@@raotouqeer9052 yes. It can make a big difference to have someone pulling for you on the other side

@@sasx1487 that is a weird saying, isn’t it?

@@_fl3x_up_ The thing is, there is nothing fast about proofs. Getting the hang of them takes a lot of time and practice. Usually, before jumping into something like Real Analysis, you’ll want to learn the basics of logic and methods such as induction. Number Theory is a great way to get started with that; it offers lots of simple problems you can try those techniques out on. A good book for learning Real Analysis is Rosenlicht’s “Introduction to Analysis.” I’ve always felt he did a really good job at introducing basic real analysis and metric spaces. If you really need to learn how to prove things in real analysis, then pick up the book, read through a theorem, put it down and try to prove it. Then, failing that, read the proof. And then try to reproduce the proof. Ask yourself at every step what theorem the author used to get there. And then the next day read the theorem, and try to reproduce the proof. If you can’t, then divide the proof into chunks, where each chunk accomplishes some particular goal. Do the same the next day. Rinse and repeat.

Certainly, unless you make your own company, you aren’t going to have that freedom. But really, I have always had the luxury of my choice of research projects, and control over my own schedule. My whole time in academia I have only had a couple hours a week where I had to be anywhere in particular, and I’ve been left alone to do my work otherwise. This includes my PhD time and my postdocs.

@@JoelRosenfeld That's great for you, indeed. I just would guess that that's not the norm.

@@DOCTORJAN714 it’s very field dependent. Most mathematicians that I know have this sort of freedom for their schedule. But those in biology and chemistry have much more rigid schedules because of the lab work

Indeed!

@@privateprivate5302 depends on what you mean by teacher. If a teacher has a bachelors degree, then they would need to go to graduate school to get a PhD. That’s 5 years on average. Then if they get a professor position right out of school, then it would take another 6 years for tenure, typically. If they need to do a postdoc for more experience, then that would be another 3-5 years or so. So about 15 years from the completion of a bachelors degree

​@@JoelRosenfeld So, you're saying that people can't start doing research into new areas until they are almost 40 years old? Try telling that to Tesla, Einstein, Edison, Ford, The Wright Brothers, and the handfuls of other people who are responsible for the modern world; even the Founding Fathers of America were mostly around 20 years old, perhaps you need to reevaluate this concept. Tenure is not the way to afford research, tenure is the way to retire from a career wherein you enriched the minds of the next generation with your own knowledge.

@@BlueNEXUSGaming No. You do research the whole time. I have brought in over a million dollars of external funding to do research into topics like control theory, formal methods in computing, data science, and neuroscience over the past 15 years. The 15 year mark is when Universities trust you enough to give you permanent employment. I’ve been doing research since I was about 24/25. And having gotten tenure, this is when everything starts to ramp up even more; bigger group, bigger grants, etc. It’s anything but retirement.

@@BlueNEXUSGaming Doing this research is very hard. It takes a lot to learn what’s important and how to do it. but it is also what leads to companies like Tesla to be able to have self driving cars. That sort of technology incubated in academia for decades before it became commercially viable.

​@@JoelRosenfeld You never mentioned any of that in the video; additionally, self driving cars was the first AI according to many accounts, electric vehicles competed with the Model-T from Ford but were seen as too girly, and Edison Motors made Electric Trucks before Tesla Motors and they used their refunded Tesla Motors Cyber Truck investment money as their startup money because they were tired of waiting. Furthermore, starting a sentence with "and" is bad grammar, comparable to including "0+" at the start a trigonometric math equation, because there is no prefix to conjoin the suffix of the "and" to; my grandmother was an English teacher, and you should have better writing literacy if you plan to write such complicated research papers while working towards tenure, it is not difficult to access a Thesaurus. Proper research doesn't require the majority of the development time, as the largest hurdle is the manufacturing process and building the infrastructure to scale, as most of the infrastructure requires funding for technology that is not yet existing, or has not had proper worksite education to safely teach people how the manufacturing machinery functions, or it might lack a proper legal team to provide copyright and trademark protections, or they lack a customer base large enough to quantitate such a large manufacturing process; eventually, someone more skilled shows up, and they simply do a better job at building a better product with a more streamlined logistics than the previous product, but the minutiae is stifling progression, and malformed information alongside half truths are not making the learning and research process any easier. Even with tenure, that won't guarantee you won't get laughed out of academia for your research, lest we forget what happened to those who suggested that the world was a spheroid, or that the Earth was not the center of the solar system, or that momentum is relative to the observer and not a universal constant, or that the speed of light was a range of speeds and not equivalent to the speed of causality because light had mass (additionally, (E)=((m×c)²)=((m²)×(c²)) vs (E)=(m×(c²))=((m¹)×(c²)) is also a difficult equation to interpret because it is commonly written as "E=mc²" and people should really use Parenthetical Brackets as their primary identifiers of the computational sequence ESPECIALLY in academia, since the result can cause a wide differential between potential sums), just to name a few.

Anyone can submit a paper for publication. The thing is that it’s hard to meet the standards for publication without training, and to find problems that are relevant and novel. That’s what a PhD program helps set you up with. Best thing to do for now is to start taking some proof based courses in your math department and see if you can connect with a professor to conduct some undergraduate research

The real novelty here is the discovery of a new proof of the Pythagorean theorem. There are honestly hundreds that we know, but the fact that a new proof was discovered by high school students is really the icing on the cake. Sine and cosine were known to Ptolemy in antiquity. In the 12th century, Indian mathematicians discovered the power series representations.

@@bennoarchimboldi6245 oh interesting, he was a student of Ken Ono. I wonder if I met him? A large number of Ken’s students visited UF while I was there

maybe! For my part, I was taught by American guys who also mispronounced his name lol

@@thelocalsage to be fair tho; maybe academia is not for you, if you have a learning disability? And what is the type of disability? No harm meant

@@satioOeinas ADHD, and while it is characteristically severe in my case I promise you that learning is not my deficit-I am a lifelong and voracious learner, and can pick up content very quickly. “Learning disability” is just what others call it, and in fact my capacity to learn is one of the few things left unscathed by it. I have consistently been plucked out apart from my peers by professors as better with cultivating deep understanding, rote memorization and arbitrary information tends to be what is harder for me. If academia is not a place for me, then academia is not a place for people who desire to garner deep insight and understanding of the world-that very well may be true, but if academia is not the place for that, then I am unsure where on this Earth is. Generally, I prescribe for most learning disabilities to the social framework for disability, as opposed to the medical framework. I think that’s important too.

@@peterhall6656 haha that’s hilarious!

Haha thanks! Not a lot of options in a hotel room! More KAN to come

@@alxlg it’s still really in the early days of this method, and we haven’t quite figured out all the limitations yet. It has great potential and I think there is a lot to explore with them.

@@aryansaxena4978 Go for it! Lots of opportunities. Two of my PhD students who graduate a couple years ago went to national labs. They make more than I did on my 9month contract after 6 years from my PhD.

As someone with a PhD in control+nonlinear dynamics and having taught at two universities soon after my PhD, PLEASE GO TO INDUSTRY. I swapped to being a contractor and it's infinitely better than academia. Pay is also better. Some people can find joy in academia, but I certainly was not one and not only that, many universities are going through brutal cuts to jobs and the like. Furthermore too much is expected of lecturers and I've experienced it first hand.

@@Vhaanzeit I understand, things were different 15-20 years back. Now it's pretty much mandatory to have at least two postdocs to even have a shot at getting a faculty position at a decent institute, at least that's the case here in India. Kudos on channelizing your energy to your needs mate!

@@JoelRosenfeld Glad to know that, and great work at your end for not luring them into academia, if someone quits their PhD programme for a good financial opportunity at my institute the professors think of them as failures and condescendingly refer to them as "well they went to industry".

@@aryansaxena4978In my opinion, a successful PhD is one that allows my student to do what they set out to. If they decide they no longer want to do it, because their priorities change, then that’s fine. I’m happy if they can find what makes them happy

Before you say June Huh he was lectured by a fields medalist

@@SCBA-if4wl If your point of comparison is a fields medalist, I don’t think there is any real hope of achieving that lol. But, don’t worry, there is plenty of room at the bottom.

@@JoelRosenfeld I'm 16 btw

@@SCBA-if4wl You have a couple years left in you then! I’m sure you’ll find your path, what is it you’d like to do?

@@JoelRosenfeld something geometric that can be applied to theoretical physics, like what Witten said for a properly developed theory of elliptic cohomology and/or moduli spaces. I am a mathematical platonist and don't really care about receiving a fields medal but I do want to contribute something significant enough that'd warrant such a prize, to use the fields medal as a metonym of great achievement

I think it's really fascinating, honestly. The Kolmogorov Arnold Representation is something that is entirely new to me, even though it's been around since the 1950s. KAN has renewed interest in this representation, and I think there is a lot to do with it from both the AI implementation side of things and from the theory side.,

@@carlgauss6383 inside a figure, just keep adding includegraphics commands. If you put one line break between the commands, they should stay on the same row (if possible) and adding a second line break puts them in a new row.

thanks

how can i make a box and in box i how place 3-4 images?

There are a lot of opportunities out there for someone with a math phd, and in unexpected places. One friend of mine who got a PhD in Algebraic Number Theory, after a little extra training, got a position working at Hello Fresh and then Spotify as a Data Scientist, and she had a 100% remote position. If you want to work in Cryptography, the National Security Agency is the biggest employer of PhD Mathematicians in the country (if not the world). Lots of places to look (and most pay better than a postdoc in academia!)

Yeah, it wasn't easy to get here, and I usually tell my PhD students that a PhD in Math or Engineering can open a lot of doors. But I also tell them that those doors don't have to be in academia. Lots of great research opportunities in National Labs and Industrial Labs too (and they pay better!)

I'll do my best to keep it up! Planning on just Fridays at 11 or so for now. At least, for the book writing. If I stream on a different day/time, I'll do something different so that people won't miss anything.

It’ll be a while before it’s ready for review. But I welcome any comments as we work through it!

@@jalepezo I do my best to explain things from the ground up! I hope my channel can help!

@@TriThom50 definitely! I plan on streaming every Friday. Today I didn’t announce it because it was my first go. Next Friday should be more structured.

Yeah, sorry. On my later videos I do.

​@@JoelRosenfeld Sorry I didn't mean to sound rude, thanks!

@@kylebierly1639 oh no worries. It comes up a lot. I’ve been working on getting better

@@jdsarfo610 lol thank you! I try hard!

@@jmguevarajordan I have seen it happen. It wasn’t pretty, and I have no idea why he didn’t just stop working with the graduate student. It was an extreme example. I would never write something like that. But even a mediocre letter can be damning.

@@JoelRosenfeld i saw again your video and thank You for your answer. The great mathematician Sinai, in his interview for receiving the Abel's prize, said that recommendation letter system in the US mathematics community is unfair. The case of your video is a clear example. You said that your friend sent too many applications which is a clear sign that his or her relation with the adviser was not good. Unfortunetaly, he or she was looking for first jobs after getting the phd and he or she needed the letters from the adviser who wrote extremely weak recommendation letters. I think that if you are doing your phd in math and your relation with the adviser is not working well then You should find another adviser or transfer to a new university.

@@jmguevarajordan I am studying in Turkey phd, I changed 3 universities just because of professors opinions. They believe that nobody can be better than their own students. Then how much you try no meaning it has. I know a lot of students that find postdoc in european universities by help of their professors just because of their nationality.

@@aysum5743 hi, your post does not say in which field you are doing your phd. In general, a phd in an académic field without a profesional profile will get you into the recommendation letters 'trap' and the network of contacts. This observation will be válid for math, physics, literature, history, geography, sociólogy, etc. If your undergraduate or college degree is in those field then the effect of advisers and professors opinions is even strongly. This has been known for long time and it is unfair. The solution or recommendation is getting a college degree with strong professional profile (for example engineering or computer science) before going into a phd program in an académic field with no professional profile. Good luck in your phd studies.

@@danielwilhelm yes even mid letters can be damning