Math Major Guide | Warning: Nonstandard advice.

A guide for how to navigate the math major and how to learn the main subjects. Recommendations for courses and books.

0:00 Intro
1:33 Calculus
4:06 Multivariable calculus
6:44 Ordinary differential equations
8:56 Linear algebra
12:46 Proof class (not recommended)
13:49 Real analysis
18:31 Partial differential equations
20:51 Fourier analysis
22:18 Complex analysis
25:19 Number theory
30:08 Algebra
36:38 Probability and statistics
39:41 Topology
43:58 Differential geometry
47:25 Algebraic geometry
51:48 Summary and general advice

Books mentioned:
Calculus:

Multivariable calculus:

Ordinary Differential Equations:

Linear Algebra:

Real analysis:

Partial differential equations:

Fourier analysis:

Complex analysis:

Number Theory:

Algebra:

Probability and Statistics:

Topology:

Differential geometry:

Algebraic geometry:

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[akashekhar]

This is a gem of a video, like finding a hidden treasure on YouTube.

[alexandersanchez]

Unless I'm completely mistaking your views from this, I think your (clear) opinions about Algebra and Grothendieck's vision of mathematics are just wrong. It was ultimately highly abstract algebraic techniques that lead to Wiles' proof of FLT--a very concrete, historical problem, indeed. While I agree with you that not everybody needs to walk that road (and I, too, think that mathematicians broadly disregard the virtue concreteness), I don't think that the solution is to dismiss abstraction and generality as useless; such an attitude is rightly seen as regressive. However, I will grant you that many mathematicians seem to have the opposite problem, and to a much greater extent. In that respect, I find your views somewhat refreshing and intriguing. Grothendieck was certainly right about one thing: you've got to be willing to stick to your guns and express your opinions! Sub earned. I look forward to seeing more of your content in the future!

[theedj]

SO REFRESHING to see someone pitching the idea of a math major interested in more than rigid formalism and proofs, proofs, proofs. Recognizing that there is a time & a place for it, and that real world applications are an important motivator for many people. So often in my quest for good mathematics resources I stumble across people who sneer at any books that include applications, as if knowing how the math might be used in the real world is somehow a bad thing. What's wrong with starting with some historical perspective, learning about the problems people were and are trying to solve, and then--once a person's curiosity is piqued--dig deeper into the theory of the thing? So thank you! I appreciate you.

[dhickey]

Thank you for the list of great resources. It's surprising how university courses have stripped away most the history and context of math topics. After doing well in first year calculus I realized I'm only at the midpoint to truly understanding it. The rest of the subject falls on the student to complete on their own -- with the help of a few good book recommendations!

[oelarnes]

The first and last time I used "differential forms" was in my highly abstract freshman year vector calculus class. I have a PhD! While I enjoyed the course I think I and certainly most other students would have been much better served with a concrete, historical approach. Thanks for your videos, I'm interested in reconnecting with the historical and technical basics of the subject and I appreciate your approach and recommendations. I already picked up the Fourier Analysis book which showed up in my algo.

edit: Now that I've watched the whole video I want to expand on how valuable I think this advice is. I take it at face value that you believe this approach is valuable for research-track mathematicians, and I am inclined to agree without being able to add any force to that. But as someone who went from a PhD into industry I think this advice is even more relevant and valuable. There is such a value to being able to formulate and solve concrete problems using a mathematician's approach, and the skills I learned in those few years of rigorous mathematics have paid continual dividends over my career. But I would have been even better off with more concrete facility in the basics, simply writing down, manipulating, and solving equations, having all the machinery of ODE, linear algebra and statistics at my finger tips, and the background in numerical methods to transform those facilities directly into code.

Looking back on my undergrad and early graduate years, I can see clearly a fetishization of abstraction and generality when the essence of mathematics is solving problems. I hope that kind of hierarchical thinking can be done away with and replaced with a relish for tackling any concrete problem with rigor and insight.

[popoubc]

As a mid-age man who graduated from a decent Math department 20 years ago, don't remember much details but still can't help reading some old math books once in a while, I really enjoy listening to your genuine thoughts and experiences with these books. I'm sure younger generation will benefit even more from your sharing. Thank you, Professor Rubin, for making this great video!

[Stefabro]

super insightful and love the inclusion of the “genetic“ books. I think modern math classes sometimes focus too much on making people memorize a bunch of random theorems and formulas. Without actually trying to teach students the intuition or motivation behind certain rules or tricks in math. I’m the the type of person who can’t memorize any rule or formula unless I have seen it proven, in a clear way.
You definitely helped me with all that extra work needed to find the right book definitely gonna check out all your suggestions. Thanks!

[tylerstevens]

my proof class helped me a ton personally, but it wasn't untill I took graph theory did I realize the importance of a well defined approach to describing mathematics and alongside direct applications. Now I apply that approach isomorphically from course to course.

[Eta_Carinae__]

I would say for LA students: understanding Hilbert spaces is a must. Opening the floodgates to functional analysis becomes so incredibly useful.

[miltonedwincobocortez]

I have been a math teacher for over 20 years. I never understood the pedagogical importance of the historical approach. Your video made me want to start studying math again, reading books with more historical and applied perspectives. I know that students really like this approaches.

[joaosousapinto]

Great video! I would definitely have benefited from hearing this when I was finishing high school. Excessive formalism, lack of direction, and lack of motivation are definitely plaguing university-level education in mathematics. I was glad to see some of the great exceptions listed in your video (e.g. "Algebraic Number Theory and Fermat's Last Theorem", but really many other examples in your list; it's mathematics with a purpose!).
I'd like to add a couple of suggestions to your list, that I honestly think would be a pity to not include. "Theory of Computation", by Michael Sipser, should fit in a modern undergraduate mathematics curriculum (e.g. proves a modern formulation of the Incompleteness Theorem); there's probably space for other courses in Logic. "Probability with Martingales", by David Williams, was also a great book; rigorous, yet practical; it's an analysis masterpiece, and also exposes basic measure theory in a way that's actually useful, leading all the way to a proof of the strong law of large numbers.

[agnaldojunior]

I think that the techniques and 'reasons why' of the theory is the most important part to learn the subject.

[axisepsilon]

Damn, I'm not a math major (Wish I did a double/triple major in math but kinda late now lol), but some of your points are indeed spot on. I took discrete math, theory of probability, and multivariable calculus this spring semester ('twas a tough semester haha but glad I got it through especially with covid lol). I really liked all of them besides calculus 3, but I wish professors have taught us "how did mathematicians came up with these theorems and definitions and what is the motivation behind them" so that we get a chance to think more like a mathematician. For example, why did mathematicians came up with vector calculus? What is it used for? What was the motivation behind the jacobian matrix? How did Gauss come up with Gaussian Curve? etc. I'm the type of guy if introduced to the motivation behind each theorem, definition, it really clicks and hence the "ah-ha moment". But with calc 3, the professor just taught us the definitions and how to plug numbers into the formula. It got so freaking dry that I end up not paying much attention and end up getting a bad grade. It is honestly my fault that I didn't do well but your way of approaching math will definitely help a lot of math and non-math majors.

[saptarshisahoo]

How say you are a fan of applied math without saying it loud for around 1hr? : This should be the title of the video.

Joking aside, great video!

[joelklein]

I am a 4th year undergraduate student in math and physics.
I appreciate a lot the general point presented in this video.
As a student with a background in physics in math courses, I feel kind of sad for fellow classmates who just don't have motivation to lean on when learning various math courses. It is mostly present at any analysis related courses, whether it's Calculus, Complex Analysis, ODEs and PDEs, Probability and so on. Even Metric and Topological Spaces which isn't necessaripy analytical, can be very baffling to many students.
The reason for that is that as Daniel expressed, abstract theories with no motivations can only take you so far.
I do believe that math is expansive enough on it's on and doesn't need other fields to justify it's existence, but I also believe that the real beauty and value of any subject truly shines only when considered in relation to other subjects from the same field and different fields.
Many pure math students lack this kind of motivation and lense to view the subjects from, and I think it's such a shame.
I encourage anyone who studies math to try and expand their point of view in some kind of way.
It can be by reading books such as the ones Daniel suggested, I haven't read anyone of them but they seem promising enough.
It can also be combining the math studies with other field, such as any science or engineering.
I don't agree though that every single course should be tought that way. There is a huge merit to learbing things abstractly, as well as computationaly, as well as more intuitively motivated. A good mixture of different approaches to teaching math will give the best value to the students, in my opinion

[manuel_do_rio]

Dear Mr. Rubin: very interesting video! I have discovered you lately and already enjoyed some of them a lot (your interview with dr. Hamkins or your apology of McKean and Moll's books (just bought a copy for myself). Here in Europe we don't really choose much within our Math major. I get the impression that you are of a very applied and analytic persuasion - closer to Morris Kline than G. Hardy. I think I have a softer spot for 'useless' math as, being from a Humanities background, I am accustomed to appreciating something because it is true and beatiful, and to not giving a damn about practicalities.

[pawankhanal]

Great video. I am studying mathematics by my own and this is very helpful

[Simon-lbiu]

I’m an applied maths grad (4 years of maths). I’ve been working in the IT industry for the last 30 years, but recently started studying maths again. I can see you have a very practical bent, and I’m the same. In the vid you don’t recommend doing proof theory; just learn how to do proofs as you go. That was my attitude as well. But recently I’ve been learning mathematical logic, and I’m really enjoying it. What’s the use of logic? Well I guess it tells us something about the systems that we use to do mathematics and their weaknesses. So far I’ve done the Godel completeness theorem (which is not that straight-forwards) and will do the incompleteness theorems soon. There are plenty of techniques to be learned as well which I suspect will be valuable in other areas.

[InfiniteQuest]

The thing I like about this approach is that you are always grounded. Richard Bouchards also emphasizes looking at exmamples and seeing what is going on way before you develop or understand the general machinery. And I would say he's done pretty well for himself following this approach.

[jmguevarajordan]

This video points out the weakness of math majors. Pure math majors will need an additional degree in order to find jobs outside academia. Something that this video says and repeat many times is the importance of solving problems over study abstract theories. In that sense the abstract algebra example in the video was excellent. The video points out several times the importance of numerical analysis and statistic, which are extremely important in order to have a complete landscape of mathematics and finding jobs, internships or summer jobs. Honestly, the best recommendation is a double major of math with computer science or business.