Is this a complete undergrad pure math curriculum?

[jimgavagan]

I was notified that these are all the proof-based pure math classes offered at my university for the undergrad pure math degree:

Linear Algebra
Advanced Calculus
Foundations of Geometry
Elementary Number Theory
Complex Analysis
Abstract Algebra

Since the essence of pure math is proofs, do these courses pretty much make up a complete undergrad pure math experience? (If so, what do you think is the best way to prepare for these courses? Or, if not, what additional courses/topics would you include?)

Thanks.

 

[micromass]

That's only 6 courses? Is it an american thing to let people get a math degree with only 6 courses? I've seen it a lot, and I really can't understand how it's possible.

When I was an undergrad (in belgium), I took

Analysis I, II
Discrete math
Differential geometry I,II
Probability & statistics I, II
Linear algebra
Abstract algebra I, II, III
Algebraic geometry
Code theory
Topology
Functional analysis
Computer programming I, II
Stochastic processes
Mathematical software
Mathematical logic
Numerical methods in mathematics
ODE's
affine and projective geometry
measure theory
complex analysis

These are about 25 courses. I think such an amount of courses is standard in Belgium. Do american universities only require you to do 6 courses?

 

[jimgavagan]

That's only 6 courses? Is it an american thing to let people get a math degree with only 6 courses? I've seen it a lot, and I really can't understand how it's possible.

When I was an undergrad (in belgium), I took

Analysis I, II
Discrete math
Differential geometry I,II
Probability & statistics I, II
Linear algebra
Abstract algebra I, II, III
Algebraic geometry
Code theory
Topology
Functional analysis
Computer programming I, II
Stochastic processes
Mathematical software
Mathematical logic
Numerical methods in mathematics
ODE's
affine and projective geometry
measure theory
complex analysis

These are about 25 courses. I think such an amount of courses is standard in Belgium. Do american universities only require you to do 6 courses?

I was specifically trying to avoid that interpretation, actually, lol - I was saying that those 6 courses are the only ones that are PROOF-BASED. I was therefore wondering if those 6 proof-based courses are sufficient for the full pure math experience, or if I should go somewhere else if I want a better experience of and exposure to pure math.

 

[micromass]

All the classes I mentioned were also proof-based, as they should be...

Seriously, if those are the only 6 proof-based classes that your college has to offer, then I really suggest going somewhere else. Because it's really lacking topology, real analysis and probability. And these are some things that every math major should know...

 

[AlephZero]

Hmm... When I did a maths degree (rather a long time ago, in the UK), every course was "proof based". There were quite a few options depending on whether you wanted to specialize in pure, applied or statistics, but post #2 looks quite similar for somebody mostly interested in pure math.

"Non-proof based math" isn't math, IMO.

 

[Shackleford]

Hmm... When I did a maths degree (rather a long time ago, in the UK), every course was "proof based". There were quite a few options depending on whether you wanted to specialize in pure, applied or statistics, but post #2 looks quite similar for somebody mostly interested in pure math.

"Non-proof based math" isn't math, IMO.

My probability professor said something similar the first day of class. He said the Calculus courses are not math courses; they're basically service courses for biology, chemistry, engineering, physics, etc. He said our probability course was going to be mostly a math course.

 

[sutupidmath]

That's only 6 courses? Is it an american thing to let people get a math degree with only 6 courses? I've seen it a lot, and I really can't understand how it's possible.
. Do american universities only require you to do 6 courses?

Unfortunately this is sort of the 'norm' if you attend a liberal arts college/university in the states. What a lot of people do to overcome this difficulty(namely the lack of more advanced math courses) is do independent studies. The main reason for not offering more courses is the lack of interest on the part of students(since you need at least 5 students in order to run a course, at least at my school) and the lack of math faculty at these institutions, since they are relatively small universities. Another factor is that in US you have to take a bunch of dimensions classes (non-math classes in this case, if you are a math major), while in Europe, if you study math then for all intents and purposes, all the courses you take are math ones, with a few electives.

Ex. My university(in the states) offers these upper division math courses: Analysis I, Abstract Algebra I, Linear Algebra, Point-Set Topology, Mathematical Probability, Vector Calculus&Complex Variables...I might be forgetting another course, but basically this is it...

(Here I have excluded courses such as: Discrete Math, Calculus 1-3, Diff. Eq, Numerical Methods, Statistics 1,2 etc. )
But I also have to take(have taken) Physics I, II, Modern Physics, Quantum Mechanics, E&M.

So, what some of us do, is get a professor to agree to give us an independent study in some of the core courses, like: Advanced Analysis (i.e Analysis II), Abst. Alg. II, Algebraic Topology etc.

To the OP: my advice is do as much math as you can. That is, try to do some Topology too at least, and a second part of your analysis and algebra course.

 

[Skrew]

OP, those are the standard.

That's only 6 courses? Is it an american thing to let people get a math degree with only 6 courses? I've seen it a lot, and I really can't understand how it's possible.

When I was an undergrad (in belgium), I took

Analysis I, II
Discrete math
Differential geometry I,II
Probability & statistics I, II
Linear algebra
Abstract algebra I, II, III
Algebraic geometry
Code theory
Topology
Functional analysis
Computer programming I, II
Stochastic processes
Mathematical software
Mathematical logic
Numerical methods in mathematics
ODE's
affine and projective geometry
measure theory
complex analysis

These are about 25 courses. I think such an amount of courses is standard in Belgium. Do american universities only require you to do 6 courses?

To be blunt, I kind of doubt you took all of those as an undergraduate unless the general education requirements were very low.

 

[deluks917]

I'm not sure about belgium but in much of Europe there are basically no general education requirements.

 

[Cider]

On the quarter system I could see that many being fit in and it not being too different from the norm, but not a semester system. Also, I don't know how you can consider a course called Mathematical software a proof-based course, and the same thing possibly for numerical methods. Same thing goes for Computer programming I, II.

 

[micromass]

To be blunt, I kind of doubt you took all of those as an undergraduate unless the general education requirements were very low.

There are no general education requirements in Europe. In fact, I tried to follow some non-math courses, and the university wouldn't let me. So we're kind of forced to follow all-math courses...

On the quarter system I could see that many being fit in and it not being too different from the norm, but not a semester system. Also, I don't know how you can consider a course called Mathematical software a proof-based course, and the same thing possibly for numerical methods. Same thing goes for Computer programming I, II.

Oh yeah, I'm sory. Programming I,II and mathematical software wasn't proof-based. But numerical methods certainly was.
Also mathematical logic wasn't proof based (something that I very much disliked!), since we were following it together with computer science majors and philosophy majors...
But the rest is all proof based...

 

[HallsofIvy]

No, even at a Liberal Arts College (which can, in fact, give you a very good education and a strong mathematics major) you will take a lot more than 6 mathematics courses. At the Liberal Arts College I taught at for many years, there were 15 specifically required mathematics courses although most math majors took several more. There were also related science or computer science courses that were required.

(I believe that, with 6 math courses, you could get a minor in mathematics.)

 

[DrWillVKN]

Similar question that OP asked:

I'm doing a joint degree in biochem, so I don't have the room for so many math courses to do pure math. For applied math, are these courses enough?

Discrete Math, Linear Algebra I + II, Numerical Analysis, Partial Differential Equations, Applied analysis, Real analysis I + II, Complex Variables, Probability + Statistics, Abstract Algebra, Programming (one low level language course).

I'm not looking to go into electronics or a computer related field, aside from learning basic programming skills for certain software packages.

Would not taking any geometry and topology courses be bad? What should I self study?

 

[Chaostamer]

That honestly sounds like a fairly solid applied math program, although I'd say that a math modeling course and a course in ODEs would probably round that out a bit more.

As for jimgavagan, it would probably benefit you to seek out more proof-based courses. My school requires pure math majors to take all of these proof classes:

Logic and Proof in Mathematics (Logic, set theory, functions, relations, and methods of proof)
Linear Algebra
Combinatorics
Advanced Calculus I (Introduction to real analysis)
Advanced Calculus II
Abstract Algebra I
Topology
Statistical Theory I
Statistical Theory II

I don't think those classes, on their own, are sufficient for any pure math student interested in grad school. I personally plan to take several other classes:

Graph Theory
Advanced Linear Algebra
Abstract Algebra II
Analysis I
Analysis II
Functional Analysis
Complex Analysis
Number Theory

Are there more classes available to you? I'd at least recommend more analysis and algebra, along with some topology. Perhaps you can take a few grad classes. I'm doing the same.

 

[grendle7]

OP, those are the standard.
To be blunt, I kind of doubt you took all of those as an undergraduate unless the general education requirements were very low.

Actually, what micromass is probably telling the truth and not exaggerating; and, I say this because the university curriculums in Mexico/Argentina are really similar to what he mentioned.

For example, if you're accepted to study pure mathematics at the University of Guanajuato (some Mexican university), all the classes you take are only related to your major. You also have to keep in mind that a mathematics major takes 5 to 6 years to finish, in Mexico/Argentina because all of the courses for your major have to be finished, to graduate; and there're LOADS of them. So, that's why I believe what micromass has written, if the university industrial complex of Belgium is similar to that of either Mexico's or Argentina's.

And, if you're interested in maybe just "taking" a biology class because you took a brief interest in genetics, you'd have to read up on it by yourself or sign up to do a double major. But, doing a double major is quite rare because it's both expensive and überigorous; unless you attend the http://www.uba.ar/ingles/index02.php, where college is free, or most Argentinian and Brazilian universities, 'cause they're free, too.

:b

I do know that Peruvian universities (like the http://www.pucp.edu.pe/EN/content/index.php) tend to have programs more like the ones in the U.S., where you have general education requirements et çetera...but their bachelors pure-mathematics graduates don't have as much success as their Argentinian/Mexican counterparts, due to the curriculum comparison and extensive exposure to mathematics; I'd say.

* * *​

Moving the topic to the university I might consider attending, here in the U.S, if I decide to not go to Argentina/Mexico for university studies ('cause I'm still in high school)...

I live in Colorado, I'm also part of the lower-middle class, so I'd go to an adequate/low-cost school: the Metropolitan University of Denver. The institution offers mathematics degrees, with a concentration, in the following areas (click on'em, to see their requirements):

• Mathematics Education Concentration
• Pure/Theoretical Mathematics
• Probability and Statistics
• Applied Mathematics
• Computer Science

 

[cfddude123]

I'm about to graduate and apply to PhD programs; I am a Stats/Math major. The classes I took and will take are

Into to Advanced Mathematics
Applied Linear Algebra
Abstract Linear Algebra
Stats/Prob. 1
Stats/Prob. 2
Methods of Applied Stats
Abstract Algebra 1
Analysis of Variance
Time Series Analysis
Applied Regression and Experimental Design
Ordinary Differential Equations
Applied Complex Variables
Partial Differential Equations
Honors Real Analysis
Honors Analysis on Manifolds
Markov Chains and Stochastic Processes
Honors Seminar (wrote a senior project)
Combinatorics
Graduate PDEs
Measure Theory
Dynamical Systems
Numerical Fluid Dynamics

I've noticed that it sometimes can vary wildly between universities, but the general consensus is that you should always take way more than required.

 

[R.P.F.]

That's only 6 courses? Is it an american thing to let people get a math degree with only 6 courses? I've seen it a lot, and I really can't understand how it's possible.

When I was an undergrad (in belgium), I took

Analysis I, II
Discrete math
Differential geometry I,II
Probability & statistics I, II
Linear algebra
Abstract algebra I, II, III
Algebraic geometry
Code theory
Topology
Functional analysis
Computer programming I, II
Stochastic processes
Mathematical software
Mathematical logic
Numerical methods in mathematics
ODE's
affine and projective geometry
measure theory
complex analysis

These are about 25 courses. I think such an amount of courses is standard in Belgium. Do american universities only require you to do 6 courses?

Wow. Algebraic geometry as an undergrad? Good for you! And also projective geometry as a single course? Sounds intense. I think the area of modern math that most frequently utilizes projective geometry is the study of elliptic curves.

I should emigrate to Belgium right away.

 

[Jorriss]

That's only 6 courses? Is it an american thing to let people get a math degree with only 6 courses? I've seen it a lot, and I really can't understand how it's possible.

It depends on the institution and what that institutions aim is. MIT only requires about 8 classes + 1 seminar as a minimum, but do you think students there are only doing the minimum?

I think most institutions requires 9-10 semester quarters or 14-18 quarter courses + the standard LD linear algebra, diff eq, calculus stuff.

 

[Jorriss]

acc. double post.

 

[dcpo]

Wow. Algebraic geometry as an undergrad? Good for you! And also projective geometry as a single course? Sounds intense. I think the area of modern math that most frequently utilizes projective geometry is the study of elliptic curves.

I should emigrate to Belgium right away.

This kind of course load is pretty standard in Europe. For my undergrad in the uk I took

In the first year:
2 courses of linear algebra, 1 of general abstract algebra, 3 of analysis, 2 of geometry, 3 of calculus, PDEs, 2 of probability, stats, and dynamics (all compulsory).

In the second year:
More linear and abstract algebra, more analysis and more ODEs (these were all compulsory), group theory, field theory, Lebesgue integration, number theory, topology, multivariable calculus, calculus of variations (these were options, of which I had to choose a set number).

In the third year:
Galois theory, Banach spaces, logic, set theory, algebraic topology, and a course on the philosophy of maths (all options).

 

[R.P.F.]

This kind of course load is pretty standard in Europe. For my undergrad in the uk I took

In the first year:
2 courses of linear algebra, 1 of general abstract algebra, 3 of analysis, 2 of geometry, 3 of calculus, PDEs, 2 of probability, stats, and dynamics (all compulsory).

In the second year:
More linear and abstract algebra, more analysis and more ODEs (these were all compulsory), group theory, field theory, Lebesgue integration, number theory, topology, multivariable calculus, calculus of variations (these were options, of which I had to choose a set number).

In the third year:
Galois theory, Banach spaces, logic, set theory, algebraic topology, and a course on the philosophy of maths (all options).

Are all the bold parts non-overlapping? I had all those materials in a year-long algebra sequence, but probably not in the same depth as you did, because it seems yours were progressing towards specialization.

 

[dcpo]

Are all the bold parts non-overlapping? I had all those materials in a year-long algebra sequence, but probably not in the same depth as you did, because it seems yours were progressing towards specialization.

There was very little overlap as I recall. In the general abstract algebra courses the focus of study was groups rings and fields, but the specific group and field theory courses extended the material without covering much of the same ground. I remember the fields course had a lot about field extensions in it, and was a prerequisite for the Galois theory course. This was all a few years ago now though, so my memory is a little fuzzy.

 

[R.P.F.]

There was very little overlap as I recall. In the general abstract algebra courses the focus of study was groups rings and fields, but the specific group and field theory courses extended the material without covering much of the same ground. I remember the fields course had a lot about field extensions in it, and was a prerequisite for the Galois theory course. This was all a few years ago now though, so my memory is a little fuzzy.

Are algebraic function fields and valuation theory also covered in the field course?

I have to say I'm very jealous. I don't like the gen-ed requirements in the states at all. I pay a huge amount of tuition here in the states and yet couldn't quite get the education I (at least I) want.

 

[dcpo]

Are algebraic function fields and valuation theory also covered in the field course?

I have to say I'm very jealous. I don't like the gen-ed requirements in the states at all. I pay a huge amount of tuition here in the states and yet couldn't quite get the education I (at least I) want.

I think algebraic function fields were touched on. I remember studying valuations at some point but I can't remember if it was in this course or elsewhere.

I'm glad I didn't have to do non-math courses, though I'm also glad I was given the option of doing some in my third year.

 

[feuxfollets]

This kind of course load is pretty standard in Europe. For my undergrad in the uk I took

In the first year:
2 courses of linear algebra, 1 of general abstract algebra, 3 of analysis, 2 of geometry, 3 of calculus, PDEs, 2 of probability, stats, and dynamics (all compulsory).

In the second year:
More linear and abstract algebra, more analysis and more ODEs (these were all compulsory), group theory, field theory, Lebesgue integration, number theory, topology, multivariable calculus, calculus of variations (these were options, of which I had to choose a set number).

In the third year:
Galois theory, Banach spaces, logic, set theory, algebraic topology, and a course on the philosophy of maths (all options).

You took 14 math classes in your first year? How is that even possible?

 

[dcpo]

You took 14 math classes in your first year? How is that even possible?

Well, as far as I'm aware there's no global standard for how much material goes into a 'class', so a theoretical answer could be that some of the classes I took were very light. The modern syllabus for the course I did is http://www.maths.ox.ac.uk/system/files/attachments/DRAFTMods_2011-12.pdf, and it looks similar to what I remember from nearly ten years ago, so you can check for yourself.

 

[Dens]

In my university pretty much anything math a natural science major would know, you should know. The rest are all considered "electives" because our school is small and courses like topology or number theory isn't offered every year

 

[Toph_fan]

Ugh, I wish I lived in Europe. First we have sucky high school math education and then we have to deal with all these geneds in college.

 

[homeomorphic]

I was notified that these are all the proof-based pure math classes offered at my university for the undergrad pure math degree:

Linear Algebra
Advanced Calculus
Foundations of Geometry
Elementary Number Theory
Complex Analysis
Abstract Algebra

Since the essence of pure math is proofs, do these courses pretty much make up a complete undergrad pure math experience? (If so, what do you think is the best way to prepare for these courses? Or, if not, what additional courses/topics would you include?)

Mainly, it's missing topology. Also, there should be more math electives that you can take, like graph theory, topics courses, differential geometry. If you added a few more of those, it would be like a typical math program in the US. Maybe you could do some reading courses and catch up.

Or, you can always teach yourself. There's a newer grad student here who is very advanced who came from a small school without too many options.

General education requirements aren't such a bad thing. It's not good to be so one-sided. Actually, there was some study on engineers that concluded that the ones who took humanities courses had more flexible thinking or something like that. I don't remember it very well.

As a graduate student, I feel more and more ignorant about the world outside mathematics. I have trouble following the news, etc. because math is just a handful.

 

[R.P.F.]

General education requirements aren't such a bad thing. It's not good to be so one-sided. Actually, there was some study on engineers that concluded that the ones who took humanities courses had more flexible thinking or something like that. I don't remember it very well.

As a graduate student, I feel more and more ignorant about the world outside mathematics. I have trouble following the news, etc. because math is just a handful.

It's good to have the option to take classes outside math, but when you make it a requirement, it can be annoying. It's not fun to shovel a humanities class down your throat when you have a stressful semester filled with math classes.

 

[nonequilibrium]

Wow. Algebraic geometry as an undergrad? Good for you! And also projective geometry as a single course? Sounds intense. I think the area of modern math that most frequently utilizes projective geometry is the study of elliptic curves.

I should emigrate to Belgium right away.

I also studied an undergrad degree in math in Belgium, and I had less math courses than Micro (went to another university, of course), so think twice before moving (although I'm quite content about my education). The math courses (math majors here were also required to take physics classes for example) I took are (in quasi-chronological order)

• Calculus I/II/III
• Linear Algebra
• Proof and Reasoning
• Statistics I
• Geometry I (Euclidean and Affine)
• Analysis I (Real Analysis + Metric Space)
• Differential Equations
• Algebraic Structures (general intro to concepts like groups etc)
• Abstract Algebra I (groups, rings, fields)
• Probability
• Geometry II (Projective, Algebraic Curves, Intro. to Diff. Geo.)
• Analysis II (Multivariable, Lebesgue, Banach, Wavelets)
• Numerical Math
• Mathematical Introduction to Fluid Dynamics (*)
• Statistics II
• Topology
• Complex Analysis
• Abstract Algebra II (Galois, Sylow, Presentation theory)
• Number Theory

I count 21. Depending on one's criteria I could also add "Mathematical Methods in Physics", where I (albeit superficially) learned about Stochastic Processes and Representation Theory.

(*) Despite the name no physicists ever took it; it's an applied math class, and a compulsory one at that.

 

[Jokerhelper]

To be fair, I'm pretty sure in most places in Europe students take an extra year in high school, while their university degrees are usually 3 years long (13 years of primary and secondary school + 3 years of university, compared to 12 years + 4 years of university in North America). So in a way, their last of high school is sort of equivalent to our year of general studies. Because of this, their degree programs tend to be more focused on their majors.

 

[Mépris]

To be fair, I'm pretty sure in most places in Europe students take an extra year in high school, while their university degrees are usually 3 years long (13 years of primary and secondary school + 3 years of university, compared to 12 years + 4 years of university in North America). So in a way, their last of high school is sort of equivalent to our year of general studies. Because of this, their degree programs tend to be more focused on their majors.

As far as I know, one starts primary school in the year they turn 5. Then there's a total of thirteen years of schooling, meaning that one ends high school at 18. In the US, 5 year olds start at kindergarten, yes?

 

[homeomorphic]

As far as I know, one starts primary school in the year they turn 5. Then there's a total of thirteen years of schooling, meaning that one ends high school at 18. In the US, 5 year olds start at kindergarten, yes?

Kindergarten is kind of like day-care. Not too much happens there.

 

[chiro]

Kindergarten is kind of like day-care. Not too much happens there.

 

[nonequilibrium]

I'm from Europe Jokerhelper, and I didn't have an extra year in high school, nor do I know of any other country in Europe that does.

EDIT: didn't see the other posts but oh well