Eating Curves for Breakfast - Numberphile
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- Опубликовано: 26 сен 2024
- This is a continuation of a video with Isabel Vogt at: • Error Correcting Curve...
More links & stuff in full description below ↓↓↓
Isabel Vogt at Brown University - www.math.brown...
Interpolation for Brill--Noether curves - arxiv.org/abs/...
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This is a continuation of a video with Isabel Vogt at: ruclips.net/video/CcZf_7Fb4Us/видео.html
I'll never get tired of seeing professional mathematicians getting passionate about their own work! :)
What may not come across, because of Vogt's modesty, is how impressive this result is. A question this simple and natural is something one would expect to have been answered already in the 19th century. (Brill-Noether theory did indeed originate in the 19th century.) And if it wasn't answered in the 19th century, then one would expect that the enormous advances in algebraic geometry in the 20th century would have polished it off. The fact that the problem wasn't solved until the 21st century indicates that the problem is very hard. Many people tried to solve it and produced only partial results, until Larson and Vogt answered it completely.
Regarding whether the theorem is beautiful in light of the finitely many exceptions, of course it is true that theorems without exceptions are prettier. However, the existence of finitely many exceptions is something that mathematicians have learned to expect, and to live with. Sometimes the finitely many exceptions have their own beauty. (The classification of finite simple groups has finitely many exceptions---the sporadic simple groups---which are very beautiful.) The existence of finitely many exceptions also usually makes the theorem harder to prove, because your argument has to take them into account somehow. Any argument that is too simple can't be correct because it won't explain the exceptions.
...and then Vogt comes sweeping in and crushes it.
Don't you think you or I could've done the same thing?
This follow-up video begs for a video to be made on Brill-Noether curves and what differentiates them within the broader family of curves in general
Roughly speaking, Brill-Noether curves are "general" curves that can be embedded in the target space. The restriction to Brill-Noether curves excludes "uninteresting" counterexamples.
Man as an amateur mathematician, and one who briefly pursued a degree in Mathematics, I'm so jealous, but also so very happy to see someone who has made it as a mathematician. Hopefully one day I'll appear in a Numberphile video for something I've found. If nothing else, that'll be a cool bucket list item to cross off.
I'm still trying to be first to comment as my bucket wish list .
I feel like the title of this video is going to be in a rap song some time in the future
where's MC Hawking when you need him?
Love her enthusiasm! Really fun videos!
Why is her enthusiasm so contagious?
"Do you wish it wasn't kind of a little bit ugly" is a great question about a piece of math :D
Brady man you called her achievement ugly 😂 she didn't lose her temper though good for her
Love her energy. :)
I could tell in her eyes that she knew this theorem might be named after her, but mathematicians are generally a humble bunch, and as expected, she would never think of naming it that herself.
Is the Noether in Brill-Noether theory Emmy Noether?
I believe it’s for Max Noether (her father)
It's actually her father, Max Noether, according to Wikipedia :)
Her father, Max Noether
No. It was actually her father, Max Noether.
Almost. It was her father Max Noether.
The only name of the all the works of the persons that appeared on this channel that I will remember forever is the "Parker Square"
Beautiful, Beautiful result!!
A brilliant individual
You are such a good interviewer
awesome mathematician
I love how excited she is to explain this all, she has a great vibe. Would enjoy a lot if she was my lecturer.
Amazing work!
I think what most mathematicians fail to grasp the profoundness of, is that with the infinitude of numbers, there are so few exceptions and they are of such extremely low values.
The fact that we can prove these theorems (even with the restrictions) using such low value numbers is absolutely mind boggling.
You really think most *mathematicians* fail to grasp this?
The largest sporadic group has an order less than 10^54. That is absolutely tiny compared to almost all finite numbers. I suspect most mathematicians grasp that.
What a delightfully strange result!
Great video on the general Vogt-Larson theorem.
Any relation to Robbie Vogt?
This is so cool. So beautiful. Great job!
This should have been part 1... (and with the duration of part 1)
Obviously, this is the "Larson-Vogt" or "Vogt-Larson" Interpolation Theorem.
Funny enough, I did heard about this theorem from Larson himself in a seminar talk, but I didn't realize she is his collaborator until now.
great! The CRC16 is rediscovered!!!
Great explaination!
Larson and Vogt and married to each other, which is a fun detail.
voght-larson interpolation theorem, obviously
But without the typos 😂
Congratulations :)
Well Done!
Great at math, not so great at drawing circles ✍️
whow! well done
so.. does this Vogt-Larson theorem have a wikipedia page yet?
Hmmm I wonder if the same tuples appear in the tropical setting! Perhaps preserved under degeneration - but tropically I could believe more tuples show up because of tropical varieties that aren’t tropicalizations of regular curves…
Unrelatedly, I’m also curious: in these exceptional cases, how else are they geometrically realized? Consider an exceptional case triple (d,g,r). Does this imply curves of of genus g embed into their W_d^r(C) in a special/unexpected way?
Many of the mathematicians I know occasionally adorn themselves with some kind of mathematical object. Do Professor Vogt's earrings have such a story?
They look like algebraic surfaces to me.
Fields Medal contender?
I need her to extrapolate more info about the exceptions.. ! #Numberphile3
Funny, I didn't understand anything about the theorem, except that It seems beautiful, and also she is a cutie pie.
Sadly, I don't really get it. I'll have to take another run at it.
A few questions for Vogt: You mentioned that the four exceptions are curves that live in a surfaces that do not pass through the right number of points. Is there anything in common between these four surfaces? Are they pretty? (Show us pictures! :D )
I'm guessing that r=1, which would cause problems given the r-1 denominator, makes no sense because you can not have an Horizon of dimension 0.
Pretty much. There aren't many curves in 1-dimensional space!
What would make someone think fo complex numbers though..it could have all real solutions for all you know..
4' 26'' 😂😂😂😂👍👍👌😉
I've had to watch the pair of videos twice because i was too confused the first time by the four switches at the back of the book shelf. WTH a library has wiring behind wood in 2023.
4:08: Is ℙ an alternative symbol for the complex numbers?
Very cool! Since two of the exceptions are (as I understand it) in 3-dimensional space, is there a way for us to kind of easily visualize those?
I would love to know this as well.
it's actually 6 real dimensions (3 complex) so probably not
I thought the '3 complex' only applied for the surface, not for the curve?
@@asthmen No, all the dimensions are complex. One is trying to fit a complex curve (2 real dimensions) through a bunch of points in complex 3-space (6 real dimensions), and their impossibility proof argues that the curve lies on a complex surface (4 real dimensions), and even the surface can't interpolate the points.
we can agree Analysis has the best tricks in the book, but Algebra is the legit magic
But number theory has a better combo of simple materials and complex situations (including conjectures that are simple to state).
This is so cool and I would love for a deeper dive into this, maybe at main channel pace. Let’s have more Isabel!
That’s so weird and cool. What is driving the exceptions!? Why is it finitely occurring and in the small numbers!? Those particular numbers