Question about undergrad math courses

High schooler here, need some math/college advice

>inb4 18 years old to post
I'm a senior and already 18

So I'll be headed off to college next school year, and I'm considering a double major in math (first major will almost certainly be CS).

What's the general "course progression" for an undergrad in math? I took Multivariable Calculus and Linear Algebra last year, but let's suppose that my college requires me to retake them. I am not taking any math classes this year.

Freshman year:
1st Semester - Discrete Math (for CS), Calc 3/Linear Algebra course
2nd Semester - Linear Algebra/Diff Eq course, probably proof based

So starting here I'm not too sure what the progression is. Help is appreciated

Sophomore year:
I have been told to choose a probability course and a combinatorics course

Junior year:
1st Semester - ODE, Real Analysis I

If you're majoring in math it'll probably be closer to 2-3 courses in your senior year. Progressions sounds fine, but you'll be taking more than less later on.

The real question that will dictate your course load is what you want to do with it.

I dont know about how things are good for a double major but for a math major that plans to continues his or her studies in a graduate program this courseload is a joke. Real analysis should be taken in your second year at the latest, and algebra in the same year. As a computer scientist, there is much more reason to take algebra rather than "real analysis II" which I assume to be analysis on manifolds. Algebra is the language of discrete math, analysis not so much, and real analysis I i.e. something like Rudins book is enough for probability studies like measure theory and fourier analysis. Also while it is nice to be acquinted with the theory, if you are not pursuing a job in academia learning math "rigourously" is not required, an employer could care less about your ability to recast stokes theorem in the language of differential forms (unless you are pursuing a job at a HFT hedgefund but I digress). As an undergraduate major in math these are some of math classes I took.

Year 1: Multivariable calculus, introductory probability, and Real analysis. I studied ODE, algebra (linear and abstract together through Artins book), and analysis on manifolds on my own time.

Year 2: Algebra 1 and 2 (these were required to be taken), applied discrete and continuous math (2 seperate classes), functions of a complex variable, and introduction to PDE

Year 3: Commutative algebra, representation theory, geometry of manifolds, number theory I, elliptic curves, and mathematics for finance. I went way way overboard this year and almost failed out, commutative algebra and representation theory were extraordinarly difficult

Year 4: Differential geometry, differential analysis, functions of several complex variables, and some seminars on lie groups and hecke algebras.

I treated my studies like a job, I worked 9-5 everyday, didnt think about math the rest of the day if I could help it.

christ we really need to make one of these things a clearly visible sticky

there's at least 2-3 people a day making
>gibs undergrad math curriculum
threads

This doesn't look like enough/not the right progression. Have you looked at the course requirements for your university. They probably have a suggested course plan for people with a focus in CS. If I were you, I'd take it at an accelerated rate and then do stuff you want in the future. If not, I would do abstract sophomore year (because I did). If it isn't a core requirement at your uni, then it should be.

Also, don't worry about restudying unless you want to test out of something.

Hello OP. The single most important missing piece of information from your OP is your country.

I hold a four-year degree in math from an American university. Your mileage and country may vary. An undergrad math curriculum in an American university looks something like this --- actually it sounds like you're well ahead of the curve (by American standards) and I won't be of much help. I spent the first half of my undergrad doing the basic calc + linear alg things that you've already had exposure to. But anyway, I proceed:

If the American in question needs it (not you in any event), the first year is commonly given over Calc I & Calc 2 (derivatives, limits, integrals, techniques for dealing with all of these, series, applications, and a bit of proof, all of which you should well know). Then, as a function of these (whether you already tested out at the particular school or not), at some point between freshman and sophomore year, calc III and linear algebra need to happen. Whether you get "credited" for already having done them is something for you do lobby for of your own accord. If you really feel that you don't need to be doing these courses again, then what you need to do is to very politely (and assertively) get in touch with your math department's chair and request to demonstrate proficiency to their satisfaction. Whether they insist on you "doing it again" is the type of thing where departmental heads ought to have latitude, pull, purview, etc. They just want to know that you won't bite off more than you can chew and fuck up early on, thus wasting time and lowering pass rates that the uni can honestly self-report in stats - reasonable, no?

2-3 courses per semester?

Sweet. People always told me that taking too much math would screw me over and unless you are trolling your schedule looks awesome. I'm not going to take as much as you did because I want to do CS but shit you got me excited.

What exactly was "Algebra 1 and 2"? Linear algebra with ODE?

I have, I didn't see anything for a course plan with a focus in CS, thus I'm trying to make one myself. Abstract sophomore year sounds great.

I'm from America. Hopefully I can show the department my high school transcript where it says I dual enrolled and took multi/linear and that will be enough.

I don't know how the fuck university/college works in the U.S, but this is how I did it;
Age 18, 1st semester:
Calculus 1, Linear Algebra, Discrete mathematics
2nd semester:
Calculus 2, Abstract Algebra (I was tricked into taking this, first lecture I knew I was fucked, shit was a struggle, but I ended up doing breddy gudd), differential equations
Age 19, 3rd semester (current): Calculus 3, Commutative Algebra (again, tricked by faggot algebraists into taking it, fuck this one especially, still waiting for exam results), real analysis (don't think there's a real analysis 1 and 2 here)

I think this is relatively standard. See if you can do something similar (but do the two algebra subjects at later semesters. Maybe abstract 3rd/4th).

Algebra 1 is mostly linear algebra, while 2 was mostly abstract like the study of rings, polynomials and such things. However, I did take alot more than most people. I did this because many of my colleagues had significant experience coming in to undergrad and I didnt want to fall behind. I was lucky to come out alive, and my third year almost ruined my grades. Remember that grades, while they may seem superficial in retrospect many years from now, are highly important to get into graduate school. Also, my studies were very unfocused and all over the place because I did not yet realize what branch of math I wanted to study. A budding algebraist or computer scientist has no business taking differential geometry. To see how rigorous or thorough your universeties courses are, compare them to the following canon of undergraduate student texts:
Advanced Calculus by Sternberg for analysis on manifolds
Principles of Math Analysis by Rudin for intro analysis of 1 real variable
Algebra by Artin for both linear and abstract algebra
Note that freshmen courses are decidedly not very rigourous and this is quite normal, my first multivariable calculus class was not proof based, nor was my first probability course really. But just because you may know how to compute determinants, does not mean you know the foundations of linear algebra from a theory perspective, which is essential for progression into more specialized math topics (involving algebra). In summary, I suggest you worry about doing the basics first. Completing Rudin and Artins book will almost sufficiently prepare for you for any number of branching paths (note no differential equation study in either though)

No that is definitely not relatively standard, this is the first time I have ever heard of someone taking Calculus 3 and commutative algebra simultaesly. Its like learning how to crawl and how to go super saiyan at the same time. It can be done since the subjects are mostly indepdent but its highly bizarre.

Right, I meant except for the algebra. Otherwise it's pretty standard, no?

I wish to pre-empt any dickwaving on this user's part , or others like him, by plainly acknowledging the lower status of my education, and recounting the rest of what I'd had to say.

It is also clear that this user is proud of his undergrad courseload, (the overboard boast about junior year), as he should be, but what OP is really about is getting a sense of what a courseload is, or ought to be. He can take my example as the low end, the other guy's example as the high end, and reason that what he'll get is either in the middle somewhere, or else however far he cares to push it - his priority is CS, as he said.

OP, you'll get college level algebra, so don't worry about that part. Basically the rest of what goes on in undergrad math is a grab-bag of maybe 8-12 of 20 or so different subjects:

let's just mention stats and probability and combinatorics in here while we're at it.
Discrete Math
Abstract (or, Modern) algebra: one or two semesters. groups, rings, fields, and so on.
Foundations: sets, cosets, logic, ZFC, some history.
Geometry: Euclidian, non-Euclidian, applications to computer graphics perhaps, in your case (matrices for rigid motions IIRC) Then specifically, later, perhaps,
ODEs/PDEs
Numerical Analysis/some applied course if that's your thing
Real Analysis
Complex Analysis
Topology

And, as the other user said, certain topics which quite frankly can be rolled into a related class. The stuff that he mentioned like lie groups and Big Boy number theory (which is what I assume he meant, and the association with elliptic curves) is the type of thing that is either high-end undergrad or else babby grad school.

All of this is also of course a function of what the uni requires for a major, and again whether you feel up to it (sounds like you are).

I'd shoot them an email. I took calc 1 and calc 2 from a university while in high school and those credits weren't accepted. After significant bitching, they let me test out of calc 1, but I had to retake calc 2.

Some universities do things like this. In general high schools do a shitty job, so it isn't the end of the world if you have to sit through a few hours of classes a week and at absolute max a few more hours ripping through some easy ass homework.

Thanks for your input, I see now that I was way too conservative with my schedule.

Thanks for those books. I think the university offers a proof based multi/linear/ODE two-semester "sequence course" from what I've heard, so I should be set freshman year.

I had misread your post to say "year" instead of "semester" which makes your study of commutative algebra even more bizarre, although the calculus much more standard. I will say that study of math as an undergraduate only has set form for those who arent math majors, whereas actual future mathematicians can very well come in to undergrad knowing all the undergrad material and then some. with this in mind I can only comment on the outliers of your courseload.

I wasnt bragging about going overboard, I was bemoaning it. The hit my GPA took prevented my from getting into my top two choices of graduate school which I will not name.

Also there's nothing wrong with taking an easy semester your first semester. I did. I got bored, but it was an easy 4.0 semester to start off on a good footing. Plus, I was glad not to be one of those people who are immensely stressed out because they thought taking 5 hard courses should be a breeze compared to the 7 or 8 they took in high school.

Good thing about math is you can always get a good book and just teach yourself if the level is right. For the physical sciences not so much...

I took calc 1 and calc 2 as AP Calculus (APs are an American thing) and students at the university have told me that APs are accepted. I'm not worried about needing to retake calc 1 or 2, but perhaps I will need to retake calc 3 (multi) and linear algebra. I'll definitely shoot them an email when I've finalized my college decision.

This university requires we take a couple bs-y classes related to ethics studies, my plan is just to do those first year to get them out of the way.

Ignore the general course progression. The math major is watered down and not sufficient for doing any real advanced math. The first real math courses you'll take are a rigorous course in linear algebra and analysis. For linear algebra Hoffman and Kunze or Axler, for analysis I recommend Rosenlicht first then Rudin's principles. From here you can basically take whatever math courses you want. I recommend real analysis and abstract algebra since they're everywhere. General topology and complex analysis are other options. Now that you know all of this you can begin learning stuff like algebraic topology, differential geometry, algebraic geometry, functional analysis. You probably won't get to this stuff in undergrad.

>first major will almost certainly be CS

You're making a terrible mistake. At least major in EE/CpE and make use of your math skills.

>Is it ever a good idea to take multiple math classes per semester

The rule is you don't want to take more than 3.

>What's the general "course progression" for an undergrad in math?

Get out of mythmatics while you can. Total waste of time. You're chasing a dream.

I think I agree with this, but I could be interpreting it incorrectly. I've already said this, but I don't think you should ignore the course progression completely. I think you should do it at an accelerated rate and then spend the rest of your major taking classes you want to take. A lot of the required courses are prerequisites anyway.

>What's the general "course progression"

The general progression is a joke since most majors are slacker and take the bare minimum. If you want to make the most of your time, it should look something like this:

>Fall 1
Vector Calculus
Matrix Algebra
Physics I: Mechanics
Chem I or Bio I
Intro to Programming in C++
Technical Writing

>Spring 1
Ordinary Differential Equations and Dynamical Systems
Intro to Proofs and Abstract Mathematics (Math department)
Physics II: EM
Chem II or Bio II
Digital Logic and Automata
Data Structures and Algorithms in C++

>Fall 2
Real Analysis
Physics III: Modern
Electrical Engineering Fundamentals
Computer Architecture
Algorithm Design and Analysis
Combinatorics and/or Graph Theory (Preferably graduate level)

>Spring 2
Theoretical/Intermediate Linear Algebra
Mathematical Logic (Mathematics department)
Real Analysis II / Analysis on Manifolds
Computer Networks
Programming Languages and Compilers I
Operating Systems

>Fall 3
Abstract Algebra I
Complex Analysis
Probability Theory (Mathematics department)
Computability and Complexity Theory
Compilers II and/or Type Theory
[CS Elective]

>Spring 3
Abstract Algebra II
Point-set Topology
Mathematical Statistics
Computer Security and Cryptography
Linear and/or Convex Optimization
[CS Elective]

>Fall 4
Numerical Analysis I (Mathematics graduate department)
Partial Differential Equations
Algebraic Topology
Computer Graphics and/or Vision
Artificial Intelligence and Machine Learning
[Elective]

>Spring 4
Numerical Analysis II / Numerical Linear Algebra (Mathematics graduate department)
Software Engineering or Capstone or elective
Macro and Micro Economics
[Elective]
[Math Elective]
[Math Elective]

>year2 fourier analysis

it's merely going to be a methods course like an engineer would take without having done measure theory or functional analysis.

proper treatment of various kernels, fourier inversion formula, L^p theory, hilbert transform and singular integrals? nuh uh

which is fine since it is only year 2 but maybe just calling it " fourier methods" or perhaps more generallly "mathematical methods" wouldn't oversell it

>course progression

calc 3/linear algebra

then a proof based intro class or some kind of transition with diff eq/abstract tied in

Then whatever you need to take for your school, there are often many paths with different combinations of upper division courses.

What school did you go to? At most schools I look at, all that math undergrads take their freshman year is the Calculus sequence. Did you take a shit ton of math in high school or something?

Pic related is going to be my math curriculum at a school that is considered good for math. Yours seems like a world more rigorous.

I don't understand how people are taking linear algebra, abstract algebra, etc. in their first semester. Looking at these threads always make my curriculum look like fucking garbage but it's the curriculum I see everywhere.

>I don't understand how people are taking linear algebra, abstract algebra, etc. in their first semester

trolls or autists

Yeah, I am pretty butthurt about this shit too. When I was a freshman in high school they wouldn't let me take algebra II and geometry in the same year. In my state (at the time) we were required to take 8 semesters of math and if I took more than one math course per semester, I would run out of courses by my senior year and would not be able to graduate. The discrepancy in high school opportunities is pretty crazy.

Here are the two curriculums I'm looking at. The first is a Math major, the second is a Math and Actuarial Science double major.

This does not include any GE requirement classes.

Am I missing anything for either? How do they both look? These are the barebones for both, should I try to include more stuff? Just need curriculum advice. Note that more than likely, I will be doing extracurricular activities and I am not because I'm not autistic.

*I will be doing extracurricular activities because I'm not autistic.

how fucking stupid are you that you have to post on an image board about this instead of looking at your university requirements and determining for yourself whether you are able to do so or not?

if you will major in math, read follands analysis and then analysis on manifolds for "calc3" (the biggest meme class since god)

>but let's suppose that my college requires me to retake them

unlikely, p = 0.10

anyone studied maths at pic related?

There are not that many differences between what you have listed and what I posted, other than I did not start off with calc 1 and 2, which is taught in most US highschools.

you are out of your mind if you think that is not perfectly normal at a lot of schools

this guy took precalculus his senior year of high school
TOP KEK

No. How the fuck did you take diff.eq at the same time as calc 2?

Yeah, it's not at all normal to take abstract algebra in your first year. Most schools don't even want you to take abstract until you've taken another (usually unrelated) upper division class in their program.

Don't be an idiot.
Most people couldn't handle math major courses, you plan to take them plus CS courses.

You're not a genius.
Don't bite more than you can chew.

Better do one major well, really well.
I am talking about straight 4.0.
You will have more opportunities with great grades.
Scholarships, internships, etc.

Just do CS.
Keep math as a hobby, perhaps take 1 or 2 more advanced math courses as electives.
Be realistic.
No one is really looking for math graduates.
Unless you want to become a teacher.

Stop fantasizing.
I assure you, no career requires you to double major in math and CS.

Jesus fucking christ, you're clearly a naive senior. Why'd you have to lay it on so thick?

...

Don't be an idiot. In all likelihood, you're not smart enough for math.

There is no reasonable way for you to read this purported 18-year old's post and impute to him that he really is that naive. Quite the opposite, for a person of 18, he seems to have a fairly clear idea of what to expect and has received decent eleboration on the theme. Just because he's all "modern algebra, wow" is perfectly to be expected for his age - it's the thing that he hasn't studied, and he's been honest about it.

The OP's post, taken at face value, very clearly indicates that he is bright enough for math if he wants to do it.

t. not the OP and qualified to pass judgment

My defense mechanisms are overloaded.
DELET THIS