im a high school student who has no idea whats going on when I watch your videos but its still nice to watch someone talk about something they are passionate about
I also didn't know crap about fractions an year ago but today I don't know crap about calculus so I did get better in maths but still have no idea what's going on and this is a natural feeling which should lead you to be passionate about your subject.
@@mastershooter64 I second this, I am just finishing Advanced Calculus (Real Analysis) this semester with this exact book…definitely not in high school tho😂
I graduated from the Harvard PhD program, so I can fill in the blanks. Harvard specializes in algebraic geometry, differential geometry, and number theory. (As a corollary, algebraic geometry exists.) Schools in this tier typically don't have a terminal masters program. You can get a masters degree either by completing one along with your bachelors degree (kind of like 4+1, except that it doesn't take an extra year, you just need to be extra good, and you actually get the masters degree), or by dropping out of the PhD halfway. Classes like real analysis, abstract algebra, and measure theory are undergraduate classes at Harvard and similar schools. A typical PhD student would complete coursework and start research within the first year, often within the first term. On the other hand, the average time to PhD graduation is not 4 years, it's more like 5 years. Four years is considered fast, five is average, and six is slow but still within the normal range. My own advisor graduated (from Harvard!) with the PhD in two years, which is ridiculous, but that's not normal.
If they start research in the first year, and finish after 5, how many research papers are they publishing? More than one, right? Do you present multiple papers for the dissertation, or does it take that long to do research?
@@bestgun9994 Surprisingly, the average is something like 1 paper. I had zero papers when I graduated. Of course some people have more papers than average, but math is so hard that it is perfectly normal to do well in your PhD without publishing any papers. We judge PhDs by reading their thesis and evaluating their actual work, not by counting administrative publishing metrics. A great example is Bhargav Bhatt: an outstanding mathematician who had a grand total of 1 paper when he got his PhD.
Thanks for this info. You ivy leaguers are on a different planet lol. I'm a junior who is going to take courses undergrad courses on real analysis and group theory in the fall. we all gotta start somewhere
Well, I must say I'm quite surprised. In France, the pure math curriculum appears to be more advanced. Measure theory and Lebesgue integration are taught as undergraduate courses, typically during the 2nd and 3rd years of study. Alongside these, there's an advanced probability course, an introductory functional analysis course in the 3rd year, and topics like general topology, along with the usual algebraic subjects, including an introduction to Galois theory. In the 1st year of the Master's program, there are typically four categories of courses: analysis, algebra, geometry and topology, and applied math. Some of the specific courses include advanced functional analysis, commutative algebra/homology, Riemannian geometry, algebraic topology, and stochastic calculus. Moving on to the 2nd year of the Master's program, students often have the opportunity to delve into niche topics that they select in consultation with an advisor to align with their academic interests. These niche courses might include subjects like K-theory and arithmetic geometry. When it comes to pursuing a PhD in mathematics in France, there are generally no formal classes. Instead, students dive right into preparing their thesis from the outset, occasionally attending seminars relevant to their research topics. On average, the duration of a PhD in mathematics in France is around 3 years. Greetings from France !
Would you say that the undergraduate education is more focused in France? I'm a U.S. student and I spent two years outside of the mathematics degree before transitioning into it later. I'm hitting Lebesgue and measures this upcoming semester.
@@AlexanderB41 which university if I may ask? I just checked TUM and some other universities and it seems like they don’t nearly cover as much Analsyis and Algebra as ETH
I'm doing my pure maths + theoretical physics undergraduate degrees in Australia, but I am looking at doing a Master's or PhD overseas, so this was quite helpful. Thanks.
I completed a masters degree called M. A. T. in math. It was for people who wanted to specialize in teaching math. Most of the math courses were harder versions of the undergraduate program. We would have harder problems or extra readings attached to the standard undergraduate course along with our education courses.
I did such a program in the US after coming from Europe. I found it very beneficial since I was well prepared due to my European training. I teach higher ed and for me, this M.A.T. program was a welcome program. It doesn't train you to become a math researcher but, if taken with interest, it DOES make you a more well rounded instructor...at least it did to me
Teaching Maths in high school? I don't know what Maths in university you need there to teach maths in high school. And I don't know why anyone doing masters would teach university maths.
@@howardlam6181people do it because they enjoy it. I plan to do it after I retire from my main job because it’s my passion, even thought I may be “cleverer” tha teacher material, that shouldn’t be a deterrent at all
I can't see topology, functional analysis, measure and Lebesgue integrals, abstract álgebra. As I know, these subjects are advanced mathematics, but in my country we have them in a four year bachelor program at universidade Eduardo Mondlane in Mozambique.
Different countries different programs. I have done a graduate program in two different countries. Certainly overlap but also stark differences. One example: One degree program had a course on Complex analysis and the other had an extensive course on non Euclidean geometry. You see, it also depends where most students graduating from an institution are going to work.
@@mathisnotforthefaintofhearttrue, we also have complex analysis, euclidean geometry and spatial geometry, differential geometry. It's a lot for bachelor degree students, but we survived😅😅
Real analysis and Abstract Algebra were both senior undergrad classes for me.. Virginia Tech 2012 .. my senior year I took Real analysis, Abstract algebra, topology, and adv DiffEq
I believe numerical analysis is similar to what I took which is numerical methods which is basically approximations through iterations to get the answer. Surprisingly useful for Thermodynamics.
I teach math in higher ed. Two of my former students went for a PhD program, one in math and one in physics. I have a Master's in math so these students have have gone way ahead of me. That's not too hard as I feel I am barely smarter than a door knob...:) Hats off for those PhD Math folks!
I’m studying a completely different degree in the UK. Really love the videos and discussions. Had an interest in math from high school I don’t have a math degree lol. 👍 Really insightful as I would like to do a PhD someday
Also found it really funny when you were saying how international students do really well on analysis and Americans do well on algebra since my UK high school teacher also found algebra a lot easier than analysis LMAO
@@jasonwong4619 it’s interesting to see how some countries emphasize analytic skills and others algebraic skills. Algebra skills we’re definitely emphasized more so when I was in high school.
i'm a geologist currently preparing for the admission exam of a masters degree in math and I'm definely keeping the intro to analysis textbook recommendations. thanks!
My daughter is a junior in HS. She’s already through AP call BC and on to linear algebra and calculus III at the local CC. I showed her your videos and she loves them.
I just got out of HS and heading to college in a few months, mathematics is my newfound passion and ofcourse it's only AFTER i graduate that I discover it...
dude, I'm an math enthusiast in the end of second year of bachelor's in math, and I'm heading to masters in math. I'm really enjoying your videos, keep it up, man
Senior high school student here 👋. I’m watching for your passion and calm demeanor on the subject, and to get more information on what working with higher level math entails. I have no experience in formal proof writing, but want to know what kinds of practical problems a graduate would work on in Real and Complex analysis. I’ll watch a video on The Riemann Hypothesis, but get lost in the understanding, and realize I have to have a least a tiny feel for what proving some of the statements are, so I’ll watch one of your videos covering a test you recently did. And then I’ll wonder, what is the day to day and overall outlook on a mathematicians life I could expect if I even remotely thought of doing a PhD. And that’s where you will come in on my endless UA-cam scrolling. Anyway, sorry for the long comment, but I just thought I’d share.
If you're curious about proofs, introductory books on proofs to my knowledge tend to be rather self-enclosed and introduce other fields of math's for interesting problems. A lot of people recommend the Book of Proof however I haven't read it, but I've found Jay Cummings book to be quite well written; it is pretty self-contained minus that some highschool equation manipulating helps in a spot or two. Also, each chapter introduces an open research problem and a field of math, pretty sure one of the chapters introduces a basic encryption algorithm and how to actually come up with it.
For my subject, numerical methods in geotechnical engineering, Neither Yale or Harvard have this subject. While stanford and MIT have two professor each for the whole of geotechnical engineering.
I'm always after things that challenge me and I have the same reasoning behind taking hard classes because It does annoy me that I don't know these things
I am a data analyst with a background in airline business development and finance. I am good at some operations research too, self studying Multi integer now. I am starting to look into math again and get used to at least some of the math stuff again…
For some time watching youtube videos like this, I couldn’t get why the authors mention “proof-based math”. Like there is another math :). I also tried to understand the difference between calculus and real analysis. All this confusion was from the different ways/programs to study math in universities (US vs Ukraine for me). All the math courses in Ukraine are proof based from scratch. Even in school, we had to learn proofs for classical Euclid geometry theorems, and algebra - everything we used was proven. Can’t say it is a good or bad thing since the university courses were extremely complex for most of the students, so maybe the US way (repeat topics with more details) makes more sense. I could forget something (graduated from the university in 2003-2008), but the structure of education looked like this: Undergraduate: 1 year Linear Algebra I, II Analytic geometry Math analysis I, II (proof based Calculus) Discrete math 2 year Number theory (one semester) Abstract Algebra (one semester) Math analysis III + Fourier series Complex analysis I ODE (2 semesters) Point Set Topology After 2 years you can choose the specialty (the list of choice depends on the university and define your additional courses) 3 year Complex analysis II Measure Theory Functional analysis (2 semesters) Probability theory and statistics (2 semesters) PDE (2 semesters) special courses (additional topics in analysis, something else) 4 year Classical Mechanics Numerical Analysis special courses (Fourier analysis, approximation theory, entire functions, something about differential operators) Masters special courses only (additional topics in complex analysis, geometry, etc) thesis PhD No course, but there was a huge list of results you should know to pass the exam in 1 or 2 years.
To me real analysis is going back through calculus 1-3 and proving everything you learned there. The emphasis is on proofs rather than the mechanics of calculus. Some schools call real analysis, advanced calculus.
got my BS in Physics. Wish I could have done grad school, would have loved to have gotten a Masters or PhD, so it's enjoyable to watch your videos even if pure math wasn't my subject.
dang，it's wheeden and zygmund… it took me a long time to really get used to the way this book leave proof of theorems as exercises. Kinda discouraging at first, but then I realize how all the proof gives is kind of short and elegant. Chapter 3 is a nightmare to me rn( still studying).
I took abstract algebra my senior year of undergrad. It was a 3 quarter course and there was only one section of it, so it always began in fall and ended in spring. it was my senior year, so I had to pass all 3 otherwise I'd have to wait all the way until the course swung around and the part I didnt pass came up again. I did pass, but at the beginning of the year there were 44 students. 12 people took the final at the end of the 3rd quarter, and only 9 of us passed.
is it me or is the math program more advanced in France in general? we get lebesgue, Lp spaces, complex analysis and galois theory during our bachelor's study
same with US, US is actually more rigorous than most European math bachelors courses. This guy just did a non math bachlors and got into math in grad school.
I'm an undergrad student in mathematics and statistics at india. Top institutions in india like iit, ISI, CMI ,UOH specialized in different fields of mathematics.
Aha - thank you! Super useful breakdown of how the years pan out across PhD. Also interesting to see the Masters course play out over two years, it's only one here in the UK but it does feel very crammed. I have 5 exams and 3 projects to complete over the next 2 month, just to give as example. How did you find getting onto your PhD course?
@@PhDVlog777 dude don’t say that I’m applying to a few ivy league schools for computer science and yeah I don’t believe I’m gonna get in but you for sure won’t get in if you don’t apply .
I'm not sure what Harvard's master/phD program is, but I do know that their highest math program is Math 55. Look it up. It's very interesting. Some see it as it being a proteaginous program.
I am a UK sixth form student and I am going to start Maths and CS at UoBirmingham (uk not alabama), in September. I enjoy hearing you speak on how university works in the US it's very interesting!
In Europe you NEED a masters before you can apply to a PhD program, so here there is no discussion of "which one is better". Usually a bachelors is three years, masters is then another two years. Where I'm at it's common to do a masters in something like engineering, without ever receiving a bachelors, it's just 5 years of grinding before you get a degree. Some places, in England for example, you can do that immediate masters in only 4 years. But yeah, anyone here who wants a phD needs to do a masters degree first.
european bachlor programs are way less rigorous though, a bachlors in math in the US is pretty much on par with a masters in math in Europe, same with engineering. And most people in US do have a masters before a phd too
@@wg4112It's the other way around. European bachelor's degrees are way more rigorous and comprehensive than American ones. More material, earlier introduction to rigor, harder coursework, higher level of abstraction, etc. At least that's how it is in Germany. 80% drop out rate.
So far one of the most important things is to have a project or research topic at graduate level, otherwise you will fall in the same situation that undergrad taking a bunch of courses but not getting the insight of graduate education which implies to start to think critically and develop your own research (although some people manage to do research at undergrad level).
Half way through my maths undergrad. Really enjoyed this video as my course is an integrated masters so I’m trying to think ahead for when it ends and I end up stranded with a masters and no idea what to do with it XD
Most impressed by the fact that you made it this far in a writing-heavy discipline and are still using a wooden pencil. After page 15 of a problem set I would kms with how often I'd have to be sharpening. Just go mechanical brotha
Hey, nice video! Did you happen to go to Wright State University for one of your degrees? It is where Steen Pedersen teaches. I attended it myself; he was by far my favorite professor! Funnily enough, we didn't follow his book in his courses (Real Variables I & II).
I am not a mathematician. Although I got an opportunity to study pure mathematics, I shifted towards communications engineering for my masters, but with an exception. I held on to pure mathematics all the time. I self studied Analysis and Algebra and now diving towards commutative algebra. Your videos showcases a parallel as what my life would have been if I had chosen pure mathematics research career, honestly I should say I missed a great opportunity. You're gifted as you could realize as what you like and you are able to do that. Not many in life gets this blessing. Good luck ✌
I took a Real Analysis course during my bachelor’s, and it was the only math class I ever dropped. That was the moment I realized I wasn’t cut out for it.
It is one of the reasons why I initially dropped math. It is a very different way of thinking from what I was use to. But wrestling with it for a while made me realize it’s not as bad as it’s seems
Hey buddy, a silly question ' why do you use pencil so much ? Is it for the convenience of erasing the mistakes or you simply like using pencil more than pen!
If I start a PhD at one university in USA and then I decided to get the master diploma, can I change from one Uni to another? Or get the master AND no More?
Interesting!!!. You said you didn't have an undergraduate degree in Mathematics. What was your first degree in? I'm also trying to get into a Mathematics master's program and I also don't have a first degree in Mathematics. Are there any tips you can give me that would help me get into such program?
I studied Environmental science. If you are in the stem fields then it is easier to transition to mathematics. Apply to many places and email faculty members that you want to work with. If you have a rapport with them then it is much easier to get accepted.
Your handwriting is really good, with that said, Have you ever considered becoming a Machine Learning Engineer, Data Scientist or NLP scientist with your mathematical background?
In the masters program, there were three sequences and you had to choose 2. If you leaned pure math, you would take algebra and real analysis. If you leaned applied, you would take real and numerical analysis. But at the end, you would have a masters in math, not pure or applied just math.
Well, here is my story: I am currently finishing my degree, I am in the last semester of Civil Engineering and throughout my degree I have really liked mathematics and I have done very well in the mathematics subjects that any engineer takes (differential calculus, integral calculus, differential equations, etc). Lately I have been very motivated to do a master's degree in applied mathematics or quantitative finance (It's funny because I'm about to be a Civil Engineer but I want really do different things with my life and I have a different prospect about what I would like to work in), I would like to be able to do it in mathematics but it scares me since I feel that I do not have the mathematical maturity and the foundations to do a master's degree in mathematics (because I come from an engineering degree), in fact, I am currently taking along two subjects from the master's degree in applied mathematics at my university: Advanced Calculus and Nonlinear Numerical Optimization (I'm not in the masters yet but I'm taking those courses as electives), in Advanced Calculus I think I'm doing very well, I was very afraid since it's a prove everything subject and we see topics like introduction to real analysis, basic topology, complex numbers, ... and I had never seen any subject of this type previously and full focused in writing proofs and that kind of stuff. My idea is to finish and get my Bachelor's degree and then do a master's degree in Applied Mathematics or Quant Finance, because I would like to be part of the front desk of a Hedge Fund or get a decent job related with math in Wall Street because I have becoming very passionate about trading in the recent years and I want to do it right, i.e., from a mathematical perspective and I know that these big companies and hedge funds like to hire people from master's degrees in mathematics or PhD's in math or physics. And I'm not wanting to do a masters because of that, I really want to do the master because I really like math and I enjoy it, as you said: "You choose the mathematics life because you just enjoy the subject; you have to enjoy the subject or else it's just going to eat you alive" and I have felt that while studying for the Advanced Calculus subject because I have spent a lot of hours trying to understand things and adapting to thinking on a different way that I was used to. I'm planning to start admission process in these universities: * Stony Brook University * Baruch College * New York University * Rutgers University Hopefully I get admitted to one of that and start my program in QF Finance/Applied Math. If anyone can give me some advice on which master/university is more appropriate for what I want to do with my life I will really appreciate that! Sorry for the very long comment but I just feel that there are others that could be in the same situation than me.
Just wondering is 9,2 a good undergraduate grade or should I try to do better I am a mathematics undergraduate.I am aiming for a masters at first and then a phd
It's too bad there's no money in mathematics for all the years and hard work it takes. In pure mathematics about all you can do is teach and research - there's no money there. In applied mathematics you can do a little better financially, but you will have to learn another discipline to go along with it to be anywhere near successful in it. Being a quant on Wall Street might be an exception money wise.
There may not be too much money in math as you assert but for me, math has made me avoid financial blunders that so many other people are making when it comes to finance.
The entire world runs on math. Most engineers who design, make use of cad and simulation software. And who do you think programmed these initially? Physicists and mathematicians. Of course in current day, simulation software is mostly used from a handful of big companies (like synopsis), and all the mathematical techniques are stripped away from the user. Most engineers just design something with cad and let the computer do the thinking when they apply external forces, or moments on their designs. So it seems to the outsider mathematics is useless but that is far from the case.
Hello from Malaysia. In Asia, we have a different structure for PhD programme, where the duration can range from 2 years to 6 years, and the requirements are 1 thesis+2 WoS journal articles + viva. In Malaysia particularly, speaking of my own experience, we do not attend or register any Maths courses during a PhD (Mathematics) programme, except a course called research methodology. Instead, we spend the first year to write our research proposal, and present a proposal defense at the end of second semester. Then, we start working on the proposal, producing papers and present a candidature defense at the end of fourth or fifth semester. Eventually, we submit the thesis at the end of sixth sem and wait for viva. Yes, we do not have a formal lecture to learn these courses as mentioned in the video, but instead we have to study on our own or we might had studied that in our undergraduate courses. I personally think that this is not a good way to complete a PhD as our foundations might be flawed, and we do not aware about it. Hence, it is easy to see that a Statistics PhD student in Malaysia might not excel enough in Real Analysis. This could be a problem as the mind and vision of that PhD student could be restricted. Maybe there is a good way to solve his research problem, but the lack of good foundation in certain field might hinder him from achieving something novel.
im a high school student who has no idea whats going on when I watch your videos but its still nice to watch someone talk about something they are passionate about
I also didn't know crap about fractions an year ago but today I don't know crap about calculus so I did get better in maths but still have no idea what's going on and this is a natural feeling which should lead you to be passionate about your subject.
@@mastershooter64 I second this, I am just finishing Advanced Calculus (Real Analysis) this semester with this exact book…definitely not in high school tho😂
@@mastershooter64 I'm a Highschooler too, i read this book and i also vouch for it.
move beyond IIT JEE Hype if you want to get into research
literally same here I'm high school student too😂
I graduated from the Harvard PhD program, so I can fill in the blanks. Harvard specializes in algebraic geometry, differential geometry, and number theory. (As a corollary, algebraic geometry exists.) Schools in this tier typically don't have a terminal masters program. You can get a masters degree either by completing one along with your bachelors degree (kind of like 4+1, except that it doesn't take an extra year, you just need to be extra good, and you actually get the masters degree), or by dropping out of the PhD halfway. Classes like real analysis, abstract algebra, and measure theory are undergraduate classes at Harvard and similar schools. A typical PhD student would complete coursework and start research within the first year, often within the first term. On the other hand, the average time to PhD graduation is not 4 years, it's more like 5 years. Four years is considered fast, five is average, and six is slow but still within the normal range. My own advisor graduated (from Harvard!) with the PhD in two years, which is ridiculous, but that's not normal.
If they start research in the first year, and finish after 5, how many research papers are they publishing? More than one, right? Do you present multiple papers for the dissertation, or does it take that long to do research?
@@bestgun9994 Surprisingly, the average is something like 1 paper. I had zero papers when I graduated. Of course some people have more papers than average, but math is so hard that it is perfectly normal to do well in your PhD without publishing any papers. We judge PhDs by reading their thesis and evaluating their actual work, not by counting administrative publishing metrics. A great example is Bhargav Bhatt: an outstanding mathematician who had a grand total of 1 paper when he got his PhD.
Thanks for this info. You ivy leaguers are on a different planet lol. I'm a junior who is going to take courses undergrad courses on real analysis and group theory in the fall. we all gotta start somewhere
holy crap! you have a big brain...
@David Jao Please tell us some anecdotes on what Noam Elkies is like :)
Well, I must say I'm quite surprised. In France, the pure math curriculum appears to be more advanced. Measure theory and Lebesgue integration are taught as undergraduate courses, typically during the 2nd and 3rd years of study. Alongside these, there's an advanced probability course, an introductory functional analysis course in the 3rd year, and topics like general topology, along with the usual algebraic subjects, including an introduction to Galois theory.
In the 1st year of the Master's program, there are typically four categories of courses: analysis, algebra, geometry and topology, and applied math. Some of the specific courses include advanced functional analysis, commutative algebra/homology, Riemannian geometry, algebraic topology, and stochastic calculus.
Moving on to the 2nd year of the Master's program, students often have the opportunity to delve into niche topics that they select in consultation with an advisor to align with their academic interests. These niche courses might include subjects like K-theory and arithmetic geometry.
When it comes to pursuing a PhD in mathematics in France, there are generally no formal classes. Instead, students dive right into preparing their thesis from the outset, occasionally attending seminars relevant to their research topics. On average, the duration of a PhD in mathematics in France is around 3 years.
Greetings from France !
Would you say that the undergraduate education is more focused in France?
I'm a U.S. student and I spent two years outside of the mathematics degree before transitioning into it later. I'm hitting Lebesgue and measures this upcoming semester.
Exactly the same in Italy in every aspect mentioned, maybe in all the EU is like that.
@@vortanoise.2625same in germany too.
American universities seem kinda trash ngl.
MIT?
@@ClintStone-t9m Hahahahahahaaaa
Have no idea what real analysis is, heard it from this guy. Just a normal high school student enjoying these videos
You make fine estimates on functions and study things like differentiation and integration of functions and measures
its basically just rigorous calculus.
lol be ready to get wrecked
Looking for all the comments of foreigners saying “Americans are so behind, we studied this in 2nd grade/year/whatever”
I mean it’s true, at ETH Zurich you learn Analysis 4 (Fourier and HIlbert theory) during the latter half of your SECOND year of your undergraduate.
@@igoranindito4727 same in Germany
@@AlexanderB41 which university if I may ask? I just checked TUM and some other universities and it seems like they don’t nearly cover as much Analsyis and Algebra as ETH
U mad?
I'm doing my pure maths + theoretical physics undergraduate degrees in Australia, but I am looking at doing a Master's or PhD overseas, so this was quite helpful. Thanks.
I completed a masters degree called M. A. T. in math. It was for people who wanted to specialize in teaching math. Most of the math courses were harder versions of the undergraduate program. We would have harder problems or extra readings attached to the standard undergraduate course along with our education courses.
I did such a program in the US after coming from Europe. I found it very beneficial since I was well prepared due to my European training. I teach higher ed and for me, this M.A.T. program was a welcome program. It doesn't train you to become a math researcher but, if taken with interest, it DOES make you a more well rounded instructor...at least it did to me
Teaching Maths in high school? I don't know what Maths in university you need there to teach maths in high school. And I don't know why anyone doing masters would teach university maths.
@@howardlam6181people do it because they enjoy it. I plan to do it after I retire from my main job because it’s my passion, even thought I may be “cleverer” tha teacher material, that shouldn’t be a deterrent at all
I can't see topology, functional analysis, measure and Lebesgue integrals, abstract álgebra. As I know, these subjects are advanced mathematics, but in my country we have them in a four year bachelor program at universidade Eduardo Mondlane in Mozambique.
Different countries different programs. I have done a graduate program in two different countries. Certainly overlap but also stark differences. One example: One degree program had a course on Complex analysis and the other had an extensive course on non Euclidean geometry. You see, it also depends where most students graduating from an institution are going to work.
@@mathisnotforthefaintofhearttrue, we also have complex analysis, euclidean geometry and spatial geometry, differential geometry. It's a lot for bachelor degree students, but we survived😅😅
@@mozturco4931 That sounds indeed like a very heavy undergrad program
Topology, Functional Analysis and Measure theory are generally optional for undergrads, depending on what their interests and needs are.
@@mathisnotforthefaintofheart i think complex analysis and non-euclidean geom is interchangeable in scope. They have a LOT of overlap
Surprised to see no topology class, given it’s important for analysis
Point set topology and metric space theory are important for analysis but its kinda basic material that could be covered in the intro courses
Real analysis and Abstract Algebra were both senior undergrad classes for me.. Virginia Tech 2012 .. my senior year I took Real analysis, Abstract algebra, topology, and adv DiffEq
I taught honors calculus at VPI back in the mid 60’s.
I believe numerical analysis is similar to what I took which is numerical methods which is basically approximations through iterations to get the answer. Surprisingly useful for Thermodynamics.
I am a high school student from India and I have a high interest in mathematics right now I’m studying metric space 😊
I teach math in higher ed. Two of my former students went for a PhD program, one in math and one in physics. I have a Master's in math so these students have have gone way ahead of me. That's not too hard as I feel I am barely smarter than a door knob...:) Hats off for those PhD Math folks!
I’m studying a completely different degree in the UK. Really love the videos and discussions. Had an interest in math from high school I don’t have a math degree lol. 👍 Really insightful as I would like to do a PhD someday
Also found it really funny when you were saying how international students do really well on analysis and Americans do well on algebra since my UK high school teacher also found algebra a lot easier than analysis LMAO
@@jasonwong4619 it’s interesting to see how some countries emphasize analytic skills and others algebraic skills. Algebra skills we’re definitely emphasized more so when I was in high school.
i'm a geologist currently preparing for the admission exam of a masters degree in math and I'm definely keeping the intro to analysis textbook recommendations. thanks!
I almost did a masters with a geologist. Very good :)
In my second year of a NZ BSc in Maths which is 3 years, so not quite the same system as the US, still very interesting to watch your videos.
My daughter is a junior in HS. She’s already through AP call BC and on to linear algebra and calculus III at the local CC. I showed her your videos and she loves them.
I'm glad she likes them! I hope she continues to love math and science, and succeeds in her studies :)
I just got out of HS and heading to college in a few months, mathematics is my newfound passion and ofcourse it's only AFTER i graduate that I discover it...
Did my mathematics masters in the UK. My program was very different to yours and it's interesting to hear about how things are done around the world
I am a high school student and very passionate about math. I try to learn it myself so that is why I watch your videos
dude, I'm an math enthusiast in the end of second year of bachelor's in math, and I'm heading to masters in math. I'm really enjoying your videos, keep it up, man
Senior high school student here 👋. I’m watching for your passion and calm demeanor on the subject, and to get more information on what working with higher level math entails. I have no experience in formal proof writing, but want to know what kinds of practical problems a graduate would work on in Real and Complex analysis. I’ll watch a video on The Riemann Hypothesis, but get lost in the understanding, and realize I have to have a least a tiny feel for what proving some of the statements are, so I’ll watch one of your videos covering a test you recently did. And then I’ll wonder, what is the day to day and overall outlook on a mathematicians life I could expect if I even remotely thought of doing a PhD. And that’s where you will come in on my endless UA-cam scrolling. Anyway, sorry for the long comment, but I just thought I’d share.
If you're curious about proofs, introductory books on proofs to my knowledge tend to be rather self-enclosed and introduce other fields of math's for interesting problems.
A lot of people recommend the Book of Proof however I haven't read it, but I've found Jay Cummings book to be quite well written; it is pretty self-contained minus that some highschool equation manipulating helps in a spot or two.
Also, each chapter introduces an open research problem and a field of math, pretty sure one of the chapters introduces a basic encryption algorithm and how to actually come up with it.
@@soupy5890 Yes! Ive been hearing about Jay Cummings' material more and more recently, I'll definitely keep it on my radar. Thank you!
For my subject, numerical methods in geotechnical engineering, Neither Yale or Harvard have this subject. While stanford and MIT have two professor each for the whole of geotechnical engineering.
I would also recommend Understanding Analysis by Abbott for undergraduate
i wish i had access to this in highschool.. i "studied" art in college.. so glad i dropped out, used UA-cam to teach my self to code..
TIME TO TALK ABOUT THE THESIS BABY!!!!!!!!!!!!!!!!!!!
im a junior in college getting my double degree in math and computer science, and u motivated me to consider a phd
Wow, what a difference
I'm always after things that challenge me and I have the same reasoning behind taking hard classes because It does annoy me that I don't know these things
I'm in my second year of math as an undergrad, and pursing graduate school. Fantastic to get a picture of this stuff now.
Love your videos, the style is so welcoming and comfy
Appreciate you laying it out like that
I am a data analyst with a background in airline business development and finance. I am good at some operations research too, self studying Multi integer now.
I am starting to look into math again and get used to at least some of the math stuff again…
I will like to also hear about your thesis please...
I take measure theory in my undergraduate study in Indonesia and we used the same book (Wheeden). It's tough bro
Univ mana bang sama s1 nya apa
@romaboo9772 UI, Ambil matematika
Are we getting the thesis video?
For some time watching youtube videos like this, I couldn’t get why the authors mention “proof-based math”. Like there is another math :). I also tried to understand the difference between calculus and real analysis. All this confusion was from the different ways/programs to study math in universities (US vs Ukraine for me). All the math courses in Ukraine are proof based from scratch. Even in school, we had to learn proofs for classical Euclid geometry theorems, and algebra - everything we used was proven.
Can’t say it is a good or bad thing since the university courses were extremely complex for most of the students, so maybe the US way (repeat topics with more details) makes more sense.
I could forget something (graduated from the university in 2003-2008), but the structure of education looked like this:
Undergraduate:
1 year
Linear Algebra I, II
Analytic geometry
Math analysis I, II (proof based Calculus)
Discrete math
2 year
Number theory (one semester)
Abstract Algebra (one semester)
Math analysis III + Fourier series
Complex analysis I
ODE (2 semesters)
Point Set Topology
After 2 years you can choose the specialty (the list of choice depends on the university and define your additional courses)
3 year
Complex analysis II
Measure Theory
Functional analysis (2 semesters)
Probability theory and statistics (2 semesters)
PDE (2 semesters)
special courses (additional topics in analysis, something else)
4 year
Classical Mechanics
Numerical Analysis
special courses (Fourier analysis, approximation theory, entire functions, something about differential operators)
Masters
special courses only (additional topics in complex analysis, geometry, etc)
thesis
PhD
No course, but there was a huge list of results you should know to pass the exam in 1 or 2 years.
To me real analysis is going back through calculus 1-3 and proving everything you learned there. The emphasis is on proofs rather than the mechanics of calculus. Some schools call real analysis, advanced calculus.
@@Lokie-cd2hw The American artificial structure called calculus is not mathematics.
got my BS in Physics. Wish I could have done grad school, would have loved to have gotten a Masters or PhD, so it's enjoyable to watch your videos even if pure math wasn't my subject.
Currently finishing russian PhD and doing thesis on hyperkähler manifolds. Thanks for your videos
dang，it's wheeden and zygmund… it took me a long time to really get used to the way this book leave proof of theorems as exercises. Kinda discouraging at first, but then I realize how all the proof gives is kind of short and elegant. Chapter 3 is a nightmare to me rn( still studying).
I'd love to hear about your thesis!
1:35 Can you get to the point?
what if you want a job though?
Keep the videos up! Super interesting
I took abstract algebra my senior year of undergrad. It was a 3 quarter course and there was only one section of it, so it always began in fall and ended in spring. it was my senior year, so I had to pass all 3 otherwise I'd have to wait all the way until the course swung around and the part I didnt pass came up again.
I did pass, but at the beginning of the year there were 44 students. 12 people took the final at the end of the 3rd quarter, and only 9 of us passed.
is it me or is the math program more advanced in France in general? we get lebesgue, Lp spaces, complex analysis and galois theory during our bachelor's study
Also in Italy is like you said
Germany too
Same holds in Belgium
same with US, US is actually more rigorous than most European math bachelors courses. This guy just did a non math bachlors and got into math in grad school.
@@wg4112 he did enviro sci if im not mistaken
I thought Folland would be the standard textbook for graduate-level real analysis.
That book is good, I wonder are there any book that define integral as outer measure of graph of the function like that
I'm doing my masters (applied mathematics) in Sweden. Currently thinking what I'll do after graduation
You Are The best , keep making good content bro
Maths at Cambridge is definitely higher level than that at Oxford
excellent
I'm an undergrad student in mathematics and statistics at india. Top institutions in india like iit, ISI, CMI ,UOH specialized in different fields of mathematics.
Aha - thank you! Super useful breakdown of how the years pan out across PhD. Also interesting to see the Masters course play out over two years, it's only one here in the UK but it does feel very crammed. I have 5 exams and 3 projects to complete over the next 2 month, just to give as example. How did you find getting onto your PhD course?
I just applied to schools within a few states of me. I applied to a bunch because I knew I was unlikely to go to the one I wanted.
@@PhDVlog777 dude don’t say that
I’m applying to a few ivy league schools for computer science and yeah I don’t believe I’m gonna get in but you for sure won’t get in if you don’t apply .
@@PhDVlog777 I got a funded PhD studentship in Pure Mathematics this week! All worked out in the end :)
I'm not sure what Harvard's master/phD program is, but I do know that their highest math program is Math 55. Look it up. It's very interesting. Some see it as it being a proteaginous program.
The title looked to me like this were a satire video for some reasons lol
I am a UK sixth form student and I am going to start Maths and CS at UoBirmingham (uk not alabama), in September. I enjoy hearing you speak on how university works in the US it's very interesting!
Good luck!
Love your accent
Thank you lol
In Europe you NEED a masters before you can apply to a PhD program, so here there is no discussion of "which one is better". Usually a bachelors is three years, masters is then another two years. Where I'm at it's common to do a masters in something like engineering, without ever receiving a bachelors, it's just 5 years of grinding before you get a degree. Some places, in England for example, you can do that immediate masters in only 4 years. But yeah, anyone here who wants a phD needs to do a masters degree first.
european bachlor programs are way less rigorous though, a bachlors in math in the US is pretty much on par with a masters in math in Europe, same with engineering. And most people in US do have a masters before a phd too
@@wg4112It's the other way around. European bachelor's degrees are way more rigorous and comprehensive than American ones. More material, earlier introduction to rigor, harder coursework, higher level of abstraction, etc. At least that's how it is in Germany. 80% drop out rate.
can you please also make a video on phd in applied maths coursework along with books
So far one of the most important things is to have a project or research topic at graduate level, otherwise you will fall in the same situation that undergrad taking a bunch of courses but not getting the insight of graduate education which implies to start to think critically and develop your own research (although some people manage to do research at undergrad level).
Where did you go to school? These subject seem very non-advanced for a masters or the first year or two of PHD.
Half way through my maths undergrad. Really enjoyed this video as my course is an integrated masters so I’m trying to think ahead for when it ends and I end up stranded with a masters and no idea what to do with it XD
Studying undergrad rn at a state school. Studying math with a concentration in actuarial science. Love your videos
How are you doing one year on? I got a masters in actuarial science 20ish years ago. It opened up loads of opportunities.
When, if ever, does a math Phd or Masters study philosophy of math?
Good thing I'm in CS.
Do you get a stipend for the first 3-4 years of your PhD?
i am passionate about calculus and want to learn more about it, do recommend some useful theoretical books . Love your content btw .
Most impressed by the fact that you made it this far in a writing-heavy discipline and are still using a wooden pencil. After page 15 of a problem set I would kms with how often I'd have to be sharpening. Just go mechanical brotha
Nah all the smartest ppl I know use wooden pencil. Non of that iPad Apple Pencil bullshit
I would get so lost and frustrated I would throw those books in the fireplace...
Hey, nice video! Did you happen to go to Wright State University for one of your degrees? It is where Steen Pedersen teaches. I attended it myself; he was by far my favorite professor! Funnily enough, we didn't follow his book in his courses (Real Variables I & II).
Let us do the Landau Conhecture Prolem
That book’s cover looks like a cave drawing
I am not a mathematician. Although I got an opportunity to study pure mathematics, I shifted towards communications engineering for my masters, but with an exception. I held on to pure mathematics all the time. I self studied Analysis and Algebra and now diving towards commutative algebra. Your videos showcases a parallel as what my life would have been if I had chosen pure mathematics research career, honestly I should say I missed a great opportunity. You're gifted as you could realize as what you like and you are able to do that. Not many in life gets this blessing. Good luck ✌
Thank you very much!
Can you please make a video about master vs undergrad?
Could we get a video on your favorite, or what you consider the most elegant, real/complex analysis theorem proof?
My favorite proof from real is the Borel Cantelli lemma and from complex I would say the Cauchy-Goursat theorem.
I took a Real Analysis course during my bachelor’s, and it was the only math class I ever dropped. That was the moment I realized I wasn’t cut out for it.
It is one of the reasons why I initially dropped math. It is a very different way of thinking from what I was use to. But wrestling with it for a while made me realize it’s not as bad as it’s seems
Hey buddy, a silly question ' why do you use pencil so much ? Is it for the convenience of erasing the mistakes or you simply like using pencil more than pen!
I don’t mind either, but pen is a little easier to read and I don’t have to keep sharpening. Which ever is available I use.
I would like to know about your thesis!
If I start a PhD at one university in USA and then I decided to get the master diploma, can I change from one Uni to another? Or get the master AND no More?
As a fellow struggling graduate math student I salute you sir
This guy has the most depressive struggling tone ever, keep going bro. Love your channel!
What do you mean meaningless if it's too fast?
you ever take anything category theory or type theory related?
Unfortunately I have not. I’ve read a little bit in cat theory in my algebra book but that is all.
Do you regret not doing more computational methods?
Interesting!!!. You said you didn't have an undergraduate degree in Mathematics. What was your first degree in? I'm also trying to get into a Mathematics master's program and I also don't have a first degree in Mathematics. Are there any tips you can give me that would help me get into such program?
I studied Environmental science. If you are in the stem fields then it is easier to transition to mathematics. Apply to many places and email faculty members that you want to work with. If you have a rapport with them then it is much easier to get accepted.
I’m from portugal 🇵🇹 and study in UK
So you didn’t study mathematics as an undergrad and still took differential geometry, advanced linear algebra?… what did you study as an undergrad?
Environmental science
In what did you major in undergrad?
Your handwriting is really good, with that said, Have you ever considered becoming a Machine Learning Engineer, Data Scientist or NLP scientist with your mathematical background?
Talked me out if it 😂
What's the subject of Mathematics MSc you indicated in here, in other words these courses are Eg Master of Mathematics in Real Analysis ...?
In the masters program, there were three sequences and you had to choose 2. If you leaned pure math, you would take algebra and real analysis. If you leaned applied, you would take real and numerical analysis. But at the end, you would have a masters in math, not pure or applied just math.
What school do you go to?
Well, here is my story: I am currently finishing my degree, I am in the last semester of Civil Engineering and throughout my degree I have really liked mathematics and I have done very well in the mathematics subjects that any engineer takes (differential calculus, integral calculus, differential equations, etc). Lately I have been very motivated to do a master's degree in applied mathematics or quantitative finance (It's funny because I'm about to be a Civil Engineer but I want really do different things with my life and I have a different prospect about what I would like to work in), I would like to be able to do it in mathematics but it scares me since I feel that I do not have the mathematical maturity and the foundations to do a master's degree in mathematics (because I come from an engineering degree), in fact, I am currently taking along two subjects from the master's degree in applied mathematics at my university: Advanced Calculus and Nonlinear Numerical Optimization (I'm not in the masters yet but I'm taking those courses as electives), in Advanced Calculus I think I'm doing very well, I was very afraid since it's a prove everything subject and we see topics like introduction to real analysis, basic topology, complex numbers, ... and I had never seen any subject of this type previously and full focused in writing proofs and that kind of stuff.
My idea is to finish and get my Bachelor's degree and then do a master's degree in Applied Mathematics or Quant Finance, because I would like to be part of the front desk of a Hedge Fund or get a decent job related with math in Wall Street because I have becoming very passionate about trading in the recent years and I want to do it right, i.e., from a mathematical perspective and I know that these big companies and hedge funds like to hire people from master's degrees in mathematics or PhD's in math or physics. And I'm not wanting to do a masters because of that, I really want to do the master because I really like math and I enjoy it, as you said: "You choose the mathematics life because you just enjoy the subject; you have to enjoy the subject or else it's just going to eat you alive" and I have felt that while studying for the Advanced Calculus subject because I have spent a lot of hours trying to understand things and adapting to thinking on a different way that I was used to.
I'm planning to start admission process in these universities:
* Stony Brook University
* Baruch College
* New York University
* Rutgers University
Hopefully I get admitted to one of that and start my program in QF Finance/Applied Math. If anyone can give me some advice on which master/university is more appropriate for what I want to do with my life I will really appreciate that! Sorry for the very long comment but I just feel that there are others that could be in the same situation than me.
maybe if you wan't to relate back to civil, you could learn about the finite element method math and do something in the structural engineering realm
Haven't bought a single book in my 5 years of college.
Just wondering is 9,2 a good undergraduate grade or should I try to do better I am a mathematics undergraduate.I am aiming for a masters at first and then a phd
i am just doing understanding analysis by Abbot. any advice ? i am not in college, i just work in some library
Going through calculus and hammocks proof book would be a good start imo. Battle and Sherbert is a good supplementary book to help.
Mad respect to you for your math grind, but I tapped out when numbers became letters and "wingding" characters...🤪
It's too bad there's no money in mathematics for all the years and hard work it takes. In pure mathematics about all you can do is teach and research - there's no money there. In applied mathematics you can do a little better financially, but you will have to learn another discipline to go along with it to be anywhere near successful in it. Being a quant on Wall Street might be an exception money wise.
There may not be too much money in math as you assert but for me, math has made me avoid financial blunders that so many other people are making when it comes to finance.
The entire world runs on math. Most engineers who design, make use of cad and simulation software. And who do you think programmed these initially? Physicists and mathematicians. Of course in current day, simulation software is mostly used from a handful of big companies (like synopsis), and all the mathematical techniques are stripped away from the user. Most engineers just design something with cad and let the computer do the thinking when they apply external forces, or moments on their designs. So it seems to the outsider mathematics is useless but that is far from the case.
Hello from Malaysia. In Asia, we have a different structure for PhD programme, where the duration can range from 2 years to 6 years, and the requirements are 1 thesis+2 WoS journal articles + viva. In Malaysia particularly, speaking of my own experience, we do not attend or register any Maths courses during a PhD (Mathematics) programme, except a course called research methodology. Instead, we spend the first year to write our research proposal, and present a proposal defense at the end of second semester. Then, we start working on the proposal, producing papers and present a candidature defense at the end of fourth or fifth semester. Eventually, we submit the thesis at the end of sixth sem and wait for viva.
Yes, we do not have a formal lecture to learn these courses as mentioned in the video, but instead we have to study on our own or we might had studied that in our undergraduate courses. I personally think that this is not a good way to complete a PhD as our foundations might be flawed, and we do not aware about it. Hence, it is easy to see that a Statistics PhD student in Malaysia might not excel enough in Real Analysis. This could be a problem as the mind and vision of that PhD student could be restricted. Maybe there is a good way to solve his research problem, but the lack of good foundation in certain field might hinder him from achieving something novel.