Very nice, Martin. Hope you make more of these. Galois Theory is beautiful but not as mystical as it gets painted by some. Permutations of roots in the splitting fields loses its mystery when you think of it as simply flipping radicals within dense subsets of the real numbers, a+root(2)b, and a-root(2)b are just reflections of each other. If the other roots of Unity are involved a similar abstract picture emerges, another axis with orderable values which get permuted. I know there’s more to it but this is essentially what it is. Would love to see you take this series all the way through the Galois results.
Your comments about why you did this video is the thing every teacher should understand. Without telling why those abstractions came about mathematics is being taught. A context would delineate the whole subject much better. An amazing video. You should do more of these
What a quirky little thing you are. I'm happy to be making the first comment, and hoping you do a few more of these MMM's.Theoretical physicist here, semi-retired, doing things for the fun of it, but never really have time or energy to read dense mathematics tomes, so I really rely on the odd rare engaging bits of mathematics on youtube to keep in the game a bit. So thanks.
I love your editing, perfect handwriting, perfect pace. I don't know anything about Galois theory so this is gold. You've earned my subscription, and I'm looking forward to the follow-up! 😊
Dude, I am not a math type, and I am only a few minutes in, and already I am extremely imptessed with every aspect of you presentation. Congratulations from an educated leyman on your project, which is first rate as far as I am concerned. EDIT. Wow, I got through to 27 minutes before brain called Time-Out !! Superb work Dude.
Sincerely thank you for trying to make this topic more approachable, I know from experience that it's not easy, and I cherish every resource I have that can show some insight on it. Edit: 31:00 This resonates with me a lot, I have asked myself many times with Galois Theory why anything shown was thought of, or how any of it follows from the axioms. Most of what I've seen was either too vague to actually show the specifics, or too technical to clearly explain the underlying material, so you're doing a great service by laying this out fully.
Hello Martin! Algebraist here. I would like to stretch out a hand and say that you did this presentation on symmetrical polynomials very wonderfully. Very clear. Very insightful. I’m looking forward to more of your videos! I might even have a thing or two to learn from you… 😉 Greetings from Sweden! 🇸🇪
Wonderful video! I've already taken a galois theory class, but I had the same frustration you described at the end: I didn't understand where all these definitions and proofs were coming from. This video reignited my intrigue for it. I especially liked your proofs. You gave just enough detail to give a full understanding without being slowed down, and you placed emphasis on the magical moments. It was really enjoyable!
Impossible not to be humbled how a 20 years old guy from the early 1800s could come out with such a deep and abstract insight into algebra. Excellent job presenting the fundamentals of that insight, Martin, so concise and clear. Congrats!!
Amazing video! I learned briefly about Galois Theory in a history of math class, and I couldn’t understand the motivation for so many concepts that were introduced. This video was engaging the whole way through and I have so much more appreciation for symmetric polynomials! Really hoping for a follow up video!
Cannot wait for the next part.. This has been an elusive topic for me and for the first time ever it has made any sense to me after watching this video. I had to subscribe immediately.. Please keep making more of these...❤
As a highschooler with an interest in cryptology, Galois Theory has been a puzzle I've been poking at for a while. Though we have not quite gotten to Galois Fields yet, this is definitely the clearest explanation I've seen of such concepts. Really hoping this will be a series
I love how you highlight the essence of galois theory and hence demystify it. Best video so far on symmetric polynomial . Incredible work !!! I'm eagerly looking forward to what comes next in group theory
It took me around 3 nights to watch the video (as I watch TH-cam usually before sleep), and I really enjoyed it. I should watch it again in the afternoon with a piece of paper! It was very interesting and excellently animated and presented. Congratulations! I would enjoy in the future a video centered in some applications, if you have time and interest in doing so! Thank you very much!
Excellent use of exposition (telling the story of it) to illuminate a frustratingly slippery path towards Galois Theory. At least now we can see where we are stepping, and place our feet more firmly on the ground before us! Thank you! PS: Great idea using a 'green board' to present your formulas! A nice compromise between the slow-but-friendly blackboard/whiteboard, and the fast-but-impersonal use of math-formula animations! Very innovative!
Hey, this is a brilliant introduction, easily missed or overlooked, but more and more enlightening the more you listen to it. The fog is lifting and the relationship between the Galois theory and the Representation / Group theory is becoming apparent. I think I am going to revisit this intro a couple of times more. Thanks!
This video is an eye opener. Back in the day I built Reed-Solomon encoder and decoders and struggled to get the key ideas of Galois theory. I didn’t understand it. Now I am feeling hopeful with your video. I must understand this so I hope you will make follow up videos on this topic. Thank you, thank you!
I must admit I have no clue what Reed-Solomon encoders are, but it is intriguing to hear they have something to do with Galois theory. Might look into that. Thank you so much for your kind words, I'm glad you enjoyed it!
I literally just finished watching through Borcherds' playlist on Galois Theory (mostly review of stuff I learned ages ago), but am excited to complement it with your more-motivated presentation
Thank you, I enjoyed watching this video very much. This video convinced me to spend more time with pure math in the future, even though I am employed as a computer scientist and hence need to spend most of my learning time with IT topics :)
This presentation, undoubtedly, stands as the quintessence of introductory discourse on this subject matter. The presenter undoubtedly possesses a prodigious intellect akin to that of Galois. Remarkably exceptional!
Great content. Thank you. I'm just stepping into this level of math, and you have deepened my understanding. There is an annoying echo on the audio though. I don't mind replaying parts to be sure of understanding the math; but having to replay six times just to catch the word "norm" was truly annoying.
I was studying Harold's Galois Theory book and had difficulty understanding a chapter after this. (Starting on Lagrange resolvent) Great timing for me! Hope you complete this video series. Appreciate your work. :)
Fatnastic stuff! I love the way you explain and summarize. The positively-biased board (black-on-white) really helps me a lot. The audio could get a little bit better, but hey, couldn't it always 😆 Thanks for the ride 🤗
I had an interesting career in ASIC and DSP design, in one chip it was based on Galois fields and Golay error correction, another was based on the Fast Wavelet Transform with DCT inside and an FFT engine, another was more mundane, building a pattern waveform generator based on sine waves but not using a high resolution Sin ROM, instead computing Sin from low resolution Sin and Cos. Math and ASIC design is a special pairing made in heaven, lots of flexibility to produce the desired result, it's my favourite type of chip design, no arguments about architecture, the math defines tjhe that. Do I remember much about Galois fields, nope, not a darn thing.
Masterpiece in all aspects - title, artful ambient space composition, scrupulous deliberate manner of presentation, deliberately stylish outfit and haircut (English artistic sophistication a la Oscar Wild? 😅 ),.. and of course fine math
Was Galois aware of the fundamental theorem of symmetric polynomials when he proved his theorem or did he also develop the fundamental theorem of symmetric polynomials along the way to proving there’s no solution by radicals for polynomials of degree 5 or higher?
Very well explained introduction to a fascinating but quite opaque subject! Great work! I am keenly looking forward to follow-up videos. What you say in the final segment is very true. In particular the Bourbaki collective has killed the human and historical element in teaching mathematics and the tone they set has made modern mathematics somewhat cleaner perhaps but much more difficult, and unnecessarily difficult, to learn. Therefore, pedagogical videos like yours, which teach mathematics in a language and from a perspective more suited to the human brain, are very important. You could improve the audio quite a bit by suppressing reflections in the room you are recording in. It sounds very echoey. (Not like the echo in a large hall or outside, but lots of very fast reflections from nearby surfaces. This is one of the main reasons for bad quality of voice recordings and it becomes very obvious once you hear examples with and without those reflections.)
Thank you for your kind words and detailed feedback! I actually haven’t heard about the Bourbaki collective before, I’m intrigued! I’m just starting out and I really appreciate the technical feedback on the audio as well :)
@@martintrifonov AFAIK Bourbaki did important work in cleaning up the foundations of algebraic geometry, which had previously gotten into quite a mess and even produced some wrong results due to a lack of rigor. The Bourbaki books were great as an underlying structure but their huge success had a detrimental effect on the pedagogy of mathematics in my opinion, and not only in my opinion, as you can read in the Wikipedia entry: >>As Cartier remarked, "The misunderstanding was that many people thought it should be taught the way it was written in the books. You can think of the first books of Bourbaki as an encyclopedia of mathematics... If you consider it as a textbook, it's a disaster."
Really appreciate this reply! As for your remarks on the bourbaki collective, this sent me down a rabbit hole of Wikipedia today, can’t believe I never heard about any of this before! As for the audio quality, I actually have a decent microphone (or at least not the cheapest option 😂), but I see I haven’t set it up properly. It was obvious to me that the quality was lacking, and I played around with some noise reduction settings - with limited success. But now I realize much more concretely where and how I could improve. Thanks for your advice :)
If we are given a polynomial of degree 3, with roots a,b,c... can we compute something like sqrt(a) + sqrt(b) + sqrt(c)? Also, can we make use of computing symmetric expressions to give us good starting points for polynomial root finding algorithms or modify the iteration steps in one? I'm imagining for example... unlikely scenario but, if you knew for example that your given polynomial has only real roots, and one root is particularly larger than the others, you could approximate that large root by computing (r1^50 + r2^50 + ... + r_n^50)^(1/50).
Im not sure I have a good answer to this - but what you are proposing loosely reminds me of Graeffe‘s Root Squaring Method - might be worth looking into :)
@@martintrifonov Looking at this again with fresher eyes, at least for the first question I asked I think the sums of powers of roots can be applied to taylor series. For sqrt(x) a taylor series centered at x=1 converges if x is in [0,2]. If we only have real, positive roots we could compute sqrt(r1) + sqrt(r2) + ... + sqrt(r_n) by squishing the roots closer to the origin by substituting cx for x in our starting polynomial, where we can pick a constant c. After computing the taylor series to an accuracy we like, we can multiply the sum of those approximations by sqrt(c) to get the final answer. It should be relatively straightforward to determine the largest magnitude a root might have and pick c to be that largest magnitude.
Nice vfx. You have screen recorded a digital art program. I can see the artifacts of the pressure sensitivity graph in the writing. I think the canvas in the video is a 3D asset, but you could have placed it and distorted it with a perspective grid. You are using a soft brush that adds up on overlaps making it look very realistic as if the liquid collected. What blending mode are you using? Is it just opacity? I am guessing you have some textures here and there or some glossy shading applied to the 3D canvas. It looked very good, I might have to steal some of your methods. The editing where the pacing of the sped up writing following you sentences was very good. Sometimes it was in parallel, but that’s what you gotta do no? I think I will use motion graphics with custom typography with this kind of pacing. It is taking a long time to develop, especially when I want to continue to multiply my tasks with pairings between glyphs and innovative characters and transitions, but hey it could become a big animated typeface for math with special dynamic customizable characters! I don’t have much experience with coding so many variables though, but I am beginning to design a custom encoding and variable axes for it… just pseudo code at this point
Nice detective work :) ! Pretty close, a few things I did a bit different. I didn't screen-record the whiteboard, rather I just made drawings on my Remarkable (E-book type) tablet. I downloaded the drawing files from the tablet to my computer, and I wrote a little code that reconstructs the paths of the lines and animates them. The drawing is reconstructed from the paths using the p5js and p5brush libraries. The latter, p5brush, creates realistic looking brush strokes. And yes, from there on you are mostly correct - some shading and composition! The editing (speeding up the pace of the drawing) was rather labor-intensive. I'm thinking this could be automated a bit in the future. Interesting, what you're describing about a dynamic animated math typeface - but I'm not sure I follow completely. Could you elaborate a bit on what this project is?
sorry if dumb, but am I correct in saying (xy)^4 + (zy)^4 + (xz)^4 is a polynomial? and cannot be said to be represented using the elementary polynomials according to the proof, because the table doesn't have any entries of the form (xy)^n?
It's a fair question! It is a symmetric polynomial, yes - and you're right to say that we can't learn how to represent it using using only elementary symmetric polynomials by the argument in the first proof. The first proof only shows how to represent power sum polynomials (x^k+y^k+z^k) using elemetnary symmetric polynomials. However, the second proof in the video shows how to do this with any symmetric polynomial, including your example. Hope that makes sense!
MMM is a catchy intro. The effort put into this video clearly shows, kudos! the proof at 15:30 wasn't crystal clear to me. can you expand (or guide me to a resource) on how this table layout proves Newton identities on the general case?
Thanks! If I find time I will make a more formal write up of the contents of the video. The manner in which I present the proof (using the table) I haven’t found anywhere else - however, it is, in essence, very similar to one possible derivation given in the Wikipedia page of Newtons Identities. If you go to the page for newtons identities, and navigate to the subtitle „Derivations > As a telescopic sum of symmetric function identities“ you can read more there.
You're right, that didn't quite turn out how I hoped it would. Will be working on the production quality (the audio too) in future videos. Thanks for the feedback!
how tf is he making the whiteboard animations?? he seems to be able to see what is being written and the writing we see doesn't seem to be an overlay....huh???
I had the same obstruction for the Galois theory : too much theory to learn before solving any equation....it is the same than a promise for a good dessert but you have to eat a lot of heavy dishes ;-) For me Lagrangian resolvents was the point I stopped, because despite I understand the main ideas (splitting fields and so on, .... more in the group theory and fields) the difficulty is to guess whether a given equation is Galois solvable and how to process in that case. There is an unknown too : can you apply the theory when the coefficients ARE NOT rationals (not integer).. is this for "algebraic equations" only ? Guessing the resolvents (or the intermediary fields) is the extra I need ....the new entries in your "bag".
th-cam.com/video/3imeTgGBaLc/w-d-xo.htmlsi=i7s_yO2lWOolAVQE&t=192 I know it's a famous fact that you can factorise a polynomial like this, but I think you should either justify it, or at least tell a name of the theorem, so it'd be easily googleable, as you do just 20 seconds earlier with the fundamental theorem of algebra.
Audio quality is not great. That makes it hard to focus upon your lecture, which I want to hear. You will well serve your viewers by attending this niggling issue.
我每年都会在油管上面寻找关于 Galois 的理论的新的更加符合直觉的解释，这个视频中的内容就是我一直在寻找的。非常感谢您的工作。
Very nice, Martin. Hope you make more of these. Galois Theory is beautiful but not as mystical as it gets painted by some. Permutations of roots in the splitting fields loses its mystery when you think of it as simply flipping radicals within dense subsets of the real numbers, a+root(2)b, and a-root(2)b are just reflections of each other. If the other roots of Unity are involved a similar abstract picture emerges, another axis with orderable values which get permuted. I know there’s more to it but this is essentially what it is. Would love to see you take this series all the way through the Galois results.
Your comments about why you did this video is the thing every teacher should understand. Without telling why those abstractions came about mathematics is being taught. A context would delineate the whole subject much better. An amazing video. You should do more of these
A very kind comment, thank you :)
@@martintrifonov Great work though! Deserved
Do Godelllllll lol
I agree with this wholeheartedly. It's something that has always bugged me about the way math is usually taught.
What a quirky little thing you are. I'm happy to be making the first comment, and hoping you do a few more of these MMM's.Theoretical physicist here, semi-retired, doing things for the fun of it, but never really have time or energy to read dense mathematics tomes, so I really rely on the odd rare engaging bits of mathematics on youtube to keep in the game a bit. So thanks.
Thanks, I’m glad you enjoyed it. I appreciate the encouragement!
I love your editing, perfect handwriting, perfect pace. I don't know anything about Galois theory so this is gold. You've earned my subscription, and I'm looking forward to the follow-up! 😊
Thanks, I’m glad you liked it :)
Dude, I am not a math type, and I am only a few minutes in, and already I am extremely imptessed with every aspect of you presentation. Congratulations from an educated leyman on your project, which is first rate as far as I am concerned.
EDIT. Wow, I got through to 27 minutes before brain called Time-Out !! Superb work Dude.
Thank you, that’s very nice - I’m glad you enjoyed it :)
Sincerely thank you for trying to make this topic more approachable, I know from experience that it's not easy, and I cherish every resource I have that can show some insight on it.
Edit: 31:00 This resonates with me a lot, I have asked myself many times with Galois Theory why anything shown was thought of, or how any of it follows from the axioms. Most of what I've seen was either too vague to actually show the specifics, or too technical to clearly explain the underlying material, so you're doing a great service by laying this out fully.
Thank you, that’s very kind - I’m glad you enjoyed it :)
Hello Martin! Algebraist here. I would like to stretch out a hand and say that you did this presentation on symmetrical polynomials very wonderfully. Very clear. Very insightful. I’m looking forward to more of your videos! I might even have a thing or two to learn from you… 😉 Greetings from Sweden! 🇸🇪
Hey! Thank you for your kind words, I'm just starting out, so it means a lot. Greetings from Berlin!
Superb presentation in every aspect. Would have been nice 50 years ago when I first encountered Galois theory.
What a nice comment, thank you. Glad you enjoyed it!
Can't believe I just found maths' Bob Ross, what a great day
The other end of it is that many, many people will not get the Bob Ross reference.
Wonderful video! I've already taken a galois theory class, but I had the same frustration you described at the end: I didn't understand where all these definitions and proofs were coming from. This video reignited my intrigue for it.
I especially liked your proofs. You gave just enough detail to give a full understanding without being slowed down, and you placed emphasis on the magical moments. It was really enjoyable!
I'm glad you enjoyed the proofs, it means a lot to hear that - thank you :)
Impossible not to be humbled how a 20 years old guy from the early 1800s could come out with such a deep and abstract insight into algebra. Excellent job presenting the fundamentals of that insight, Martin, so concise and clear. Congrats!!
Humbling indeed. Thank you for the kind words!
Amazing video! I learned briefly about Galois Theory in a history of math class, and I couldn’t understand the motivation for so many concepts that were introduced. This video was engaging the whole way through and I have so much more appreciation for symmetric polynomials! Really hoping for a follow up video!
Awesome video!! Would love to see a follow up
I am already familiar with the material but I still watched pretty much all the way through. It is just satisfying to watch
Glad you enjoyed it! Thanks, that really means a lot :)
It was weirdly satisfying returning to the subject
Cannot wait for the next part.. This has been an elusive topic for me and for the first time ever it has made any sense to me after watching this video. I had to subscribe immediately.. Please keep making more of these...❤
Thank you, thats such a nice comment, its really encouraging :) I have some more planned, stay tuned!
As a highschooler with an interest in cryptology, Galois Theory has been a puzzle I've been poking at for a while. Though we have not quite gotten to Galois Fields yet, this is definitely the clearest explanation I've seen of such concepts.
Really hoping this will be a series
If you haven't done so already you'll need to familiarize your self with abstract algebra and a decent understanding of proofs beforehand.
Great job. Well done. Better in terms of didactic value than many university lecturers!
Thank you, I'm glad you enjoyed it! :)
This was such a wonderful watch. I can't wait to see what else you are planning to make.
That was brilliant! Hoping for more Chapters!
Thanks!
I love how you highlight the essence of galois theory and hence demystify it. Best video so far on symmetric polynomial . Incredible work !!! I'm eagerly looking forward to what comes next in group theory
Fantastic video. Very clearly presented and motivated. Thanks.
Learning math is way easier with this kind of content, thank you!!
Thank you for a lucid presentation of how to develop theories.
Glad it was helpful!
Will try again later. Great presentation.
It took me around 3 nights to watch the video (as I watch TH-cam usually before sleep), and I really enjoyed it. I should watch it again in the afternoon with a piece of paper! It was very interesting and excellently animated and presented. Congratulations! I would enjoy in the future a video centered in some applications, if you have time and interest in doing so! Thank you very much!
I'm glad you enjoyed it, your kind words mean a lot! Thank you!
Excellent use of exposition (telling the story of it) to illuminate a frustratingly slippery path towards Galois Theory. At least now we can see where we are stepping, and place our feet more firmly on the ground before us! Thank you!
PS: Great idea using a 'green board' to present your formulas! A nice compromise between the slow-but-friendly blackboard/whiteboard, and the fast-but-impersonal use of math-formula animations! Very innovative!
Hey, this is a brilliant introduction, easily missed or overlooked, but more and more enlightening the more you listen to it. The fog is lifting and the relationship between the Galois theory and the Representation / Group theory is becoming apparent. I think I am going to revisit this intro a couple of times more. Thanks!
Thanks, that’s really encouraging! Glad you found it useful!
Very nice presentation. ⭐️
Very clean and straight forward explanations of a very advanced and intricate subject. Well executed. And i like the format of the video.
Thank you, I appreciate the kind words! :)
So clear! I love the video!
I need more of those! Really helpful
Loved your intro. Decided to stay. Thank you for this.
Thats very kind, thank you! :)
Thanks a lot. Really inspiring
What an awesome video. You deserve many more views!
This video is an eye opener. Back in the day I built Reed-Solomon encoder and decoders and struggled to get the key ideas of Galois theory. I didn’t understand it. Now I am feeling hopeful with your video. I must understand this so I hope you will make follow up videos on this topic. Thank you, thank you!
I must admit I have no clue what Reed-Solomon encoders are, but it is intriguing to hear they have something to do with Galois theory. Might look into that. Thank you so much for your kind words, I'm glad you enjoyed it!
This is amazing! Please continue and make more videos!
Thank you for your kind words :) I will!
Love your work! Thoroughly enjoyable.
I literally just finished watching through Borcherds' playlist on Galois Theory (mostly review of stuff I learned ages ago), but am excited to complement it with your more-motivated presentation
I hope you enjoy it!
this inspired me to finish my linear algebra pset -- awesome content
Thank you, I enjoyed watching this video very much. This video convinced me to spend more time with pure math in the future, even though I am employed as a computer scientist and hence need to spend most of my learning time with IT topics :)
Thank you, that’s very kind - I’m glad you liked it :)
Brilliant video! ❤🎉😊
Wonderful!
This presentation, undoubtedly, stands as the quintessence of introductory discourse on this subject matter. The presenter undoubtedly possesses a prodigious intellect akin to that of Galois. Remarkably exceptional!
This is great!
This is a really beautifully edited and crafted little video ❤️ I hit the bell 🔔
Thank you, that’s really kind :)
Great content. Thank you. I'm just stepping into this level of math, and you have deepened my understanding.
There is an annoying echo on the audio though. I don't mind replaying parts to be sure of understanding the math; but having to replay six times just to catch the word "norm" was truly annoying.
Thank you
I was studying Harold's Galois Theory book and had difficulty understanding a chapter after this. (Starting on Lagrange resolvent)
Great timing for me! Hope you complete this video series. Appreciate your work. :)
Thanks, Im glad it helped! The Lagrange Resolvent chapter was very illuminating, that’s where the book really takes off :)
Fatnastic stuff! I love the way you explain and summarize. The positively-biased board (black-on-white) really helps me a lot. The audio could get a little bit better, but hey, couldn't it always 😆
Thanks for the ride 🤗
Thanks, that’s very kind :) will work on the audio!
I love your approach to math youtube videos, keep it coming 💪
Thank you, very nice of you to say :)
very nice
Excellent.
I had an interesting career in ASIC and DSP design, in one chip it was based on Galois fields and Golay error correction, another was based on the Fast Wavelet Transform with DCT inside and an FFT engine, another was more mundane, building a pattern waveform generator based on sine waves but not using a high resolution Sin ROM, instead computing Sin from low resolution Sin and Cos. Math and ASIC design is a special pairing made in heaven, lots of flexibility to produce the desired result, it's my favourite type of chip design, no arguments about architecture, the math defines tjhe that. Do I remember much about Galois fields, nope, not a darn thing.
Amazing!!!
Excited for your next videos!
Masterpiece in all aspects - title, artful ambient space composition, scrupulous deliberate manner of presentation, deliberately stylish outfit and haircut (English artistic sophistication a la Oscar Wild? 😅 ),.. and of course fine math
Such a kind comment, you're being very nice - thank you :)
How did you move from quadratic to quintic without explainig cubic and quartic? :)
If this becomes a series the quintic would be the next chapter! :)
17:18 unmatched parenthesis on the second line
What video software are you using? It is excellent.
Thanks! Mostly DaVinci Resolve, and additionally I wrote some code that allows me to create the whiteboard animations easily
Was Galois aware of the fundamental theorem of symmetric polynomials when he proved his theorem or did he also develop the fundamental theorem of symmetric polynomials along the way to proving there’s no solution by radicals for polynomials of degree 5 or higher?
Suggestion:
check out the semirings of polynomials, they're pretty cool too.
Will do!
Very well explained introduction to a fascinating but quite opaque subject! Great work! I am keenly looking forward to follow-up videos.
What you say in the final segment is very true. In particular the Bourbaki collective has killed the human and historical element in teaching mathematics and the tone they set has made modern mathematics somewhat cleaner perhaps but much more difficult, and unnecessarily difficult, to learn. Therefore, pedagogical videos like yours, which teach mathematics in a language and from a perspective more suited to the human brain, are very important.
You could improve the audio quite a bit by suppressing reflections in the room you are recording in. It sounds very echoey. (Not like the echo in a large hall or outside, but lots of very fast reflections from nearby surfaces. This is one of the main reasons for bad quality of voice recordings and it becomes very obvious once you hear examples with and without those reflections.)
Thank you for your kind words and detailed feedback! I actually haven’t heard about the Bourbaki collective before, I’m intrigued! I’m just starting out and I really appreciate the technical feedback on the audio as well :)
@@martintrifonov AFAIK Bourbaki did important work in cleaning up the foundations of algebraic geometry, which had previously gotten into quite a mess and even produced some wrong results due to a lack of rigor. The Bourbaki books were great as an underlying structure but their huge success had a detrimental effect on the pedagogy of mathematics in my opinion, and not only in my opinion, as you can read in the Wikipedia entry:
>>As Cartier remarked, "The misunderstanding was that many people thought it should be taught the way it was written in the books. You can think of the first books of Bourbaki as an encyclopedia of mathematics... If you consider it as a textbook, it's a disaster."
Really appreciate this reply! As for your remarks on the bourbaki collective, this sent me down a rabbit hole of Wikipedia today, can’t believe I never heard about any of this before!
As for the audio quality, I actually have a decent microphone (or at least not the cheapest option 😂), but I see I haven’t set it up properly. It was obvious to me that the quality was lacking, and I played around with some noise reduction settings - with limited success. But now I realize much more concretely where and how I could improve. Thanks for your advice :)
If we are given a polynomial of degree 3, with roots a,b,c... can we compute something like sqrt(a) + sqrt(b) + sqrt(c)?
Also, can we make use of computing symmetric expressions to give us good starting points for polynomial root finding
algorithms or modify the iteration steps in one?
I'm imagining for example... unlikely scenario but, if you knew for example that your given polynomial has only real roots,
and one root is particularly larger than the others, you could approximate that large root by computing
(r1^50 + r2^50 + ... + r_n^50)^(1/50).
Im not sure I have a good answer to this - but what you are proposing loosely reminds me of Graeffe‘s Root Squaring Method - might be worth looking into :)
@@martintrifonov Looking at this again with fresher eyes, at least for the first question I asked I think the sums of powers of roots can be applied to taylor series. For sqrt(x) a taylor series centered at x=1 converges if x is in [0,2]. If we only have real, positive roots we could compute sqrt(r1) + sqrt(r2) + ... + sqrt(r_n) by squishing the roots closer to the origin by substituting cx for x in our starting polynomial, where we can pick a constant c. After computing the taylor series to an accuracy we like, we can multiply the sum of those approximations by sqrt(c) to get the final answer.
It should be relatively straightforward to determine the largest magnitude a root might have and pick c to be that largest magnitude.
Nice vfx. You have screen recorded a digital art program. I can see the artifacts of the pressure sensitivity graph in the writing. I think the canvas in the video is a 3D asset, but you could have placed it and distorted it with a perspective grid. You are using a soft brush that adds up on overlaps making it look very realistic as if the liquid collected. What blending mode are you using? Is it just opacity?
I am guessing you have some textures here and there or some glossy shading applied to the 3D canvas.
It looked very good, I might have to steal some of your methods. The editing where the pacing of the sped up writing following you sentences was very good. Sometimes it was in parallel, but that’s what you gotta do no? I think I will use motion graphics with custom typography with this kind of pacing. It is taking a long time to develop, especially when I want to continue to multiply my tasks with pairings between glyphs and innovative characters and transitions, but hey it could become a big animated typeface for math with special dynamic customizable characters! I don’t have much experience with coding so many variables though, but I am beginning to design a custom encoding and variable axes for it… just pseudo code at this point
Nice detective work :) ! Pretty close, a few things I did a bit different.
I didn't screen-record the whiteboard, rather I just made drawings on my Remarkable (E-book type) tablet. I downloaded the drawing files from the tablet to my computer, and I wrote a little code that reconstructs the paths of the lines and animates them. The drawing is reconstructed from the paths using the p5js and p5brush libraries. The latter, p5brush, creates realistic looking brush strokes. And yes, from there on you are mostly correct - some shading and composition!
The editing (speeding up the pace of the drawing) was rather labor-intensive. I'm thinking this could be automated a bit in the future.
Interesting, what you're describing about a dynamic animated math typeface - but I'm not sure I follow completely. Could you elaborate a bit on what this project is?
sorry if dumb, but am I correct in saying (xy)^4 + (zy)^4 + (xz)^4 is a polynomial? and cannot be said to be represented using the elementary polynomials according to the proof, because the table doesn't have any entries of the form (xy)^n?
It's a fair question! It is a symmetric polynomial, yes - and you're right to say that we can't learn how to represent it using using only elementary symmetric polynomials by the argument in the first proof. The first proof only shows how to represent power sum polynomials (x^k+y^k+z^k) using elemetnary symmetric polynomials. However, the second proof in the video shows how to do this with any symmetric polynomial, including your example. Hope that makes sense!
@@martintrifonov thanks, should have finished the video first
@@martintrifonov Just finished the video, I see that the polynomial I suggested can be expressed as ((s4)^2 -s^8)/2. Thanks for the knowledge
so fun to watch
Would it be rude to point out that his solution for the roots of the quadratic is wrong?
Which solution is that?
0:34 @@martintrifonov
the one that says -4 x 2 x 5, and should say -4 x 2 x (-5) @@martintrifonov
Right, I see. Oops! Thanks for pointing it out! Ill add it to the video description.
Found this after 9 months. I feel lucky now.
0:35 Wrong sign under the root.
Oops. My mistake. Thanks for pointing it out!
sub. i plan to someday really understand Galois theory. your video helps.
Glad to hear it, thanks! :)
ayyy, this is pretty gooodddd
9:04 - Groovy! ^.^
perfectly placed
MMM is a catchy intro. The effort put into this video clearly shows, kudos!
the proof at 15:30 wasn't crystal clear to me. can you expand (or guide me to a resource) on how this table layout proves Newton identities on the general case?
Thanks!
If I find time I will make a more formal write up of the contents of the video. The manner in which I present the proof (using the table) I haven’t found anywhere else - however, it is, in essence, very similar to one possible derivation given in the Wikipedia page of Newtons Identities. If you go to the page for newtons identities, and navigate to the subtitle „Derivations > As a telescopic sum of symmetric function identities“ you can read more there.
Great video! My only suggestion would be a differently colored background as your hair blends in and is a bit distracting. Looking forward to more.
You're right, that didn't quite turn out how I hoped it would. Will be working on the production quality (the audio too) in future videos. Thanks for the feedback!
Improve audio quality please
Working on it!
good maths and serious drip
If Saturday Night Live did a skit on math I am trying to learn.
I wish I had a whiteboard who could understand what I wanna write on it just like your whiteboard 😂
how tf is he making the whiteboard animations?? he seems to be able to see what is being written and the writing we see doesn't seem to be an overlay....huh???
Мавроди?
Bob ross style 😂😂😂😂 nice ❤❤❤❤❤
👍
I had the same obstruction for the Galois theory : too much theory to learn before solving any equation....it is the same than a promise for a good dessert but you have to eat a lot of heavy dishes ;-) For me Lagrangian resolvents was the point I stopped, because despite I understand the main ideas (splitting fields and so on, .... more in the group theory and fields) the difficulty is to guess whether a given equation is Galois solvable and how to process in that case. There is an unknown too : can you apply the theory when the coefficients ARE NOT rationals (not integer).. is this for "algebraic equations" only ? Guessing the resolvents (or the intermediary fields) is the extra I need ....the new entries in your "bag".
I like your analogy with dessert - it does feel like that sometimes :)
th-cam.com/video/3imeTgGBaLc/w-d-xo.htmlsi=i7s_yO2lWOolAVQE&t=192 I know it's a famous fact that you can factorise a polynomial like this, but I think you should either justify it, or at least tell a name of the theorem, so it'd be easily googleable, as you do just 20 seconds earlier with the fundamental theorem of algebra.
You’re right, I should have! Thanks for the feedback, I appreciate it :)
I hope your middle name is Sam or something so you can start making videos on Saturday/Sunday.
Audio quality is not great. That makes it hard to focus upon your lecture, which I want to hear. You will well serve your viewers by attending this niggling issue.
Noted, working on it. Thanks for the feedback!
that drip
zamn
Made it to 10:22 then I got lost.
That's valuable feedback, it's good to know which moments might be hard to follow, so thanks!
Hogwarts math!
I am in love with you. And I am a guy.. I am questioning my sexuality..
Sound is pretty low quality. Also bad diction/unclear pronunciation. Content is ok