# Mathematical subjects you wish you learned earlier

I am learning geometric algebra, and it is incredible how much it helps me understand other branches of mathematics. I wish I had been exposed to it earlier.

Additionally I feel the same way about enumerative combinatorics.

What are some less popular mathematical subjects that you think should be more popular?

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What is "geometric algebra"? – Kevin Lin Jul 28 '10 at 19:16

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– Jonathan Fischoff Jul 28 '10 at 19:26

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I fear this is going to degenerate into "List subjects that you like." – Nate Eldredge Aug 17 '10 at 1:04

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@Nate: I agree. Every time I understand something in mathematics (i.e., every good day I have) I wish I understood it much earlier, of course. This includes material from subjects I first "learned" long ago... – Pete L. Clark Feb 4 '11 at 22:19
Everything I like. – Jonas Teuwen Feb 5 '11 at 11:52

## 19 Answers

Category theory and algebraic geometry.

I spent a lot of time in undergrad studying things that were kinda nifty, but way too classical to be of any use/interest beyond "fun math". When I got to grad school, category theory was assumed and made some of my courses much harder than they should've been.

In the words of Ravi Vakil, "algebraic geometry should be learned slowly over a number of years". I currently NEED algebraic geometry, so I don't have this number of years. I wish I would've started that a long time ago. Additionally, both of these topics would've helped me learn the things I was thinking about anyways, in particular commutative algebra.

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What is a good way to get an overview of algebraic geometry and the problems it solves? – Jonathan Fischoff Jul 22 '10 at 20:10

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For a general overview of ALL of alg geom, I have no idea. But here are some specifics. If you care about enumerative algebraic geometry, tinyurl.com/28bkunk . If you care about real solutions to algebraic systems, arxiv.org/abs/0907.1847 . If you care about representation theory, tinyurl.com/2dkss6h . If you care about mirror symmetry or string theory, arxiv.org/abs/math/0601041 . If you care about number theory, math.mit.edu/~poonen/782/782notes.pdf . If you care about PDE's or analysis in general, math.stanford.edu/~dbaskin/sdgs-microlocal.pdf . :D – BBischof Jul 22 '10 at 22:46

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– BBischof Jul 23 '10 at 3:46

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So we have recommendations for algebraic geometry and geometric algebra. :) – Oscar Cunningham Sep 17 '10 at 12:50
@BBischof Lots of caring. – Gustavo Bandeira Dec 16 '12 at 11:46
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Though I'm sure it's not unpopular, I don't think many people learn it early: Group Theory. It's a real nice area with a lot of cool math and some neat applications (like cryptography).

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As a physicists I totally support that answer. – Lagerbaer Feb 4 '11 at 19:28
A firm understanding of Group Theory radically changes the way you define problem/solution space in Computer Science. – awashburn Feb 17 at 23:11

Lattices and order theory. While these concepts are so ubiquitous, they seem to be banned from mathematics courses. Also, if you know something about order theory, many concepts from category theory turn out to be quite familiar. (E.g. view a poset as a category, then product resp. coproduct become infimum resp. supremum, the slice and coslice category are up and down set, etc.)

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I wish I'd understood the importance of inequalities earlier. I wish I'd carefully gone through the classic book Inequalities by Hardy, Littlewood, and Poyla early on. Another good book is The Cauchy-Schwarz Masterclass.

You can study inequalities as a subject in their own right, often without using advanced math. But they're critical techniques for advanced math.

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Can you elaborate a bit about its importance? My only acquaintance with inequalities is back in high school olympiad stuff. Perhaps I'm not an analysis guy, I never really used those olympiad stuff after I entered university. – Soarer Jul 28 '10 at 20:14

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Inequalities are everywhere in analysis. Often you'll bootstrap a simple inequality, one you may have seen in high school, into a sophisticated inequality. For example, you might take ab <= (a^2 + b^2)/2 and parlay that into a theorem about operators on Banach spaces. – John D. Cook Jul 28 '10 at 20:51
One of the major reasons a lot of students struggle in undergraduate analysis is becuase they don't have command of basic inequalities. Proving limits rigorously is VERY confusing without this skill. I know I'M sorry I didn't learn it before that. – Mathemagician1234 Jan 29 at 5:20

I wish I'd learned logic much, much earlier. Obviously young students couldn't handle much depth, but at least a basic introduction to a few concepts would be nice. Just understanding the concept of axioms and deductive rules would put all of math into some perspective. When I finally understood that math was constructed with formal definitions and proofs (or, for example, that there was more than one useful way to axiomatize), I felt I'd been kept in the dark my whole life, doing something (math) that I had absolutely no understanding of.

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Graph Theory is a fantastic field. First, the fact that abstract concepts can be readily visualized makes it engaging for new students. And second, I believe it provides solid foundations into mathematical thinking like proofs, and engages the student to explore other related fields.

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I don't know what level of mathematics you are referring to, but here's my opinion after recently finishing my university's undergraduate curriculum.

Firstly, I would like to second Jan Gorzny's reply of Group Theory.

Second, I wish that I had learned linear algebra earlier. The topic usually has two semesters: matrix algebra and then an early proof-based introduction to vector spaces and linear transformations. The real work in this topic can't begin until after both of these classes are completed.

I also wish I had been exposed to topology earlier than I was. Of course there are two "standard" approaches here, and I suppose the approach that introduces general topology before advanced calculus would have better suited my tastes.

Here is a good book that may give some more insight into the heavily debated area where your question lies: Thomas Garrity, All the Mathematics You Missed But Need to Know for Graduate School

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Nobody learns linear algebra in two semesters. Hoffman and Kunze or bust. – user126 Jul 21 '10 at 1:17

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where by "nobody," Harry, you mean "most math majors in the United States who take a class covering any kind of abstract linear algebra." No need for "I've learnt [x subject] [faster/better/with harder books]" on this eminently reasonable question and answer. – Jamie Banks Jul 21 '10 at 1:25
I have taken more than two semesters of Linear Algebra, followed by a reading course using HK (a great book). But there is still much, much more to this subject - only accessible after one can make it through a book such as HK. – Tom Stephens Jul 21 '10 at 1:38
Thanks Tom the book looks interesting. I'm always looking for non-rigorous heuristics to mathematical subjects. – Jonathan Fischoff Jul 21 '10 at 2:51
@Tom I was initially very excited by that book, but later grew to dislike it. I found that in fact, much of what is in there is less relevant than other important things. Additionally, the style of presentation leaves much to be desired. I was sad when I realized that it didn't help me much. Instead I would recommend the Berkeley problems book. That material is absolutely essential, and by solving problems you can be sure your solid on the topics. – BBischof Jul 22 '10 at 14:38
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Theory of computation, information theory and logic/foundation of mathematics are very interesting topics. I wish I knew them earlier. They are not unpopular(almost every university have a bunch of ToC people in CS depatment...) , but many math major I know have never touched them.

They show you the limits of mathematics, computation and communication.

Logic shows there are things can't be proved from a set of axioms even if it's true--Godel's incompleteness theorem. There are other interesting theorems in foundation of mathematics. Like the independence of continuum hypothesis to ZFC.

Theory of computation showed me things that's not computable. Problems that takes exponential time, exponential space, no matter what kind of algorithm you come up with.

Information theory proves the minimum amount of information required to reconstruct some other information. It pops up in unexpected places. There is a proof of there are infinite number of primes by information theory (Sorry I can't find it, I can only tell you it exists. I might find it later).

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I've removed the article link, as it does not belong on a site about mathematics (and because I think the rest of the answer has value, more so without the link, and the point of a CW format is to compile the best answers possible, which can mean substantially changing or combining answers). If anyone disagrees with this decision or wishes to discuss taking such action, please do it over on meta. – Jamie Banks Jul 21 '10 at 7:09
Information theory is typically only relevant to the comp sci sort, but I strongly agreed with the foundation/logic of mathematics/Godel's Incompleteness theorem. – Noldorin Jul 21 '10 at 8:08

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@Katie, I agree with your uptake of the CW format, but not with your stand on the legitimacy of the link. The article obviously does not really deal with predicting the future, at least not in a constructive manner. The axiom of choice does have some interesting non-intuitive results which seem like "prediction", for example: xorshammer.com/2008/08/23/set-theory-and-weather-prediction – Tomer Vromen Jul 21 '10 at 17:02

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I'm confused by the removal of the link with the rationale that it does not belong on a math site. The link is to an article which was recently published in the American Mathematical Monthly, which is, I believe, of all American periodicals devoted exclusively to math, the one with the largest circulation. What was found to be objectionable about this article? – Pete L. Clark Jul 29 '10 at 7:20

Statistics is the topic in which I am still poor and it is still useful to me which I learned so late and that's why I am poor in Statistics.

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I wish I'd learned about special functions earlier. The subject is a treasure trove of results that were commonly known a century ago but now few people know.

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I don't really think that graph theory is a "less popular mathematical subject," but I certainly wish I had been exposed to it earlier.

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I did mathematics as an undergrad, and I thought that differential equations were boring and pointless. Type of diff eq -> existence and uniqueness proofs for solutions -> rinse and repeat. Yawn.

But now I find my lack of knowledge of differential equations is hampering my learning some interesting parts of physics that I'd like to know more about...

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My answer is: fundamental concepts and methods of both first order logic and set theory. I really wish I learned them much earlier, since all mathematics is based on them.

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Non-linear Dynamics and Chaos!

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Not unpopular, but I wish I had studied the theory of Rings and Fields, and basic Topology earlier, because in my opinion both these branches appear in many interesting subjects studied relatively early during one's undergraduate studies. For example, the whole concept of a minimal polynomial of a linear transformation $T$ is more intuitive (at least for me) when viewed as the generator of the ideal of polynomials such that $P(t)=0$. Metric Spaces (studied in the introductory Topology course at my university) appear as early as in calculus, and are generally a basis to many definitions there. Also, many algebraic structures can be viewed as topologies, which can sometimes give a new insight or assist in proofs (a favourite of mine is a topological proof of the infinity of the primes, which can be found in Proofs from the Book).

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Topology was the first real math subject I learned. And I struggled through a text before I had taken a proofs and logic class so it was also how i learned to write proofs.  Now taking my second real analysis course and its a breeze. – AnonymousCoward Feb 5 '11 at 2:29

Information theory.

Incredibly deep field - it will have you perceive the world in a completely new way.

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Riemann's Explicite Prime Counting Formula: $$\pi_{0}(x) = \operatorname{R}(x) - \sum_{\rho}\operatorname{R}(x^{\rho}) - \frac1{\ln x} + \frac1\pi \arctan \frac\pi{\ln x}$$ and all the theory of Dirichlet functions behind it...

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Integrals.

My high school didn't cover them, so in 1Y uni, it was all new to me.

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Linear Algebra for sure. I would also add Statistics, though I would rather learn both well and in good depth than necessarily early.

Would second the answer on Logic.

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