I study mathematics at the university of Bergen (Norway). Your videos make me even more interested in what I am to learn in the coming years, especially topology! Would also love to see the proof of Liouville's theorem.
@@SLiMYbrolaf awesome!!! Topology is one of my top 3 math areas. I’m working on the proof of the Liouville’s theorem. I mean, there are very well known ways to prove it of course, but I’m trying really hard to find a way that does not depend on previous results (corollaries, theorems, etc), or at least that I can prove them as well in the same video, so that almost no pre-requisites will be necessary to follow along 😎
At minute 1:30 we use a variant of triangle inequality, namely I a+b I >= IaI - IbI and then we apply the triangle's generalized inequality for the first "n" terms of the sum to obtain your inequality. Curiously enough, most of the arguments in your proof hold inside the field of reals too, such as the continious image of a compact is a compact. The only complex ingredient is Liouville's theorem which is specific to the field of complex numbers. Down With Globalism!
@@GicaKontraglobalismului that’s an awesome observation! I haven’t noticed it. Indeed, Liouville’s theorem relies heavily on the Cauchy’s integral theorem and Cauchy’s integral formula, which are concepts exclusive to complex analysis. I’m actually working on it right now, so thanks for pointing it out!
This is such a beautiful concept: There are many ways of proving this … it’s remarkable that one of the most ‘walkable’ invokes the canonical sense of smoothness for the first tensor space : C: the algebraically closed field you are indeed working over to induce a fractal notion of ‘primes’ There are ways that introduce the Zariski topology of Grothendieck’s motive - cohomology, as it were, to express the role the concept of Topology plays therein…. There do exist truly more canonically topological methods in the sense of @1:30 : of course this is a metric analysis exercised here …. These arguments shine important clarity on the proofs that are more sincerely a study of topology of a tensor space and how its meaning is distorted when used in the context of augmented meanings (of operator spaces and so forth: the Einstein equations e.g. , as canonical relationship between the various notions of curvature that are derived from a powerful notion of curvature: R_ij and so forth….) My personal take is that there are so many meanings associated to metric bounds of this kind: A priori, it is orthogonal to the pursuit of the topology concept’s meaning insofar as there are cultural consequences to global attempts at picking a point of intersection … amidst mutually irrelevant notions of what it means… This proof is using a particular notion of smoothness: the singularity- distortions of holomorphic. Indeed the canonical projection is not even, .. ! and I’m afraid one is indeed reminded of the notion of smoothness to imply the’4 dimensionality ‘of the so called tangent bundle: ‘Linearity in y’ ! What’s algebraic about a smooth parameter? Is sin(x) infinite dimensional now? Nigel Hitchin’s ‘The Smooth Jacobian’
@@dibeos I don't think this name is appropriate. We should not call it like that just beaause its proof requires facts from complex analysis. It is a theorem about the ring of complex coefficient polynomials not about all complex-variabled functions. Also, I would suggest that the cauchy integral formula is the fundamental theorem of complex analysis as it proves that complex differentiability is a more powerful property than real differentiability and that it is equivalent to being analytic.
@@RU-vid_username_not_found yes, it makes sense. You have a point there, but you have to admit that “the Fundamental Theorem of Algebra” is not very appropriate as well, since there are other concepts that are more “fundamental” in algebra
@@dibeos >> *you have to admit that “the Fundamental Theorem of Algebra” is not very appropriate as well* Wait, I have never claimed otherwise. Edit: in fact, my main comment is about rejecting this choice of title.
It has been a while since the last time I saw this proof and it is 4:00 here ... But why proof by contradiction? In other words, f(z) non constant was used somewhere? I don't think so ... Anyway, the inequality at 1:28 is just a consequence of |x+y| ≥ ||x|-|y|| ≥ |x|-|y| isn't it? I will sleep now.
@@dibeos hummm. This ... what I meant is that we shouldn't assume p(z) is non-constant. Just assume p(z) has no roots, then define f(z)=1/p(z) and prove f(z) is constant, so p(z) is constant. This means "p(z) has no roots" implies "p(z) is constant" The contrapositive of this is "p(z) non constant" implies "p(z) has some root" So the statement "p(z) is non constant" shouldn't appear in the proof. That's what I meant. I really have to sleep. 4:28 now.
@@ValidatingUsername oh, I just looked it up now. Yeah it seems a little over the edge hahaha maybe we, mere mortals, are just too stupid to understand his high level math 🧐
@@dibeos No, I think it’s just a person with influence trying to wrap their mind around too many areas of math and engineering all at once but that doesn’t preclude him from light ribbing like a school ground joke about messing up the 7x tables on the test.
@@Satisfiyingvideo-uu9pw hahaha yes!!! You deserve it, you already asked me many times!!! Hahah I promise I’ll make one, but give me a little bit of time!
@@dibeos but i want also a video on rsa encryption. I am your subscriber and i watch your every video, i am working on math. And invent some formula. I want to talk with you privately, do you have any facebook account
