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.
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 just had my applied math PhD interview. I'll hopefully hear back soon, but regardless, I've enjoyed watching your videos leading up to this point. I think you're doing a service for the subject.
@@charlie3k Lol not my way for the PhD, but I got a good job thanks to my math background, so there's that. Still thinking about applying again in the future!
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.
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 RUclips 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.
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 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...
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
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
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.
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 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.
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
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.
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 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.
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😅😅
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!
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?
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).
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 .
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.
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 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 ✌
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).
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.
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.
Ime, they should teach numerical analysis instead/before calc1. Miss freshman me with any course that introduces you to the epsilon delta definition of a limit in your first day of class, lol. (majored in physics)
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).
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.
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
recently discovered your channel and have been liking your videos bro. i have recently added a math degree to to the degree i am currently studying because math is amazing! :) like yourself, i am an american. however, i am also canadian, so i get to see the similarities and differences between the university programs of the two nations. the Canadian education system closely follows that of the Americans! i think where it diverges significantly is when it comes to the canadians/americans and the europeons. thats my rudimentary perception so far, anyways.
"I didn't like the material and the book we used was not good". Same experience, in my case Coding Theory was packaged with a brief intro into Algebra which kept it somewhat interesting. I'm not sure what it is about Coding Theory taught at the undergraduate level but I think primarily the textbooks doesn't motivate the same level of rigor that typically keeps a student engaged.
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!
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.
Hmmm this is really interesting... I wonder how hard these courses have to be since in my german university we have real analysis 1 and 2 and linear algebra 1 and 2 in the first year of the bachelors program and after that you can basically do abstract algebra, measure theory and probability theory in the third semester topology and functional analysis in the 4th and after that partial differential equations, Abstract Algebra 2(a lot of noether theory), Differential Geometry and even some algebraic topology in the 5th and 6th semester (along with your thesis). And they dont really skip over any important details as well, so i really wonder what exactly you do in the bachelors program? ^^.
Usually, 1st and 2nd semesters are Calculus I, II, respectively. In second year, students usually take Elementary Linear Algebra, Math Proofs/Discrete Maths, Calculus III, ODE, and Probability&Statistics classes. They are super easy compared to European university's math classes. Real math starts from junior year with proof-based courses like Real Analysis I, Abstract Algebra I and math electives. In my uni, Complex Analysis, Topology, Intermediate Probability Advanced Linear Algebra, Real Analysis II, Abstract Algebra II, PDE are considered as electives so we may or may not take those classes. Normally, people who want to pursuit masters degree in Maths take those pure math/advanced classes. Other students usually take Finance, Actuaries, Stats related courses in their junior/senior years because they are more useful than pure math classes.
Easy program. At U of Md, you have to do three areas instead of two. It is like analysis, algebra, and [topology or PDE or ODE] for the first year. You have it easy. Must be an easy school.
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.
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
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.
Is it just me or do Americans take soo few classes. Or are they very big? What's the deal exactly? I don't know how 5 classes is supposed to be enough for an entire year. We get like 15 classes each year, 7/8 each semester.
@blondymonk1535 I think it's similar to my country where most graduate students are working while there's no rush to finish the graduate program as soon as possible.
In my school as a computer science major if I take another 3 math classes I can achieve a minor in mathematics. Do you think this would be a good idea to go along with my CS major?
It’s a good idea to go along with any major because it’s makes you better at figuring things out and it’s an enriching way to challenge yourself shrug 🤷♂️
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
start with "understanding analysis" by stephen abbott, start right now! abbott is great for self-learning!
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
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
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
Looking for all the comments of foreigners saying “Americans are so behind, we studied this in 2nd grade/year/whatever”
I just had my applied math PhD interview. I'll hopefully hear back soon, but regardless, I've enjoyed watching your videos leading up to this point. I think you're doing a service for the subject.
Did you get it
Update?
How did it go? Don't leave us hanging!
@@charlie3k Lol not my way for the PhD, but I got a good job thanks to my math background, so there's that. Still thinking about applying again in the future!
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.
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 RUclips 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!
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 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...
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.
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
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.
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 :)
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.
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
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
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.
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 wish i had access to this in highschool.. i "studied" art in college.. so glad i dropped out, used RUclips to teach my self to code..
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.
I'm in my second year of math as an undergrad, and pursing graduate school. Fantastic to get a picture of this stuff now.
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
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.
TIME TO TALK ABOUT THE THESIS BABY!!!!!!!!!!!!!!!!!!!
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!
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
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?
Love your videos, the style is so welcoming and comfy
This guy has the most depressive struggling tone ever, keep going bro. Love your channel!
The title looked to me like this were a satire video for some reasons lol
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).
Appreciate you laying it out like that
Wow, what a difference
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 :)
Currently finishing russian PhD and doing thesis on hyperkähler manifolds. Thanks for your videos
I'd love to hear about your thesis!
Keep the videos up! Super interesting
im a junior in college getting my double degree in math and computer science, and u motivated me to consider a phd
I would also recommend Understanding Analysis by Abbott for undergraduate
You Are The best , keep making good content bro
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.
excellent
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 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!
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).
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 take measure theory in my undergraduate study in Indonesia and we used the same book (Wheeden). It's tough bro
Studying undergrad rn at a state school. Studying math with a concentration in actuarial science. Love your videos
I would get so lost and frustrated I would throw those books in the fireplace...
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.
Mad respect to you for your math grind, but I tapped out when numbers became letters and "wingding" characters...🤪
I thought Folland would be the standard textbook for graduate-level real analysis.
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.
i am passionate about calculus and want to learn more about it, do recommend some useful theoretical books . Love your content btw .
I will like to also hear about your thesis please...
i'm a 16 yo highschooler, I don't even know myself what am i doing here in this channel XD. but i like watching your videos though.
Ime, they should teach numerical analysis instead/before calc1. Miss freshman me with any course that introduces you to the epsilon delta definition of a limit in your first day of class, lol. (majored in physics)
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).
As a fellow struggling graduate math student I salute you sir
Let us do the Landau Conhecture Prolem
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.
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
Dude you read my mind I was just thinking about this lol
5,000 views surpassed. Let's see that thesis video, son!
Good thing I'm in CS.
Love your accent
Thank you lol
I'm doing my masters (applied mathematics) in Sweden. Currently thinking what I'll do after graduation
I would like to know about your thesis!
Are we getting the thesis video?
recently discovered your channel and have been liking your videos bro. i have recently added a math degree to to the degree i am currently studying because math is amazing! :)
like yourself, i am an american. however, i am also canadian, so i get to see the similarities and differences between the university programs of the two nations. the Canadian education system closely follows that of the Americans! i think where it diverges significantly is when it comes to the canadians/americans and the europeons. thats my rudimentary perception so far, anyways.
I’m from portugal 🇵🇹 and study in UK
"I didn't like the material and the book we used was not good".
Same experience, in my case Coding Theory was packaged with a brief intro into Algebra which kept it somewhat interesting. I'm not sure what it is about Coding Theory taught at the undergraduate level but I think primarily the textbooks doesn't motivate the same level of rigor that typically keeps a student engaged.
can you please also make a video on phd in applied maths coursework along with books
Can you please make a video about master vs undergrad?
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 don't know why I'm here. 😆
I'm just an accountant studying up for my cpa exams.
Freeman Dyson, who dd not have a PhD, thought it was a waste of time.
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
Hmmm this is really interesting... I wonder how hard these courses have to be since in my german university we have real analysis 1 and 2 and linear algebra 1 and 2 in the first year of the bachelors program and after that you can basically do abstract algebra, measure theory and probability theory in the third semester topology and functional analysis in the 4th and after that partial differential equations, Abstract Algebra 2(a lot of noether theory), Differential Geometry and even some algebraic topology in the 5th and 6th semester (along with your thesis). And they dont really skip over any important details as well, so i really wonder what exactly you do in the bachelors program? ^^.
Usually, 1st and 2nd semesters are Calculus I, II, respectively. In second year, students usually take Elementary Linear Algebra, Math Proofs/Discrete Maths, Calculus III, ODE, and Probability&Statistics classes. They are super easy compared to European university's math classes. Real math starts from junior year with proof-based courses like Real Analysis I, Abstract Algebra I and math electives. In my uni, Complex Analysis, Topology, Intermediate Probability Advanced Linear Algebra, Real Analysis II, Abstract Algebra II, PDE are considered as electives so we may or may not take those classes. Normally, people who want to pursuit masters degree in Maths take those pure math/advanced classes. Other students usually take Finance, Actuaries, Stats related courses in their junior/senior years because they are more useful than pure math classes.
American higher education is a baby version of the European system
Easy program. At U of Md, you have to do three areas instead of two. It is like analysis, algebra, and [topology or PDE or ODE] for the first year. You have it easy. Must be an easy school.
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.
bro how tf did you know i am highschool
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Do you get a stipend for the first 3-4 years of your PhD?
Meanwhile Europeans taking Real Variables in 1st year and Measure theory in 2nd year
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
a PhD is just a research-based masters on steroids
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.
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.
"Calculus with steroids" lol imagine Calculus made me stroke, those Real-s will make me catatonic 🤣🤣🤣🤣🤣
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.
Is it just me or do Americans take soo few classes. Or are they very big? What's the deal exactly?
I don't know how 5 classes is supposed to be enough for an entire year. We get like 15 classes each year, 7/8 each semester.
@blondymonk1535 I think it's similar to my country where most graduate students are working while there's no rush to finish the graduate program as soon as possible.
Do you post for science as well if can that would be better 👍.how about science?
In my school as a computer science major if I take another 3 math classes I can achieve a minor in mathematics. Do you think this would be a good idea to go along with my CS major?
It’s a good idea to go along with any major because it’s makes you better at figuring things out and it’s an enriching way to challenge yourself shrug 🤷♂️
Probably the best minor you can take alongside CS as long as you dont mind the challenge.
Yeah its worth. The rigor of the three extra classes shouldn't be too unbelievably hard.
Highschool student here 🖐️
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