truly inspiring that some people have spent thousands of hours mastering these courses and concepts in order to use all of those hours in trying to understand further unknown concepts in mathematics
@@Entropize1 I got it. Is it enough to an ultimate comprehension of the last theorem? PS in your (philosophic) opinion, Physics and Mathematics could reach an end game of knowledge?
@@KRYPTOS_K5 Unfortunately, I'm not sure what you're asking. What I can say is that if you want to understand FLT from ground zero, you have a lifetime of work ahead of you. Regarding your second question: no, I don't think it's possible.
I feel like this strategy of learning all of the mathematics needed for FLT before having even read the proof is not the best. I think it would be a lot more motivating and perhaps more efficient to first read the theorem starting with some minimal amount prerequisites (I would be curious to know what you would consider that to be). You could then black boxing everything difficult and focus on the main ideas and intuitions with the help of your videos, only then going back and filling in the holes with the extra references in a second/third/fourth pass to reach a desired level of depth in understanding. Kind of like a mechanic who reads a millions books on engines without ever having popped open a hood vs. a mechanic getting their hands dirty early on to understand what is needed to be studied and how it's used based.
Yes, you are spot on. This video is a bit of a meme to be honest (but only a bit). I do not recommend actually learning all of this before starting the proof. This video is more "here's most all the things you should learn in total if your goal is to have as few questions as possible when learning FLT and to understand the maximal number of details." In reality, provided you have been through a solid undergraduate curriculum (whatever that even means), and a reasonably good MA curriculum (again, whatever that means), I'd say you want my "top 8": 1. Vakil/Hartshorne 2. Neukirch Ch 1-3 3. Class Field Theory (Milne/Neukirch) 4. Silverman 1 5. Silverman 2 6. Diamond and Shurman 7. Diamond, Im 8. Snowden's Torsion Theorem Course (because its proof uses a lot of the tools you'll need to be familiar with before getting into FLT; it's a good warmup) Then I'd pick up Cornell Silverman Stevens and the DDT notes and start going through the FLT proof. But, see, there are technically many prerequisites one ought to have to begin reading the above 8, and my video simply tries to take a lot of that into account. My video also tries to a) anticipate the questions you will inevitably have if you start reading the proof too early and b) anticipate the questions you will have if your background isn't as thorough as it should be, even if you have gone through a fairly typical education. Here are some things I recommend blackboxing (initially or forever). This list is not comprehensive; it consists of what I could think of in 2 minutes flat while sitting on the couch talking to my wife: 1. Carayol's Theorem on Vanishing Cycles 2. Langlands-Tunnell (these are the biggest two black boxes by far) 3. Faltings' Theorem 4. Ramakrishna's Thesis 5. Deligne-Serre 6. Jacquet-Langlands 7. The Open Image Theorem (although I actually think you should grapple with this sooner rather than later) Perhaps I should make a video elaborating on the information I have just given.
Personally i liked this video. So many different topics, and gives me motivation to learn more about those. But a video like the one you suggest would be interesting too.
Honestly i’ve actually Always wondered how you solve these awesome theorems. But it was only a thought. And now there’s a 20 min vid about everything you need to know
I have an undergratuate math degree, and I was happy with myself through maybe the first slide. "Crazy ammount" is right. I stopped being able to say "I can do that" very very early for how much longer the prerequisite list went.
I think I’ll hold out for the hope that a simpler proof will be found before I die. Seriously though, I’m not a mathematician so forgive my naïveté, is there any hope that someone will ever come up with a simpler proof? Maybe one that could be understood by, say, an undergrad in math? When I think about how simple it is to understand the theorem and how monumentally difficult it is to understand the proof, I get the impression that it’s just a matter of time before someone comes up with another approach that will be much shorter and much easier to follow.
Not really. The theorem is false in Q_p, destroying any hopes of a congruence based proof, which is what most people think of when they think of "simple proof." There are simplifications to the method I presented, but the main narrative remains the same. There is also the Khare-Wintenberger proof, but I am not knowledgeable of how that works, and it is still quite complicated.
I think it's pretty exciting to have a chance for someone to give such a comprehensive and concise set of options for learning a broad subset of mathematical ideas! Overwhelming eventually gives way to engaging if you get bored enough :)
Depressive??? On the contrary. This is one of the most exciting and uplifting videos I have seen in a long time. If anyone is even remotely interested in math, this video is a goldmine. Everyone knows about Fermat's Last Theorem and it comes up in discussions now and then. It is an extremely simple problem and often people wonder why it took so long to solve. This video is phenomenal and very helpful to see the math required to prove FLT. It is the first time I have seen it listed in a structured way like this. Thank you for making this video! I will recommend it to my friends.
Maybe a good mathematician, but a bad narrator or even worse sound editor. Or simply an very outdated sound system for recording. So much fading and wobble in sound makes it almost incomprehensible...
... the proceedings of my investigation into the peculiar case of Monsieur Pierre de Fermat's last unproven theorem suggest that even if a simple proof exists, its discovery would be met with false rebuttals, ridicule and unanimous disbelief.
...and consequently, as was indeed the case with Wile's discovery of his proof, it is a far better approach to be completely secretive until a point of certainty is reached that at least some merit would be earned for novelty or effort, even in the event of still being completely in error. A difficulty then immediately arises that any simple proof of FLT will be so underwhelming and deflating that the greater majority of high schooled mathematicians with a passing interest in a simple proof will overwhelm and moderate any public narrative with their incredulity such that a (largely) correct simple proof will never reach the attention of the infinitesimal minority of those able to sufficiently understand Wile's proof to make an authoritative determination as to its veracity. The latter, being mostly already convinced anyway that there is no such simple proof, will never take notice of yet another simple proof of FLT to be bothered with it, lest they put their professional reputation and career at risk. This is the current state of any purported simple proof of FLT, and it is extremely unlikely to ever change, despite Wile's proof's failure to explain "Why is it so?" (ie. axiom of choice) apart from the apparently patently obvious observation that "A circle is a elliptic curve in disguise". Thus any simple proof of FLT, and especially one that can also show "Why it is so." will be lauded only in the Hall of Ridicule and thereafter quickly lost to science, if not utterly banished from, the journals of western academic literature.
The likelihood of there being a quick proof of FLT is incredibly low, since it is false in Q_p. Moreover, the interesting part of FLT is the Modularity Theorem, which a quick proof of FLT would not provide.
@@Viewpoint314 Of course! I was joking. But there is still the possibility that Fermat found an alternative proof involving some simple procedure: Wouldn't that be wonderful?!
@@r.w.emersonii3501 It would! but the boring answer is that he probably just made an error somewhere, still his intuition is incredible to just "feel" the right answer in this case
What book would you recommend for someone who is not that bright and only has time to read one book, but nonetheless is interested in fermat's last theorem?
@@Entropize1 Well, i'm welcoming myself to the No Club lol. I didnt finish my Maths degree and now being in my mid 50s and moving to Europe soon, from Australia, wondered whether to go finish it for curiosity sake as i love maths yet there is so much online re lectures etc, i dont know if i could handle stringent academia at my age plus i like debating things as am very unconventional. Back in the days of torrents, i once downloaded a file with maths textbooks only to find there were literally like 5000 of them. I went, oops lol That is a lot of textbooks :-)
@@johnbyrne1022 because multiple experts have come together, each of whom know all the details of various parts of the proof, and have collaboratively exchanged results that allowed the result to be proven. This happens all the time in mathematics. Even though no one person knows all the details, all of the details are written down somewhere in a published journal or book.
He didn't, because half of this math didn't exist when Fermat "proved" his theorem. We don't actually have his original proof, he just said it as a fact, added a footnote that he can't fit the proof into the book because it's too long and promptly died before passing it forward. It took about 3 centuries after that for the theorem to be confirmed, so unless Fermat was simply that ahead of his time and just didn't tell anyone, he probably made a mistake somewhere
@@callejohansson5732 We can't rule out that possibility, but ever since Fermat's death mathematicans, some of the greatest ones among them (Euler, Lebesque, HIlbert to name a few) have tried to crack that proof without success (aside from proving few specific cases) Do not underestimate the amount of work that went into solving this problem, if there was another way achievable by 17th century math alone, I'm 100% sure somebody would've found it. Fermat was still human, it's way more reasonable to assume that he simply made an error than that he was a better mathematician that virtually all number theorists (and more) after him combined
