College Algebra Pre- Calculus Calculus 1,2,3 Differential Equations Probability and stats Discrete I,II Linear Algebra Real Analysis Modern Algebra Geometry Topology Optional( complex analysis, Number theory, Time series,
I would say Topology is a must have! It helped me a lot to understand many fundamental concepts and their connections in between. Stuff like Optimazation or Numerics is cool too (if you are into aplied mathematics)
Would you say I should take Euclidean Geometry before Topology? I didn't have a good background in Euclidean, and I don't have too much space for superfluous classes since I'm double majoring.
@@bottlecap6169 I kind of think euclidean geometry is more of a math history class. You get to cover axiomatic systems and talk about ideas about the 5th postulate. On the other hand, when I took this course we never talked about manifolds or topology. In my opinion, topology is the modern language in which we talk about shapes. There is some distinction between geometry and topology based on some very complicated explanation about whether classification spaces are discrete or continuous. But in general it is important to realize that even the subjects we call geometry like riemannian geometry or smooth manifolds are still based in the language of topology, that is when we discuss geometry we are discussing local aspect of topological spaces. There is no "geometrical space" but there is a formal "topological space". So this is all to say that I think topology is a better use of your time for modern mathematics. You should check out the text by janich on topology. I haven't read it but there is a nice exert at the beginning where it describes topology as a nice language to deal with counterintuitive aspects of shapes. So I highly highly recommend topology instead of that class on euclidean geometry. That euclidean geometry is an old language. Topology is a modern language that will even help you with the old stuff. Topology is the modern language of shapes.
@@bottlecap6169 nope, General Topology can be studied without any pre reqs besides a proofs course. With a proofs course that you've done well, you can do Real Analysis, Complex Analysis, Point Set (General) Topology, and Advanced(Abstract) Linear Algebra, and Abstract Algebra. I say proofs, but I also mean logic etc which most proof books have already.
All channels like math scorcer are good channels I love channels that actively encourage learning through book recommendations, practice problems, or informative videos of such. You have earned my SUB, I hope to see more amazing videos in future. Would love to see videos doing certain problems if you have time to make such videos like solving specific integrals or teaching basic theorems or stuff.
Thanks for the list! I'm considering going back to school to get a math degree and currently self studying but practically starting over from scratch because it's been so long. Been at it for a year and about to start calc I. Seeing the subjects planned out like this is very motivating.
@@אהלןסהלן I live in Azerbaijan. Not the country where there is a rule of law. Corruption and there is no free court system. The competition isn't fair, because you just do not need to be talanted to earn money, you need good patron. Companies actually do not need lawyers, they will operate without lawyers well, too. I felt like I have useless job. Besides, dealing with people with lots of personal problems is quite depressive for me :) I gathered money and want to switch my career to applied math and computer science. I loved and love problem solving a lot. It was my biggest mistake in my life to choose law degree despite my school teachers pushing me to get math degree back then when I was 17 years old. My father also is retired math professor who taught in local and foreign universities. When I do math I find peace inside me, it helps me to focus. I just dont want to die as a lawyer. :) I love guitar and math.
I love this list, although it hurts not see any topology courses🥲 I personally had a much less consistent list of courses, since I picked most of my courses based on how much did the course sound interesting to me, and that way I found the topics that interested me the most, and I think I would recommend this way to every new student, of course after he took the basic courses that are necessary in basically every field in mathematics. Great video!
Point set topology is taught with an analysis course. Analysis is essentially all topology (even starting from epsilon delta proofs; it's just convergence in the standard metric topology on R)
@@ffc1a28c7 yeah we had a segment about metric spaces in our second analysis course. But if you don't take topology you're missing the really juicy parts like the construction of quotient spaces and the fundamental groups:)
Great video! Here's my version for the first 3 years of undergrad: sem 1) analysis in R(sequences, derivatives and all that but done properly as a math course, not like calculus), algebra 1 (just the basics about groups, rings etc), topology-1 (up to Van-Kampen theorem). sem 2) Lebesgue measure and integral, linear algebra (Linear algebra done right fits), Algebra-2 (Group Theory). sem 3) ODE-1 (also done as a math course, see V. I Arnold's textbook), Differential Geometry-1 (Just basic stuff about manifolds and all that, probably John Lee's "Intro to smooth manifolds is a suitable chioice, but some chapters could be skipped"), Algebra-3 (Representation theory "Fulton, Harris" sem 4) Differential Geometry-2 (More advanced stuff like bundles, cohomology, see "Dubrovin, Novikov, Fomenko "Modern Geomtery" vol 2,3), ODE-2 (Arnold "Geometry methods in theory of ODE"), Lie Groups and algebras and their representations. 5) Functional analysis-1, PDE (again, done as a math course, probably Simon, Reed "Methods of modern mathematical physics", Algebra-4 (Attiyah, McDonald's book on commutative algebra), complex analysis (again, as a math course with theorems and proofs) 6) Functional analysis-2 (this subject is infinetely big, 1 year is just for the basic introduction), PDE-2, algebraic topology, homological algebra-1. P.S I'm finishing 2 year of undergrad, and how I would like my math courses map would look like, but it didnt, so I have a lot ot catch up on my own.
'Linear Algebra Done Right' goes crazy hard. Was surprised to see you call it a textbook for advanced linear algebra since it requires no linear algebra background (and the first sub-chapter is literally on complex numbers). That explains why it hasn't helped me much in my first year uni linear algebra class lol.
I was confused at first with the college algebra 1,2,3 but then you explained it can also be trig and then pre calculus and that made sense. At my college I did that; college algebra, trigonometry, and precalculus.
i can speak to recommending probability theory, it's crucial if you're intending to work in a field reliant on applied mathematics. knowing probability theory allows you to conceptualize how to approach problems people often face in these fields
Probability theory is essential for anyone wanting to get into Quantum mechanics. Will give you a massive advantage over everyone else if you’ve taken classes in probability and statistics.
Love the video and your channel. As an older person (58) learning math as a hobby, I would love to see you match this list with the books you would use in order. The biggest problem I'm having is to choose which books to read with the limited time/money that I have. I'm also a book collector, which compounds the problem. So if you're on a desert island and want to learn math, what's your ideal list, with the caveat of stopping when you're ready for anything at the post-graduate level.
It is a good question… I would say Royden and Fitzpatrick’s Real Analysis for measure theory and other related topics, and Martin Issacs Algebra book for graduate level algebra. I haven’t found my ideal complex analysis book yet but still looking. For functional analysis, definitely the Kreyszig book, and for linear algebra, I would go with Axler.
I'm also learning math as a hobby - brushing up again on my linear algebra, geometry and real analysis so that I can "re-connect" with my favourite subject of differential geometry 🙂
At my current university, they split up the Calculus Sequence into 4 classes rather than 3 classes. Calculus 1 & 2 are the same, but 3 & 4 is pretty much equivalent to your usual Calculus 3 at other colleges. I rarely see this done at other universities outside of a couple community colleges, and other Universities in the region of the US I live in. Also, the general course plan for the Bachelor of Arts and Bachelor of Science has you taking multiple math classes after Calculus 2. It's set up as: -Semester 1: Calculus I -Semester 2: Calculus II -Semester 3: Calculus III Linear Algebra -Semester 4: Calculus IV Differential Equations I After the 4th semester, the degree programs split off in terms of courses, as the BA is less rigorous than the BS in terms of Mathematical Content. But as a BA student I am taking all of the BS courses as I am not a fan of science lol.
What does Calculus IV cover at your university? I know that at most other universities in America, Multivariable Calculus and Vector Calculus are together crammed into Calculus III.
@@lorax121323 It's essentially just multivariable calculus but stretched out into 2 semesters rather than being crammed into just one semester. I go to engineering school, and I think a lot of the reason behind it is to have the Engineering majors grasp the topic of multivariable calculus a little easier than if it was all done in 1 semester. Generally, the Math Curriculum is mainly set up more so for Engineers rather than Mathematicians until you get to your upper-level Mathematics courses like Advanced Calculus, Intro to Topology, or Real Analysis.
It's very strange, here in Germany (pure mathematics bachelor) we dont have Calculus 1,2,3 in university, everything is prove based and it starts with more advanced courses: 1 semester: Introduction to Real Analysis, Linear Algebra I 2 semester: Real Analysis, Linear Algebra II (which is advanced Linear Algebra) 3 semester: Measure and Integration Theory, Numerical Linear Algebra, Introduction to Probability Theory 4 semester: Introduction to abstract algebra and number theory, Complex Analysis and ODE, Numerical Analysis, Probability Theory 5 semester: Topology, Functional Analysis, Abstract Algebra, PDE 6 semester: 2 of the following: Diff. Geometry, Alg. Topology, Alg. Geometry, Functional Analysis 2, Alg. number theory, advanced PDE, advanced numerical analysis
If you focus on geometry/topology, your master might look like this: 1) Lie groups and algebras, algebraic geometry 2, algebraic topology 2, 2) homotopy theory, representation theory, geometry of schemes 3) homological algebra, representation theory 2, hodge theory 4) your thesis
Understand most people graduate highschool with at most Algebra 1 and a little geometry in the US. From my experience most highschools dont even teach calculus and its normally busses to highschools that do OR dual enrollment in some community college. US school system is ass and a 4 yr degree is about 100k USD so America doesnt like smart people.
It's true that many American schools offer the college algebra/precalculus/calculus sequence as primarily a remedial tool, which takes up space in the first year. However, all top schools in the country are on par with the best European schools; even if the course order is flexible, fundamental undergraduate topics are treated early, such as taking first courses in real analysis/linear algebra/abstract algebra at the freshman level.
@@musashimiyamoto9035 That's bullshit. Most people who intend to major in mathematics out of a US high school have generally finished at least precalc in high school, perhaps even calc 1 and 2.
I don’t know the country you are in, but where we are probability and statistics are included in the undergraduate degree. So they are not included where you are ?
Yes for sure, some of the things you’ll come across at undergrad are distribution types, random variables, Bayes theorem, linear models, Markov chains, statistical models, then even decision theory at a more advanced level. I think this video was more of a pure maths focus though, probability and statistics is a whole different world.
This is my hardcore plan for self study undergraduate mathematics before studying physics 1. Openstax Algebra and Trig 2. Openstax Calculus 3. Thomas’s Calculus Early Transcendentals+Solutions 4. Book of proof 5. Real Analysis Jay 6. Algebra and Trig sullivan + Solution 7. Introduction to linear algebra Gilbert 8. Vector Calculus Susan + solution 9. Differential Equations Zill + Solutions 10. Calculus of Variations Elsgolc 11. A First Course In Integral Equations Abdalmajid + Solutions 12. Complex analysis Zill + Solutions 13. A Student's Guide to Vectors and Tensors Fleisch 14. Numerical analysis burden 15. Contemporary Abstract Algebra+Solutions 16. Understanding Topology Ault 17. The Probability Lifesaver 18. Mathematical Methods for Physics and Engineering Riley + Solutions
@@pyrenn no i'm graduated from physics. you dont have to take all this math now. start with algebra and trig first then calculus and then university physics.
I don’t know if you’re still doing this, book after an intro to proofs book (I’m thinking how to prove it, etc), consider doing something like Spivak before doing actual Analysis. You’ll learn a lot & actually get good at proofs. Everything else will be a lot easier
I’m going to give my IGCSEs in November (if you don’t know, it’s what British curriculum students have to sit for at grade-10) and of course planning ahead and I want to do something in the field of mathematics and your videos make me feel like you are right in front of me talking and giving advices in a very chilled way which I just love about you!! Ty!! Keep going ✨
this feels strange coming from a uk background; in first year of my undergrad degree we had mandatory classes on basic real analysis, linear algebra and group theory (among other things like dynamics, probability etc.) for anyone doing a maths degree. Second year had stuff like complex analysis, more linear algebra, topology, ring theory, and lebesgue integration. Third year i got into more galois theory, functional analysis etc. Feels weird to be able to do a full undergrad degree without doing any much serious algebra work beyond linear algebra
Even a lot of Canadian universities are way ahead (we even have stuff like riemannian geometry, full on model theory, category theory, operator theory, representation theory, and lie algebras at the undergrad level at my school). I have no idea how people can justify having a bachelor's in math having not taken more than a dozen math course (and of those, only a handful proof based ones).
This guy has a lot of remedial stuff in here that most STEM math major wouldn’t need. Most high school graduates will have some familiarity with calculus but they generally all need more in college. I think the difference is that math majors and STEM folks take a lot of computational type classes early in since lower level classes are generally the same for math and engineering majors. Also seems like we have a lot more general education in the US and less specialization going into college. Europe and the world at large seems to specialize earlier.
Wanted to revisit this after watching it for the first time when it came out. I just finished Calc 1-3, Diff Eqs and Linear Algebra at community college, and on Monday I start grad school in Statistics. I didn’t follow the path you laid out here, but this was an inspiring video.
Right now im studying maths in my first year in Germany and we have the courses linear algebra 1 and 2 as well as analysis 1 and 2, and cover the themes of the books you mentioned for "Real Analysis" and "adv. lin. alg.". Im courious as to why you cover these themes only in your 3rd/4th year. On the other hand, things like ODE, PDE or number theory are strictly for the 2nd and 3rd years in germany, not for the first years.
At my university (years ago), you can take exams prior to starting freshman year for College Algebra and Trigonometry. If you pass, you don't have to enrol for these subjects when the semester starts. You've received credits for them already. We also had Set Theory in college, though no Topology. Set Theory was usually taken along with Linear Algebra and Abstract Algebra (to help lift your overall GPA as Abstract Algebra can be tough - haha). Finally, you cannot call yourself a "true" math undergrad unless you pass Vector Calculus, Abstract Algebra and Differential Equations (ODE, PDE). By this time, close to 60% of your cohort would have dropped out already.😄
@@epicm999 I heard of a group of people who tried to go for an ivy league university math degree as a challenge haha, it could be, though is doing a degree worth it just for it being considered difficult? And is it the only challenge worth taking on and no other?
@@אהלןסהלן I'm right next to being able to take Vector Calc and am already planning on taking ODE. You do bring up a good point though lol, it's not worth doing a degree for the challenge unless the degree gives you opportunities.
Interesting. In Finland the first year curriculum consists generally of: Fall: Proofs, Calculus 1 & 2, Linear Algebra 1 (+ compulsory general studies like Finnish, Swedish & English) Spring: Linear Algebra 2 & 3, Advanced Calculus (proof based Calc 1 & 2), Series Spring Electives: Logic 1 & 2, Probability and Statistics, Applied Linear Algebra Knowledge of trigonometry and elementary number theory is assumed.
My school is teaching calculus, number theory, group theory, linear algebra, stats, mechanics, complex numbers all in one year, and I'm doing physics and product design (they didn't offer engineering) A-levels at the same time. This was last year when I was 16 y.o.
I picked up Axler's LADR as a freshman into my math major and I strongly agree with 10:40! I fondly remember struggling as much as I could through the first chapters. Holds a special place to me
I am still a sophomore 😅, but I have an opinion I would put a modern algebra course instead of the number theory second course, and I would put a course in topology instead of complex variables second course. I think the core courses for undergraduated students should be in different areas as the student can, and the optional courses should be courses that prepare the student for his master degree.
Topology is, in my opinion, a must for every math student at the undergrad level. I know some people struggle with it but to be honest, when one grasp the fundamentals of topology most subjects start making a lot more sense, like Complex Analysis, I would even go as far as to say that it should be a prerequisite for taking Complex Analysis. Also, I'm kinda partial to saying that math undergrads should know a little bit of probability theory from both a measure theoretic and a non measure theoretic perspective, not because I particularly enjoy probability but because it gives the students who decide not to and/or are unable to enroll into a master a better shot at getting a good paying job.
I would think those are covered in the differential equations courses. In my ODEs class we did Laplace transforms, and I think the PDEs/Advanced Engineering Math covers Fourier transforms
It‘s interesting how different education systems are. Here everything up to and usually including Calc3 is covered in school, university starts with Real Analysis, Linear Algebra and Differential Equations. Which was actually kind of an issue for me because I started my degree 3 years after finishing school so I had to re-learn integration on my own while being asked to prove its underlying ideas in the Analysis class. The additional work was a struggle but I‘m glad I discovered my passion for math and theoretical physics. Better late than never.
I would say I wish I saw complex analysis sooner, it is absolutely beautiful and it is really helpful for studying modular forms which is a magical subject
Man… modular forms are something I aspire to understand. I was chemical engineering in undergrad so I got through diff EQ. Right before things get deep. Now I’m self learning and modular forms are a ways away from me now😂
@@Dark_Souls_3 There are something’s you just need to take for granted when learning modular forms (for me it’s been Fourier/harmonic analysis) but in the end they’re just some funky well behaved periodic complex functions. That for some mystical reason are connected to number theory! What are you self learning at the moment if I can ask?
First of all, if you did every course you listed up to @11:54 at a college you would be there over five years as an undergraduate. I mean College Algebra I-II-III and Calculus I-II-III are sequential, so that completing those sequences is over two years right there.
I would recommend taking calc 3 if you are a chem major. I ended up dropping it a few weeks in, but think it would of been helpful. Usually just calc 1-2 are required.
@@eshankulkarni2843 It would of been helpful in electricity and magnetism course of physics (2nd semester physics usually) and also in quantum chemistry class (upper elective I took). Both used double and triple integrals, but b/c calc 3 wasn't prereq, we didn't have to solve any on exams. But would of been helpful to know these things to understand better. Again, I would of just preferred I hadn't drop the course. It didn't put me at a disadvantage not having taking it though so I wouldn't worry if you skip it.
Hi I bought the book of how to think about Analysis by lara alcock and I loved it. Thank for the suggestions. I have 22 analysis book including fizpatrick book and bartle but I loved this book more than anything else.. People praise alot about the real analysis book by jay cummings (this is also the Amazon's best selling book), if you don't mind could you do a review on that book as well I want to say this but I love your channel and the way you talk and think are alot similar to math sorcerer( he is also a youtuber)
Thank you for the kind words, I will check out the Cummings book, but will need to read it thoroughly before offering my take. Also, I appreciate the comparison to Math Sorcerer, I like his videos :)
For all other business and biology chemistry degrees, math is really useful up to Calc AB-BC levels. After calc 2 (bc) math sort of becomes its own thing, unless you are doing physics. Computer programming is also more math related depending on which branch you go.
This is a great list. I thoroughly enjoyed your video. I can't emphasize enough a strong background on linear algebra and modern algebra, for now my goal is to learn galois theory and class field theory, hopefully sooner than later, but my real objective is to what is the deal with elliptic curves, galois representations and L functions. Any recommendations for this path would be appreciated.
Thank you. My background is in analysis so my algebra recommendations are not the strongest. But some authors I would recommend for studying algebra that I liked were Isaacs, Hunderford, and Conrad.
I'd recommend checking out J.S.Milne's website - he has book-length notes series on topics ranging across field/Galois theory, commutative algebra, algebraic number theory, class field theory, and more. I'm looking to go into number theory, and owe a lot of my foundational knowledge to those texts. Another wonderful book series is Kato's "Fermat's Dream" (Iwanami Tracts in Mathematics). Hopefully these are helpful to you.
To get into L functions and elliptic curves, you'd need a strong background in algebraic number theory for which the book by Stewart and Tall could be a good place to begin. You can also look at the later sections of the book by Ireland and Rosen after going through the first book. For all this, you'd need a good foundation in abstract algebra for which you might look at the books by Artin, and Dummit and Foote. And after all this background, you can look at Silverman's book on elliptic curves. After you've learned these you should also look at objects called modular forms which is (sort of like) the bridge between elliptic curves and L functions. That's how I've learnt the subject (until now). Hope that helps :)
Hi had a great time watching your video, I am intended to learn Differential Geometry on my own, and currently taking a Modern algebra Course and a Course on topology. Can you pin a list adding the geometry path from the functional analysis, please? as I am learning from youtube it is difficult sometimes on which course should I take next for the same
Unfortunately I don’t have much to recommend with geometry at the grad level. I’ve been trying to find great books in geometry. One of my instructors is using a book by Pogorelov however it’s in the neighborhood of calculus 3. He’s says you don’t need much passed calculus 3 to study geometry but I still search.
I have worked differential geometry and smooth manifold theory, and I know you didn't ask me but, it really depends on your foundations. If you have only taken courses like Linear Algebra and Calculus I would recommend you check Do Carmo's "Differential Geometry of Curves and Surfaces", while the approach has been kinda superseded by the study of manifolds, it still a really good book and it gives you some insight into how the ideas of Differential Geometry originated. If you have already taken a course in Topology, Group Theory, and feel comfortable with Calculus and Linear Algebra I would recommend you check either Lee's "Introduction to Smooth Manifolds" or Tu's "An Introduction to Manifolds", they both develop the theory in a very approachable way, with lots of examples. The main difference between manifold theory and (classic) differential geometry is in how they study the same objects, in manifold theory we study local properties that allow us to draw conclusion about the whole object while in differential geometry we study global properties that allow us to study the local properties, there are theorems that guarantee they are both equivalent in most important cases.
@@PenaflorPhi Hi thank you so much for these suggestions, I will start with Tu's "An Introduction to Manifolds" and follow up more. do you have a twitter handle, would like to connect and discuss more if possible :)
bro my community college had a placement test that if you scored high enough on it would automatically skip you straight to calculus. it was multiple choice and i just guessed for most of the questions bc i was very lost and i scored a 86 just from random educated guesses and now i’m taking this calculus class and i’m failing so hard bc i just don’t remember how to do algebra. i’m literally so fucked dude my parents are going to kill me idk how to tell them i got a 47 on my midterm
Ask yourself why you failed. If you failed because you didn’t study enough, then own up to it and try harder next time. It is possible to come back. I failed a graduate level linear algebra test once, and was able to come back from it through hard work. If you studied your butt off and still failed, then it is not your fault. I would consider taking college algebra again and then retake calculus.
Do you guys not do any Topology, Differential Geometry, Representation Theory or Algebraic Geometry in America? I would consider Topology and Algebraic Geometry a must have for anyone in Algebra or Geometry and Differential Geometry for anyone in Analysis and PDE's.
We certainly do. The content of the video is just one specific recommendation regarding certain courses to take. Many schools (and all competitive ones) in the US study topology as early-on as possible, diff geo in some form, and rep theory/alg geo whenever the algebraic prerequisites are met. There are various courses at as early as the sophomore level across different institutions that introduce all of these aforementioned topics.
I'm currently taking my 1st year as undergrad (double major in math and comsci) These are the predefined curriculum (almost) that our university uses. But it also have applied math subjects every semester like physics and statistics.
Where is undergraduate level group ring and possible field theory in Abstract Algebra. No undergraduate course can miss it. And a student who studies abstract linear algebra also gets confused with out group ring theory.
Is measure theory metric spaces? I notice you didn't stress statistics so much (unless probability covers this) or something like combinatorics (you did mention graph theory). And maybe something like Lie Algebras and Differential Geometry (maybe non-Euclidean geometry covers this). Then there's topology and algebraic geometry which I know little about. Good list though.
Not quite. Metric spaces generalize the notion of distance (think absolute value between points), while measure spaces generalize the notion of length, area, or volume.
I appreciate the video, but as a math student, I'm shocked that you didn't put down at least one topology course. Point-set and algebraic topology are both really important, as well as differential geometry and algebraic geometry. I feel like excluding all of these gives people the wrong impression about what math is.
Do you have any texts on the algebraic manipulations of sums, multiplications (capital pi), and other stuff that is not too readily taught in any course curriculums. I wish to start tackling MIT-level integrals and more advanced topics in math but need some texts to aid. Any suggestions on texts will be wonderfully appreciated.
