so well-done. Thanks😻😻

[@azizbekurmonov6278]

Aleph is back ! Good see you Thanks for the lesson

[@Math4e]

So good to have you back!

[@gi99hf60]

4:25 should be g(1,-1) or any other non-zero yielding (x,y)

[@Kyzyl_Tuva]

Great video. So nice to see a new video from you. Thank you

[@felipegomabrockmann2740]

excellent quality of explanation. Please more videos on this topic.

[@funktorial]

hey this was a really well done video! the level of abstraction seemed just right, and that's a difficult needle to thread

[@scalex1882]

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.

[@anshumanii]

Happy to start learning Algebraic Geometry from you 😊

[@smallmimibigmimi]

Why is g not equal to 0 @4:26?

[@stecardile15]

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... ?

[@andrelima5029]

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.

[@jamiepianist]

What a great educator and math experience!

[@moularaoul643]

AMAZING!!! Thank you so much!!!

[@roboto12345]

This was so cool. You motivated me to keep my self studying....thank you

[@arnabdasphysics]

Great introduction! Very thoughtful and wise presentation.

[@Ruktiet]

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?

[@joelsleeba2524]

Thanks for suggesting the books in the end. Might take a look into the subject soon enough

[@signorellil]

More videos on Algebraic Geometry please!

[@tracyh5751]

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.

[@theflaggeddragon9472]

I used both and they complement each other beautifully IMO

[@dogedev1337]

the algebraic geometry notes by Ravi Vakil are great too and freely available on the internet

[@AmoghA]

At 4:27, how is g(1,1) = -2? Should'nt it be 0? Or am I understanding something wrong?

[@MasterHigure]

Having basically only had Hartshorne through my university courses, a few recommendations on the lighter side is always welcome.

[@maxwellguars444]

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?

[@visionary4040]

4:28 should this be g(-1,1)?

[@Taku-j7s]

It’s a very inspiring video, thank you for making it!

[@aaronwolbach9880]

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.

[@extraterrestrial46]

After so long, nice seeing you, great video

[@jimwarb]

At 4:26 why is g(1,1) = -2?

[@anisomorphism]

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)

[@philipoakley5498]

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;-) ]

[@thea.igamer3958]

When the world needs him, he comes !!!!!

[@afzalsoomro7950]

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.

[@literallyjustayoutubecomme1591]

For algebraic geometry you need commutative algebra(study of commutative rings with unity), and the more topology you know the better

[@KrasBadan]

4:24 what? If g(x, y)=y-x, then g(1, 1)=1-1=0. What am I missing here?

[@kingarthur4088]

it's a mistake, but the point is you can input something else on the curve (e.g. 1,-1) and make it not return a 0, i.e. y - x isn't 0 as a function on the curve

[@zy9662]

Bump

[@zy9662]

⁠@@kingarthur4088but (1, -1) is zero on y+x, so one of the factors is zero and he said that both factors have to be nonzero for a point on the curve to be reducible

[@kingarthur4088]

@@zy9662 true, but for a function to be zero on the curve, it has to be zero on _every_ point on the curve; y - x isn't zero because of 1,-1 and y + x isn't zero because of 1,1

[@zy9662]

@@kingarthur4088 thank you but that’s completely different from what he said, he even stated that for irreducible curves both factors need be zero when evaluated on the same point on the curve, but by the look of it and after your explanation, seems that reducible curves just mean that they can be factorized regardless of the property of having zero as a product of nonzero factors

[@MrJaffjunior]

Can someone explain why in 4:26 g(1,1) = -2 ?

[@strangeWaters]

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.

[@pc_phage8500]

I went through blood sweat and tears trying to teach myself and learn algebraic geometry and had to give up, the same with quantum field theory. Please make some more videos at the 5th grade level!❤

[@miltonmontiel853]

Super cool, I've been waiting for this

[@Taric25]

Your explanation at 4:22 is nonsense. 1 - 1 = 0, not -2. You cannot multiply two nonzero real numbers together and get zero. You completely made that up out nowhere, and it's wrong.

[@ruizhenliu9544]

At 8:46, shouldn't (0) in SpecZ be a generic point? It looks like a closed point in your picture.

[@SirZafiro]

Yeah, I guess that depends on how you like to plot generic points. Remember a wiggle or cloud is just an useful convention, lol.

[@Un1cFunaai]

Isnt there an error at 4:26? g(x,y) = y - x and g(1,1) = 1 - 1 = 0. Or am i missing sth?

[@jhonnyrock]

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

[@Robert-ro6gl]

I enjoyed the book recommendations in conjuction eith the video thanks.

[@Grassmpl]

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?

[@TheoremsAndDreams]

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.

[@Grassmpl]

@@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.

[@Yuvraj.]

As an engineering student, this felt extremely approachable. Good work!

[@Math4e]

At 4:23, why do you say g(1, 1) = -2? Isn't g(x, y) = y - x? This means g(1, 1) = 1 - 1 = 0.

[@Shape4995]

This was such a good video! I’d love to get some more algebraic geometry content at some point.

[@user-xf6ig9ur2y]

Wait, what. About 4:28 it is stated that g(x,y) = y - x and g(1,1) = -2. Clearly g(1,1) = 0.

[@oreo-sy2rc]

Yes, I don’t get that either

[@Jojo87171]

this is so insanely good

[@SM321_]

A video about the weil conjectures would be great 😊😊🙏

[@TykoBrian7]

LOOK WHOS BACK?????❤❤❤❤

[@amercado3000]

Perhaps I am not understanding this well, but you define g(x,y)=y-x, then should g(1,1)=1-1=0 4:25 .

[@harshaindukuri603]

One word: beautiful!

[@kapilsharma1721]

Very nice explanation

[@KieranOklahoma]

I don't understand the statement at 5:30. I can calculate points on the curve given by the top function, and plug them into each of the two terms in the bottom function, and one of the two will always be zero. What am I missing?

[@ethanbottomley-mason8447]

One of them will always be zero, but individually, each of those functions are not always zero. I.e. there is a point on the curve which makes the function on the left nonzero and a different point which makes the function on the right nonzero.

[@punditgi]

Excellent video! 🎉😊

[@liamgauvreau]

The goat has returned

[@johnkieffer5854]

What is the book displayed at the beginning?

[@golden_smaug]

Now I'll take this course next semester

[@kapoioBCS]

I would suggest before tackling algebraic geometry to first master basic commutative algebra (like Miles Reid Undergraduate Commutative Algebra)

[@gradf8678]

ahhh you are back!!

[@khaledfarrag9754]

Fantastic work

[@KristianiMyrselaj]

Whats the name of the book showed in the video?

[@christiankathoofer2006]

What is the book called you referred to

[@as-qh1qq]

4:26 correction: at (1,1), g _is_ 0. Perhaps u meant (-1,1)

[@angelortiz6406]

This video isamazing!!! Very clever!

[@somethingsomething2825]

At least someone explaining what Grothendiek worked on

[@shohamsen8986]

at 4:24, you define g(x,y)=y-x. Then u write g(1,1)=-2=/=0. But g(1,1) is 0. Subsequently u say that f is not 0, g is not 0 but f.g is 0. After this you say product of two pieces is 0, then 1 is 0.

[@vigilantradiance]

there is a typo, it should say g(1,-1) = -2, but I don't think this is a problem--f and g are zero at some points on the curve, but they are not equal to the zero function on the curve, so f is not the zero function in the coordinate ring, and neither is g, but f*g is. I think that's the part that is special for the coordinate ring of a reducible curve, as this isn't possible for the coordinate ring other curve at that part of the video.

[@shohamsen8986]

@@vigilantradiance that would make the most sense

[@cybergoth2002]

awesome video, hoping you do some homological algebra soon

[@DavidAspden]

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!

[@Lawfair]

Is algebraic geometry the same as (American) high school geometry? Or is there a different formal name for high school geometry?

[@zy9662]

It’s the evolution of Cartesian geometry

[@98danielray]

clearly not

[@xuehaoding9849]

4:28 is not correct, 1-1=0

[@ZanderzMcCluer]

If possible, could you do a video on what is differential geometry?

[@gnaistvlogs]

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.