The shocking connection between complex numbers and geometry.

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A peek into the world of Riemann surfaces, and how complex analysis is algebra in disguise. Secure your privacy with Surfshark! Enter coupon code ALEPH for an extra 3 months free at surfshark.deals/ALEPH.
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SOURCES and REFERENCES for Further Reading:
This video is a quick-and-dirty introduction to Riemann Surfaces. But as with any quick introduction, there are many details that I gloss over. To learn these details rigorously, I've listed a few resources down below.
(a) Complex Analysis
To learn complex analysis, I really like the book "Visual Complex Functions: An Introduction with Phase Portraits" by Elias Wegert. It explains the whole subject using domain coloring front and center.
Another one of my favorite books is "A Friendly Approach To Complex Analysis" by Amol Sasane and Sara Maad Sasane. I think it motivates all the concepts really well and is very thoroughly explained.
(b) Riemann Surfaces and Algebraic Curves
A beginner-friendly resource to learn this is "A Guide to Plane Algebraic Curves" by Keith Kendig. It starts off elementary with lots of pictures and visual intuition. Later on in the book, it talks about Riemann surfaces.
A more advanced graduate book is "Algebraic Curves and Riemann Surfaces" by Rick Miranda.
SOCIALS
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Twitter: @00aleph00
___
MUSIC CREDITS:
The song is “Taking Flight”, by Vince Rubinetti.
www.vincentrubinetti.com/
00:00-00:54 Intro
00:55-04:30 Complex Functions
4:31-5:53 Riemann Sphere
5:54-6:50 Sponsored Message
6:51-11:06 Complex Torus
11:07-11:50 Riemann Surfaces
12:11-13:53 Riemann's Existence Theorem

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16 июн 2024

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Комментарии : 223

@Aleph0 Месяц назад

Thanks for watching! If you have any resources you'd like to recommend, feel free to comment them down below. If you'd like to continue your learning, I recently started a math / machine learning newsletter! Every week, I send you the best links (e.g: videos, blogs, articles) to learn topics in math and ML. Sign up here: forms.gle/Rt1f5StAj3yZtakE6

@caspermadlener4191 Месяц назад

12:05 Little spelling mistake, but Reimann is not going to mind. I recommend RU-vidr Richard Borcherds, who has multiple series about these.

@Mad_mathematician224 Месяц назад

𝘽𝙧𝙤𝙩𝙝𝙚𝙧, 𝙄 𝙬𝙖𝙣𝙩 𝙩𝙤 𝙡𝙚𝙖𝙧𝙣 𝘿𝙄𝙁𝙁𝙀𝙍𝙀𝙉𝙏𝙄𝘼𝙇 𝙂𝙀𝙊𝙈𝙀𝙏𝙍𝙔...... 𝙖𝙣𝙙 due to absence of right guider, I am unable to learn it...... I am from India🇮🇳....... Where are you from?

@deadlock_problem Месяц назад

@@Mad_mathematician224 bro what are you begging for, you have access to the internet. Courses: google -> differential geometry -> MIT OpenCourseWare Textbooks: google + pdf -> download links -> books Simple as

@just.a.random.ava.-_- Месяц назад

Dude just wanted to thank you soo much for your videos, they've helped me gain a profound interest in maths at higher levels even though I'm still in school lol. Also, I'd love yo here your thoughts about topics like other Millienuem(spelling wrong ik) problems or even the Langlands Project. Thanks again for everything!

@Aleph0 Месяц назад

@@just.a.random.ava.-_- I'm very glad to hear that! There's definitely more number theory / Langlands videos + Millennium problem videos coming up soon, so keep your eyes peeled :)

@timothypulliam2177 Месяц назад

The reason exp(1/Z) contains an essential singularity is, if you expand the function as a Taylor series, you will get infinitely many powers of (1/Z). In essence, the singularity can't be removed by multiplying by Z. Therefore, it is "essential"

@DanGRV Месяц назад

Another fact about essential singularities: A function with an essential singularity takes all complex values (or all complex values except one value) infinitely many times in every open neighborhood of the essential singularity (Picard's Great Theorem)

@EebstertheGreat Месяц назад

Or more directly, as z goes to 0 from the positive real direction, 1/exp(1/z) goes to 0, but as z goes to 0 from the negative real direction, 1/exp(1/z) goes to infinity. So 1/exp(1/z) can't be continuously extended to 0 even in the real line, let alone the complex plane.

@peabrainiac6370 Месяц назад

@@EebstertheGreat that's true of functions with poles like 1/z^n at 0 too. The point is that the singularity exp(1/z) has at 0 is not that simple, in the sense that it can't be removed by multiplying it with some z^n - hence the name essential.

@Aleph0 Месяц назад

Love this explanation! It's "essential" because you can't get rid of it by multiplying by Z. Brilliant.

@TheRevAlokSingh Месяц назад

This def includes removable and poles of any order, just number of terms that diverge, and 0 if removable

@mohammedbelgoumri Месяц назад

No better way to start a day than an aleph0 upload

@diaz6874 Месяц назад

What time zone are you in?

@mohammedbelgoumri Месяц назад

@@diaz6874 Australia, was 6am for me when this dropped

@mohammedbelgoumri Месяц назад

@@diaz6874 Australia, was 6 am for me when this dropped

@brendawilliams8062 Месяц назад

@@diaz6874 glue 6 am to 2 pm. Geometry in algebraic disguise.

@jakobr_ Месяц назад

Riemann’s existence theorem: “Bernhard Riemann exists.”

@samiaario8291 Месяц назад

Do one on Donaldson theory!

@billcook4768 Месяц назад

Uh, I don’t know how to break this to you… but about Riemann existing…

@billcook4768 Месяц назад

Now can you explain Riemann’s mapping theorem.

@jakobr_ Месяц назад

@@billcook4768 Riemann’s mapping theorem: Bernhard Riemann was a cartographer. (This theorem is known to be false)

@omargaber3122 Месяц назад

When the world needs him he will come back

@StCharlos Месяц назад

When the world needs someone, Surfshark brings him back

@wilderuhl3450 Месяц назад

I needed this video today and he didn’t disappoint.

@phenixorbitall3917 Месяц назад

Amen

@omargaber3122 Месяц назад

😂 @@phenixorbitall3917

@omargaber3122 Месяц назад

😂@@phenixorbitall3917

@dougdimmedome5552 Месяц назад

One of my favorite things in complex analysis was just seeing that elliptical curve come out of nowhere with the Weierstrass p-function, I felt like I was seeing a fraction of what Wiles saw every day while proving the modularity theorem enough to prove Fermat’s last conjecture.

@hybmnzz2658 Месяц назад

The Weierstrass p is goated. It's the e^z of the cubic world. A question for someone who knows more than me: does Faltings theorem or something related imply there can't be anymore interesting functions for degree 4 equations and up which parameterize the curve and respect some group law?

@hyperduality2838 Месяц назад

Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!

@robertcrumplin Месяц назад

@@hybmnzz2658 I don't think I'd say Falting's theorem says anything much about interesting group laws, but maybe I see why you ask this question. Here are some simpler points about group laws you can say. The degree-genus formula tells you that for degree >4 the genus of the curve in P^2 is atleast 3 (which in particular is > 1). If the curve is additionally defined over the rationals, the K-points C(K) are finite. So if C were a group scheme defined over Q, then C(K) would have to be a finite subgroup. There is no obvious reason this gives a contradiction though, but actually theres a much easier reason why any 1-dimensional group scheme over Q is actually genus 1: The group scheme structure allows you to give a trivialisation of the tangent bundle (as the translation action of C on itself is transitive on Q-points). The only smooth connected curve over Q with trivial tangent bundle is genus 1, since the degree of the tangent bundle is 2g - 2.

@SGin01010 Месяц назад

it’s the main argument of my thesis, I’m so happy to see a video about Riemann Surface ❤️

@jogloran Месяц назад

I love how you give equal time to "zee" and "zed" 😅

@primenumberbuster404 Месяц назад

Finally, more Algebraic Geometry content

@hyperduality2838 Месяц назад

@dimitriskliros Месяц назад

i don’t often comment on uploaded videos, but i feel this video is so good that i just wanted to say thank you, and keep up the good work.

@StratosFair Месяц назад

Aleph 0 is back with yet another banger ! Nah but seriously as a grad student in applied analysis/probability/statistics and little knowledge of pure maths, i enjoy these videos so much as they give me a glimpse of the beauty of what's on "the other side". Please keep them coming !

@MonaSchmidtInc Месяц назад

This is an extremely good motivation for the elliptic curve equation(s) that I see everwhere, and a very nice explanation why complex tori are elliptic curves (and not just the other way around)! I'm a bit baffled by your way to write a zeta though...

@acidnik00 Месяц назад

9:00 sick blotter design, bro :)

@macoson Месяц назад

I've heard that blotter with Weierstrass elliptic function on it, kicks stronger

@jenbanim Месяц назад

It'll have you seeing a point at infinity

@gnaistvlogs 28 дней назад

This is one of my favorite results in mathematics. I used this categorical equivalence (along with the equivalence to algebraic function fields) in my master's thesis on prime Galois coverings of the Riemann sphere back in 2007.

@Roxas99Yami Месяц назад

Honey wake up, Aleph 0 just uploaded a new video

@GhostOnTheHalfShell Месяц назад

my math is such a rust bucket. i need to dust off a bunch of old books, but then recapitulate several semesters just to be sure i had enough of the definitions fixed in my head

@MrMctastics Месяц назад

Get some flashcards and set aside an hour a day. Start with something you love. You got it buddy ❤️

@xyzct Месяц назад

Check out 3Blue1Brown

@magnus0re Месяц назад

Been waiting for a new video from you. Just checked a few days ago. And there it is. I'm already intrigued.

@kernel8803 Месяц назад

Love the channel and the content, no pressure, but I have been eagerly awaiting the course that you talked about developing/releasing.

@hyperduality2838 Месяц назад

@1chillehotdogpro199 Месяц назад

"Sorry not now babe Aleph 0 just dropped"

@randomchannel-px6ho Месяц назад

Something that gets lost in Riemann's immense contribution to humanity was the shockingly forward thinking idea he introduced that the microscopic spacetime may be nothing like the 3 + 1 we know so well, over a hundred years before Dirac postulated the same thing which is basically where theoretical physics is now.

@hyperduality2838 Месяц назад

Space is dual to time -- Einstein. Time dilation is dual to length contraction -- Einstein, special relativity. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!

@Zosso-1618 Месяц назад

Oh I was just watching your video on the continuum hypothesis! Nice to see you back!

@hyperduality2838 Месяц назад

Continuous (classical) is dual to discrete (quantum). Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!

@ianmichael5768 Месяц назад

Respect. The printed cut outs are beautiful.

@lukiatiyah-singer5100 Месяц назад

Thanks for the video, very well explained! On this topic, I found the book by Serge Lang on elliptic functions very helpful, but also Gunning's lectures on Riemann surfaces for every thing beyond genus 1

@Shape4995 Месяц назад

Great to see more algebraic geometry!

@phat5340 Месяц назад

Always glad to see you return

@antonius872133 Месяц назад

Great video! I would love to hear some more about this Weierstrass p function.

@giovannironchi5332 Месяц назад

I would like to undersrand better if the etale space of the sheaf of holomorphic functions on a Riemann surface give another Riemann surface

@beardymonger Месяц назад

Great amazing content, I admire the effort that went into making this!!! I would add a short section about the inversion 1/z (with animation) to explain the essential singularity at infinity.

@hyperduality2838 Месяц назад

Exponentials are dual to logarithms. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!

@angeldude101 Месяц назад

The fact that there _is_ a connection between complex numbers and geometry isn't shocking at all (a very obvious connection is spinny), but I can say that I wasn't aware of this particular connection.

@hyperduality2838 Месяц назад

@headlibrarian1996 Месяц назад

I don’t know if it’s important, but in the complex torus example the interval is first written as closed [0,2pi] and later in the example it is written as open [0,2pi).

@Npvsp Месяц назад

I'm a simple person: first I like the new Aleph0 video, then I watch it (even hours later). Trust is everything!

@user-ky5dy5hl4d Месяц назад

Your senses play you wrong.

@Npvsp Месяц назад

@@user-ky5dy5hl4d I ignore what you mean, but considering it’s Aleph0, he has all my trust for he is a brilliant mathematician.

@tommytwotimes2838 Месяц назад

love your content. Please make a video about riemann hypotheses or more about the millenium problems. The biggest unsolved problems in math

@hyperduality2838 Месяц назад

@hanzhang3589 Месяц назад

10:00 Probably a really dumb question, but how does a square which is 2D become an algebraic curve which is 1D?

@nucreation4484 Месяц назад

I think it's because the curve on the right is actually in C2. Like how in the previous example t from the interval which is in R gets mapped to the circle in R2 by associating the points (x,y) on the circle with t on the interval via the trig functions ie x= cos t and y = sin t. ... In the same way, each complex pair (X, Y) on the "curve" described on the right is associated with a complex number z in the square via the functions X = P(z) and Y = P'(z).

@xyzct Месяц назад

"Complex analysis is algebraic geometry in disguise." Given that analytic functions can be described as glorified polynomials, that kind of gave a hint. (Am I seeing that correctly?)

@chobes1827 Месяц назад

You're exactly right about that. The big idea is really that if you look at analytic and meromorphic functions ("glorified" polynomials and rational functions respectively) that satisfy very natural conditions, they turn out to be polynomial or rational.

@xyzct Месяц назад

@@chobes1827, thanks! Wow, what a fun video. It's always so satisfying to see new connections that are sitting _right there._

@hyperduality2838 Месяц назад

Space is dual to time -- Einstein. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!

@xyzct Месяц назад

@@hyperduality2838, it's you!!! You and I have had many great conversations across RU-vid! I actually have a spreadsheet where I have kept a running list of your awesome examples of duality. I have found them incredibly profound ... and helpful.

@Jaylooker Месяц назад

I wonder how Riemann’s existence theorem relates to the circle method

@Rozenkrantzz Месяц назад

Absolute banger as always. I'm interested in making educational math content as well and I've been using you as inspiration for my pedagogy.

@brian.westersauce Месяц назад

Any chance your name is Steven

@primenumberbuster404 Месяц назад

@@brian.westersauce no his name is Brian.

@Rozenkrantzz Месяц назад

@@primenumberbuster404 no his name is buster

@hyperduality2838 Месяц назад

@carlosperezfranza5864 Месяц назад

Nice, could you make something embracing all the symmetries of our beloved R3 smooth sphere?

@-minushyphen1two379 Месяц назад

At 3:20, doesn’t the zeta function have an essential singularity at infinity? Edit: Oh, you meant that the functions on the left are *not* meromorphic at infinity

@hyperduality2838 Месяц назад

Points are dual to lines -- the principle of duality in geometry. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!

@erictao8396 Месяц назад

Great video!

@ErkaaJ Месяц назад

I would love a video on GAGA theorem (Serre), which is really a continuation on the topic in this. It is remarkable how Riemann's work in the late 1800's is the foundation for modern algebraic geometry.

@Aleph0 Месяц назад

That’s a great suggestion. GAGA is definitely on the list for a future video!

@user-ni2or3wr3h Месяц назад

Great video Do you have any plans to make a video about p vs np?

@kumargupta7149 Месяц назад

I wonder content of this type is also available love your content ❤.

@phnml8440 Месяц назад

Mom! Mom! New Aleph0 video dropped🎉

@kgangadhar5389 Месяц назад

Can you please add Thanks option to your videos.

@nathanhenry7711 Месяц назад

Awesome video!

@Cosmalano Месяц назад

I recently learned Riemann surfaces are used in string theory which I find really cool. I also am 90% sure they come up in the 2-spinor formalism of GR but it’s never clicked for me

@hyperduality2838 Месяц назад

Action is dual to reaction -- Lagrangians are dual, forces are dual. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!

@codybarton2090 6 дней назад

Great video explained very well ❤

@MattHudsonAtx 6 дней назад

neat to see Dhruv's name again in an unexpected place

@laposta-eu Месяц назад

Beautiful animations but I would have liked more explanation of the basic concepts.

@cycklist Месяц назад

Thank you for saying zed :)

@SydiusVideo Месяц назад

Thank you!

@kyspace1024 Месяц назад

Kind of hope you could maintain the handwriting style. You in fact inspired me to do all-handwriting demos.

@daviderady Месяц назад

Love the video!

@Aleph0 Месяц назад

thanks davide!

@whatitmeans Месяц назад

lets say z=x+iy... Where relations like f(z) = e^(z^2/(z^2-1)) unitstep(1-z^2) fall inside complex analysis? If y=0 then f(z) it is a smooth bump function, which are not analytic so at least in the real line f(z) cannot be represented as a power series, which rule it out of conventional complex calculus (this is why I call it a relation instead of a function). There is a branch of mathematics that study this kind of complex-valued objects?

@chobes1827 Месяц назад

This kind of thing falls more into the realms of real analysis in multiple dimensions. Functions that aren't analytic aren't complex-differentiable. You may be able to define such functions using complex numbers, but the algebraic structure of the complex numbers isn't really relevant for understanding these functions. It's more useful to rewrite these functions from R^2 to R^2 and study them using tools from real analysis (which includes standard multivariable calculus).

@whatitmeans Месяц назад

@@chobes1827 and how it is done? do you know how this kind of analysis is named?... At least for me is not obvious how you will make happen in R^2 all the oscillating effects that rises from Euler identity e^(it)=cos(t)+i sin(t) without it, my example f(z) it is just a 2D smooth bump function, but I think it is not his complex behaviour since in their exponent the z^2 term will left some terms dependent in the imaginary unit "i", leading to oscillating behaviour

@hyperduality2838 Месяц назад

@chobes1827 Месяц назад

@whatitmeans You basically just write that g((x,y)) = (Re(f(x+iy), Im(f(x+iy)) and then study g as a map from R^2 to R^2. Points in the complex plane are still just pairs of two real numbers, and you can always identify them as such. If you study some complex analysis, you'll learn how this works because you need to think about complex functions this way in order to derive and use the Cauchy-Riemann equations. All of the oscillating behavior ends up being expressed with rotation matrices, and it's all completely doable despite the expressions being a bit messier. For example, if r is |z| and theta is the angle formed between z and the positive real axis, then e^iz becomes e^r * rotation by theta as a map from R^2 to R^2.

@Williamtolduso Месяц назад

i neeeed the next video!!

@DelandaBaudLacanian Месяц назад

"imaginary numbers" shouldve been called "orthogonal" numbers, then people could maybe understand how it's related to geometry

@carywalker7662 Месяц назад

Love it.

@tomkerruish2982 Месяц назад

Take it up with Descartes.😂

@angeldude101 Месяц назад

I call them "spinny numbers", because they are the best tool for the job of making 2D objects go spinny. (Naturally there are also 3D spinny numbers, which are rather famous, or more accurately infamous.)

@Stylpe Месяц назад

And "complex numbers" could just be "2D numbers"

@zenshade2000 Месяц назад

Yeah, I've never understood the "mystery" of imaginary numbers. It's just a mental construct that lets us model periodicity in a precise manner.

@DanielRublev Месяц назад

Very cool! Hope to see more facts from this profound theory.

@brendawilliams8062 Месяц назад

Without algebra please

@DanielRublev Месяц назад

@@brendawilliams8062 did you get the idea of bridge from the video?

@brendawilliams8062 Месяц назад

@@DanielRublev it’s harder with algebra

@DanielRublev Месяц назад

@@brendawilliams8062 maybe

@quiversky4292 Месяц назад

Very interesting! I never got into complex analysis in uni. Can I suggest you just stick with Canadian ‘zed’? I think American viewers will understand :)

@darkshoxx Месяц назад

Up next: the GAGA theorem

@chianchen776 12 дней назад

My first time watching produced a question that, isn’t then the conclusion made in the end (analytical object can be bridged to algebraic perspective) requires the step glossed over in “projective complex space” CP2 thing, that dealt with how 0 is mapped? I might watch the second time but maybe someone was impressed with the same question as me in the comment section.

@felipegomabrockmann2740 Месяц назад

excelent video

@carlosgaspar8447 Месяц назад

at 5:00 the unit circle is labelled at points +/-1 and +/-I. wouldn't it make more sense if it was +/-1 and +/-i^2? thx.

@royronson8872 Месяц назад

I hit the like exactly at 13 seconds

@mabeteekay1403 Месяц назад

can you please do some concepts in representation theory , lie groups and that sort of math , great channel ❤❤

@hyperduality2838 Месяц назад

@ReadingDave Месяц назад

This math might be above my level, but it makes me hopeful. I was just wondering how to approach classifying ranges of relations as Rational or Irrational.

@smoothacceleration437 Месяц назад

This is a great video for going to sleep. HIghly recommend to any insomniac.

@Happy_Abe Месяц назад

Why not just define f(infinity) to be the limit as z approaches the point at infinity of f(z) where we can take |z| approaching infinity in the real case and consider all possible paths of z that do this. Why would these two limits not be the same when they exist?

@robertbarta2793 Месяц назад

Wow. More!

@lukekhh Месяц назад

I think the way you glued the ends of the cylinder together at 7:20 will get you a Klein bottle

@lukekhh Месяц назад

PS great video ❤

@gregsarnecki7581 Месяц назад

So is this like the Langlands program, just for Complex Analysis and Algebraic Geometry, as opposed to Number Theory and Geometry? Just trying to get my head around these different branches of Mathematics of which I clearly know so little!

@hyperduality2838 Месяц назад

@DiegoMathemagician 16 дней назад

12:15 it says Reimann but it should be Riemann. Other than that, thank you for the great video :D

@ArduousNature Месяц назад

beautiful

@melm4251 Месяц назад

i need to do a repair on a jacket pocket but the best way to patch it would be with a riemann surface... sadly i can't find one in craft stores

@probablyrandom31 Месяц назад

Nice!

@oshaya Месяц назад

At 11:55, Rain Man, oups Reimann, made his way in.

@heeraksharma1224 Месяц назад

5:33 Why do we need to explicitly evoke f(1/z)? Will lim z->inf f(z) not work? Also, to check if a function is meromorphic at inf, is there no other way than to see this other than checking singularity of f(1/z)?

@chobes1827 Месяц назад

The notion of taking a limit as a value approaches infinity isn't well defined in the complex plane the same way it is for the real line. On the real line, there's only really one way we can make a variable approach infinity (by making the variable bigger and bigger). In the complex plane, variables can grow infinitely along an uncountably infinite amount of paths that move in different directions. We need to make a statement about what happens as z grows infinitely large in any of the possible directions. We're interested in what happens as |z| approaches infinity along any possible path. Working with lim |z| -> infinity is technically sufficient to formulate the definition of a function being continuous, holomorphic, or meromorphic at infinity, but it's tricky to reason about a variable growing larger across the entire plane. We use the fact that as |z| approaches infinity, |1/z| approaches 0 to make the behavior we're interested in easier to reason about. By looking at the behavior of f(1/z) when |z| is small, we can study the behavior of f(z) as |z| approaches infinity by reasoning about the behavior of a function on a small disk, which is much more manageable than thinking about f's behavior as z grows larger in any of the possible directions.

@heeraksharma1224 Месяц назад

@@chobes1827 thank you for your reply. That makes sense.

@hyperduality2838 Месяц назад

@heeraksharma1224 Месяц назад

@@hyperduality2838 How high are you?

@hyperduality2838 Месяц назад

@@heeraksharma1224 Points are dual to lines -- the principle of duality in geometry. If Riemann geometry is dual then this means that singularities (points) are dual. Black holes = positive curvature singularities. White holes (the big bang) = negative curvature singularities. The definition of Gaussian negative curvature requires two dual points:- en.wikipedia.org/wiki/Gaussian_curvature The big bang is an infinite negative curvature singularity -- a Janus point/hole. Two faces = duality. The physicist Julian Barbour has written a book about Janus points/holes. Topological holes cannot be shrunk down to zero -- non null homotopic. Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Dark energy is repulsive gravity, negative curvature or hyperbolic space (a pringle) -- inflation. The big bang an explosion is repulsive by definition -- negative curvature. The point duality theorem is dual to the line duality theorem -- universal hyperbolic geometry. The bad news is that Einstein threw his negative curvature solutions in the proverbial waste paper bin of history!

@Matematikervildtsjov Месяц назад

Great video as usual! Minor correction, at 11:59, you made a typo in "Riemann" (Reimann).

@Kelikabeshvill Месяц назад

Great job, but can you do it even simpler? like without using the jargon at all.

@pierrekilgoretrout3143 Месяц назад

wow!

@johnchessant3012 Месяц назад

9:10 elliptic curve?

@hyperduality2838 Месяц назад

@ben1996123 Месяц назад

yes that's why they are called elliptic curves, because ℘ and ℘' are elliptic functions

@kwccoin3115 28 дней назад

I understand if one could not understand something we do not study or spend too much time on it. But is it bad?

@IamRigour Месяц назад

New Sub

@darkshoxx Месяц назад

Interesting choice to talk about the complex torus and the p function and y^2 = x^3-x and NOT mention the term Elliptic curve 😉 Guess you didn't want to overload the video with even more topics

@hyperduality2838 Месяц назад

@TheBasikShow Месяц назад

Just checking but like. The arrow in that last image doesn’t go both ways, does it? Sure, every Riemann surface is an algebraic surface and that’s cool, but like. There are three-dimensional algebraic surfaces, but there are no three-dimensional Riemann surfaces, right? So there are some varieties that are not Riemann-able.

@Icenri Месяц назад

From here to Taniyama-Shimura!

@Happy_Abe Месяц назад

So every compact Riemann surface is an algebraic curve but is the other way true that every algebraic curve can be realized as a compact Riemann surface? If not these fields aren’t the same, just that these surfaces can be viewed equivalently in both but not all algebraic curves can be studied using complex analysis and not everything in complex analysis is a compact Riemann surface that can be studied in algebraic geometry. Therefore, I’m not sure I understand what the video is trying to conclude about them being the same and I’m just trying to understand that last point.

@hyperduality2838 Месяц назад

@DeathSugar Месяц назад

Oooh, are we going to do Ricci flow at some point? Or only Langlands stuff with fancy graphs like those meromorphics?

@paperstars9078 Месяц назад

So my complex analysis exam is algebraic geometry in a trenchcoat?

@codybarton2090 6 дней назад

So it’s a neat twist on a binary system ? Just 1,0

@AdrianBoyko Месяц назад

Is the final statement of this video false? Shouldn’t it be “SOME OF Complex Analysis SOME OF Algebraic Geometry”? Or do I need to watch the video again?

@zaccrisp9988 Месяц назад

Example? Or is it that only if you make the right comparison or equality?

@SAMathlete Месяц назад

I love these video topics where the thesis is "X and Y look like completely different things, but when you achieve enlightenment all is one,"

@hyperduality2838 Месяц назад

Thesis is dual to anti-thesis creates the converging or syntropic thesis, synthesis -- the time independent Hegelian dialectic. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!

@zray2937 Месяц назад

Ah yes, another glimpse of a mathematical world that is far too complex for my little mind.

@oscargr_ Месяц назад

Please do Bernhard the honor of spelling his last name correctly. It's *Riemann* (@ 12:00)

@hyperduality2838 Месяц назад

Riemann geometry is dual. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!

@oscargr_ Месяц назад

@@hyperduality2838 Always so to the point.🤔🤣

@hyperduality2838 Месяц назад

@@oscargr_ Points are dual to lines -- the principle of duality in geometry. If Riemann geometry is dual then this means that singularities (points) are dual. Black holes = positive curvature singularities. White holes (the big bang) = negative curvature singularities. The definition of Gaussian negative curvature requires two dual points:- en.wikipedia.org/wiki/Gaussian_curvature The big bang is an infinite negative curvature singularity -- a Janus point/hole. Two faces = duality. The physicist Julian Barbour has written a book about Janus points/holes. Topological holes cannot be shrunk down to zero -- non null homotopic. Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Dark energy is repulsive gravity, negative curvature or hyperbolic space (a pringle) -- inflation. The big bang an explosion is repulsive by definition -- negative curvature. The point duality theorem is dual to the line duality theorem -- universal hyperbolic geometry. The bad news is that Einstein threw his negative curvature solutions in the proverbial waste paper bin of history!