How Do We Solve Difficult Problems in Mathematics?
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- Опубликовано: 6 авг 2024
- #SoME3
In this video, we discuss how we tackle difficult problems in mathematics, and look at historical examples of groundbreaking new discoveries in mathematics that emerged from trying to solve a difficult problems.
I made a follow up video to address an error in the video:
• Number Systems Invente...
Chapters:
00:00 Intro
01:09 Main Problem
10:51 FLT and Algebraic Number Theory
14:42 Quintic Equation and Group Theory
22:06 PDE and Fourier Analysis
27:36 Outro
Corrections:
11:48 It is true that Euler provided the first proof and invented Z[√-3], but first algebraic proof using Z[ω] was by Gauss
27:03 I forgot to write the coefficients c_n for the series
As Geogre Polya said: "If you cannot solve a problem, find a simpler one that you can solve and solve that one instead."
He also wrote the classical book "How to solve it" which is relevant still today.
Otherwise, my advice would be to go deeply and see a problem and its components in as many different ways as possible.
Try to understand what the concepts mean on a deeper level behind the formulation of the definition. Play around with them. Try to see connections with other concepts.
Another more recent book I like, regarding problem solving and thinking is "The 5 elements of effective thinking" by Burger and Starbird.
It is a rather short book, but contains good advice on how to structure your thinking to be more creative and solve problems.
Beautifully said
I want a PDF of this book. Haha.
"I fear not the man who has practiced 10,000 kicks once, but the man who has practiced one kick 10,000 times" ~ Bruce Lee
"It is better to solve one problem five different ways, than to solve five problems one way" ~ George Polya
Translation can also be viewed as 'rotation' about a vanishing point, i.e. a point at infinity (c.f. "PGA Ep 1: The Reflection Menace" by Bivector)
Also a sheer in higher dimensions. I'm not a graphics programmer, I think that's how they pack a translation and transformation in a single matrix.
@@gritcrit4385They use an additional dimension and use homogeneous coordinates with adding a column to the rotation matrix which is the translation vector, and then a line with [0, …, 0, 1]
when i saw the thumbnail i thought it was about category theory
I’ve been getting lots of category theory videos recently in my recommended
Now I wonder if this comment didn't trigger the algo to think this video is category theory adjacent
Everything ever is about category theory.
@@snex000As much as everything is about physics or your comment is about biology.
@@grivza Those things are also about category theory.
Absolutely incredible how intuitive you made the discovery of Fouier analysis seem!
Your pronunciation of names is as epic as your grasp of group! Amazing!!
This method is very similar to the A^-1MA method of changing matrix basis. you can kind of abuse the symmetry of a plane to simplify a problem; shifting the coordinate space by any finite value sends each point in the coordinate space back to that same coordinate space.
It kinda only works because they are isomorphisms in the Euclidean group or the general linear group if the field isn’t the reals
13:22 “Just wouldn’t fit into the margin of this video”
Seems no one caught this joke lol
I didddddddddddd 😂
Very interesting! 🎉😊
Excellent video!
Great video !!
Hilbert Spaces! I am very excited ☺️
Love your videos ❤
Nice nice nice video, thank you
Thank you for the great content.
Please continue to upload good videos~~~
I can see a Korean person here
I got obsessed with what road you filmed on. I used to work near there.
This video is awesome! I am currently a physics & computer engineering student and have only seen Fourier series in diff eq course, though I feel I hadn't made justice to it. For example I don't actually even know why a Fourier series converge, and how this somehow relates to eigenvectors of observables (hermitian operators) in quantum mechanics. I have two pictures of uncertainty principle that haven't connected to each other yet... One is Fourier transform, other is about commutator operation.
I am also hyped about groups, categories etc. but don't know in which order I should approach all these. I would love to see your opinions on how to best get started
Hi, these 2 videos by fellow creators math content are honestly best resources I have found on RUclips about the uncertainty principle,
ruclips.net/video/MBnnXbOM5S4/видео.htmlsi=fAwYAuLO4PEyhhCh
ruclips.net/video/jnxqHcObNK4/видео.htmlsi=KEZk5nTuQNUQj36s
and Fourier series indeed are the eigenfunctions of the 1 variable Laplacian, aka the second derivative.
so when you have linear equations like heat equation, wave equation or Schrodinger equation (without potential) that has the second derivative as the spatial operator,
since (d/dx)^2 (u) = Lu, if u happens to be sin or cos, so sines and cosine (or really imaginary exponents) are the eigenfunctions of the second derivative,
so for example these equations reduce to pretty much an ODE, (eg, Lu = ih(d/dt)u which would have solution u = (initial position) e^[(L/ih) t] ),
and in quantum mechanics, we would call it the time evolution operator.
I will have a video released on this topic and it will be covered in detail, in the perspective of Hilbert space and eigenfunctions of elliptic operators,
but I think it will be several videos down the line and I cannot give a exact date when it will be finished, but if you would like a resource on this,
ruclips.net/video/ToIXSwZ1pJU/видео.htmlsi=rDqvXviQvEmyM7XH is an amazing resource on why Fourier series would solve the heat equation, and is the eigenfunction of the second derivative.
but ofc fourier series isn't the eigenfunction of a different spatial operator, say (-(d/dx)^2 + x^2)u, which is called the quantum harmonic oscillator,
and the eigenfunctions of this operator are gaussian times Hermite polynomials, if you have seen them before. and you can just tack on the time evolution operator to this as well and call it a valid solution to a linear equation.
And these distinct eigenstates really are the different spectrum of frequency/energy/etc that a wave or particle can take.
As for convergence of Fourier series/transform, one intuition we can have is that smoother a function is, faster the series/transform will converge. Why that is so, well that requires a fair a bit of rigorous argument, but one intuition I can give is that less smooth a function is, more noisy a signal would have to be, which requires a lot of higher order frequencies.
Since you are studying both physics and computer science, groups/rings/fields/advanced linear algebra/functional analysis are all amazing subjects to study due to how applicable they are to physics and compsci, and different universities have different curriculum and introduces them in different order, and you will get a different flavor depending on how they are ordered.
but I would personally delay learning about category theory until you have sufficient exposure to various fields of mathematics. It is beautiful in my opinion, but it is an abstraction on top of abstraction, but I think you will find a lot of joy in finding the connection yourself first. Say, if you study group theory and topology and linear algebra, you would find that there are recurring patterns in these seemingly unrelated subjects, and start realizing how interconnected all these different subjects are, but without that experience first, starting with the abstract is rather pointless in my opinion. abstraction is invented because there are recurring patterns, not the other way around
@@EpsilonDeltaMain Somehow I didn't get this reply notification, but thank you so much for your crafted answer! I will both check the links that you've shared in depth and take your general advice about which topics to advance into seriously
I would love to have 18:16 as an animation for buffering/loading
i am waiting for generalization of fouries functions
Would the base 7 and the heavy side of 21 vectors of a 6 sided cube help maybe 2:49 with lines of connecting joints as notes to equal x y z
u r a genius
13:19 you evil man lmao
13:21 😆
Margin callback
Anyone else finding themselves playing a mental game of Geoguessr at 2:37?
wow
haha
Why cant we automate it so we can rotate and translate the same time
with matlab
the mic sounds so wet lol hard to listen to for some people that have misophonia
i found that getting the right answer while doing the complete wrong calculations will still get half marks for the question...
the prof wrote on the exam: i have no idea how you got the right answer so i'll give you half marks...
"Easy math"
Btw are u Korean
yes I am, is it my accent?
@@EpsilonDeltaMainYes.
응원합니다~
Give up and let the mathematicians do it? I think that's the answer
I dont
Supari nikaal ke baat karna vro
Dude, your English REALLY needs improvement. You frequently leave out articles and make plural words into singular words.
IMHO opinion your a not expressing Euclidean geometry correctly. Euclidean is a coordinate system in 4 dimensions, not 2.
Same with so called gravity wells are represented in 2d when they are not 2D
You have a fundamental misunderstanding of what a geometry means.
its a mathematical abstraction that can exist in any dimension. So you can have a Euclidean geometry in any n-dimension.
Just because our spacetime has (3,1)-dimensional pseudo-Riemannian geometry does not mean thats the only geometry that can ever exist.