What are...Riemann surfaces?
Вставка
- Опубліковано 10 лют 2025
- Goal.
I would like to tell you a bit about my favorite theorems, ideas or concepts in mathematics and why I like them so much.
This time.
What are...Riemann surfaces? Or: Avoid choices!
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Slides.
www.dtubbenhaue...
Thumbnail.
en.wikipedia.o...
Main discussion.
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
Background.
mathworld.wolf...
ncatlab.org/nl...
math.berkeley....
mathematica.st...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
en.wikipedia.o...
Mathematica.
demonstrations...
resources.wolf...
mathematica.st...
demonstrations...
demonstrations...
Pictures used.
Picture created using reference.wolf...
Pictures created using reference.wolf...
en.wikipedia.o...
Picture from en.wikipedia.o...
en.wikipedia.o...
upload.wikimed...
Pictures from www.dtubbenhau...
UA-cam and co.
• Riemann surfaces
• The Idea of a Riemann ...
• Imaginary Numbers Are ...
• Riemann surface of f(z...
• Where is the Riemann S...
#geometry
#topology
#mathematics
Love your channel :)
A video on the hodge conjecture would be awesome 😁
Noted! I am currently thinking about how to UA-cam the millennium prize problems. Turns out that the Hodge conjecture is quite tricky to explain, or at least I think it is the or one of the hardest ones to explain from the seven conjectures. In other words, as for now, I do not have any good way of compressing it into a 10-15 Minutes video. But this is certainly work in progress.
Ah, and thanks for the kind feedback!
Amazing. Riemann surfaces.... This is a very complex, yet interesting topic. The thumbnail is amazing!! I just made a video yesterday on Topology. Best wishes.
I will check it out!
Anyway, Riemann surfaces are at the border of analysis and topology which makes them very fascinating. Glad that you liked them!
@@VisualMath this is my college profile. I will comment you from my personal profile. I will send you my channel link. I am particularly interested in Riemann surfaces as it relates to my subject, Einstein's General Theory of Relativity.
Sounds good and yep, Riemann surfaces play a crucial role in relativity!
Thank you
Welcome, I hope you enjoyed the video ☺
how do you make these beautiful slides? please post a tutorial, thanks
Glad that you like them! Your feedback is very much appreciated.
The slides are created using latex beamer code that I have polished over the years. In other words: time helps ;-)
I will not make a tutorial on latex anytime soon, sorry for that. What I will do, and should have done ages ago, is that I will make the source files freely available so that everyone can download and use them. I hope that will be useful for someone. Thanks for (implicitly) hinting me towards this idea.
So how can we download the latex beamer code .I'm very much interested sir
Cohen Forcing please. So far I have understood the gist of it. I.e. "Think of an object. If there is no obvious objection to its existence, then it does exist in the universe of sets, for that universe is vast. Use the notion of a poset to create increasingly consistent approximations to this object, and we obtain a forcing axiom for that object. "
The more technical side includes (generic) filters, which I would appreciate if you could explain in more detail. Please, I need it for an important paper I am working on, and I do not find the explanations online satisfactory so far... Woodin made a pretty good explanation of it in his lectures on UA-cam, but I would find it pedagogically helpful, if you could complete some gaps in my understanding, which you usually do with your presentations. Thanks a lot. Please also explain how this technique is able to prove independence results in mathematics, where other techniques have failed...Cheers
Still work in progress, sorry; I am just slow.
@@VisualMath can you please provide a rough assessment of my current understanding (as displayed in previous msg). I want to make sure I don't have false info in my brain...
My understanding is too much a patchwork to make any comment, sorry, but I will have a look into it.
BTW, comment for this video: In high school we were taught that such "multi-valued functions" are not in fact functions, but RELATIONS. Thus, if you have more than 1 OUTPUT, for a single INPUT, it is not officially a function, but a relation...
Fine with me; I do not think that the name is all that important. Unless you use some "really strange" name ;-)
Anyway, the point for me is that "function" has "several inputs can go to one output", but not vice versa. This makes the definition unbalanced, and that always causes trouble. At the same time, "function" is what we learn from the start, and that is then didactically wrong.
The problem is that we need to rewind math history to fix this; too bad, I guess that is not happening ;-)
@@VisualMath you brought up interesting points here, which I totally agree with. You were referring to surjections, right? I. E. multiple inputs for 1 output allowed. Anyway, one of the papers I am working on is precisely what you mentioned, about problems in foundations of mathematics. Perhaps I can even run it by you some time, if you're interested in giving me some feedback.
I'm a former pure math/philosophy/physics student at UNSW.
I know Norman Wildberger personally, and he has looked at some of my work and was most impressed (at the time in his office).
That was about 10 years ago now.
My work is close to publication and I need some mathematical eyes to look through it.
Any feedback is tremendously appreciated.
I can email you some time soon, after you've uploaded the material on Cohen's proof, and I'll make it clear who I am. Thanks again for your time.
Oh fabulous, UNSW is close by! Feel free to email me. I can't make any promises as life throws dirt from time to time, but we have to see.
You did a specification and not very well. Here the detailed explaination:
What produce multiple values in math is called "multivalued function" and what connects the points of starting set and ending set is a binary relation. The " = " symbol is officially the "equivalence relation" and that is however a kind of relation (is a binary relation with more constraints!)
RECAP:
1) Functions are single valued and the others are multivalued (like log, arg, n-sqrt etc...)
2) Functions and MV functions connect the elements of domain and codomain with relations.
3) Functions uses binary relations called also "one-to-one functional relation" or "many-to-one functional non-injective relation".
4) MV functions uses binary relations "one-to-many" or "many-to-many".
5) The equivalence relation ( = ) is a binary relation that is reflexive, symmetric and transitive.
What you wrote may lead to error like: "A student is a person" and "a person is a student" (worng). The same with relations. Both functions and MV functions uses relations. One is the generalization (relations) and the other is the special case (equivalence in functions). However is well accepted for MV functions to write the set of solutions with = symbol like :
MV function ( .... ) = { , , , }
example: sqrt(9) = { +3 , -3 }
Sources: Wikipedia "multivalued function", wikipedia "binary relation", wikipedia "equivalence relation", wolfram mathworld "multivalued function".