# Pure and Applied Math

1. Aug 4, 2009

### PieceOfPi

Dear PF members,

I am about to begin my junior year as a math major, and I feel like I made one observation. That is, if I want to study math at advanced level, I need to choose either pure or applied mathematics. I have also noticed that if I get a title of "mathematician," that implicitly implies that I am a pure mathematician, and if my research is leaning toward applications, I would be rather referred to as applied mathematician than just a mathematician. Notice that this is only my observation, and I could be wrong about this, so feel free to give me a counterexample to my observation.

For me, if I am going to study math in graduate school, I would like to study both pure and applied math. To me, applications interest me a lot--I find it cool that something like linear algebra, for example, can have so many applications outside of mathematics even though what you're learning in class is somewhat abstract (e.g. vector space with >3 dimensions, inner-product space, etc.). I find it really fascinating, especially because I was originally a science major (biochemistry) when I first entered college. And of course, I find the theory of mathematics to be beautiful--I just took a class in abstract linear algebra, and I really enjoyed that class. I also find some of the stuff in number theory to be beautiful as well (e.g. Fermat's Little Theorem, Chinese Remainder Theorem, etc.). So to be honest, I really like the theoretical and applied aspects of math, and in fact, I am afraid that I would get bored if I could only study one of those two.

So I was wondering what to study in grad school (if I am going to one), and looked up some of the websites of applied math departments. What I found out was that most of the applied math departments (in U.S.) seem to recommend taking ODE, PDE, numerical analysis, and etc during undergrad, but not many of them recommend taking courses like analysis, algebra, topology, or those types of rigorous math courses (in fact, I think Cornell was the only school that explicitly stated that they recommend analysis and algebra before grad school). This made me think that maybe I won't see many pure math in applied math department, which is disappointing to me. But as I stated before, if I go to a grad school, I would be only studying the pure math and not much applications (based on my observation). So my ultimate question comes down to this: If I want to study both pure and applied math, what should I do?

Any thought is appreciated--you don't really need to answer to my last question as long as you're replying me with something related to this topic in general.

2. Aug 4, 2009

### n!kofeyn

Hi PieceOfPi. That's a good question, and one that doesn't really have any specific answer, and one also that I've struggled with. I'll just talk, and hopefully something helps. At my school, there is a pure program and an applied program, but they are basically one in the same. This means that once you are accepted into one, it is very easy to switch over into the other program, and there are only some minor differences during the first two years in terms of course and qualifying exam requirements. I myself am very interested in doing some type of mathematical physics, in particular stuff in quantum field theory (which I don't know yet :), but I really like being a mathematician and knowing the pure side of things. So in my case, I am going to try and stay in the pure program while picking up some physics classes along the way (which may not count as coursework, I'm not for sure at this point), so that I may have the physical ideas behind the mathematics I want to do. There is that stigma of a difference between applied and pure mathematicians, but I think it's best to be kind of stubborn and ignore those who try to point it out. I think it depends a little on what field you'd like to apply the math to and what school you will go to, as they all treat their programs differently. If you want more information on the specifics of how my school does it, let me know.

In the end, I think it is a shame there is this distinction, but I feel like getting a pure math education while taking some applied and other science courses (i.e. not ignoring the applications) is the best route rather than being more applied and trying to pick up the pure math on the side. This just feels more natural to me and is what I am modeling my course of action upon. I think a good example of someone is undoubtedly a pure mathematician, but does some very unique and heavy applications is http://www.math.upenn.edu/~ghrist/index.htm" [Broken], one of the greatest pure mathematicians of the 20th century, who chose to study theoretical physics later in his career.

Right there are two examples of topology being applied, which is usually considered purely pure :). Theoretical physics uses higher algebra very heavily, among functional analysis and other things. So I don't like when people say applied mathematics they automatically think numerical analysis. For my example, I feel like what I want to do is on the border of pure and applied mathematics, but numerical analysis is lower on my needed tool belt than algebra, topology, and analysis.

Last edited by a moderator: May 4, 2017
3. Aug 5, 2009

### mordechai9

I can tell you the answer straightaway -- if you like both pure and applied math, then you belong in applied mathematics!

"Pure" mathematicians are characterized by an interest in mathematics, exclusively, in and of itself. To some degree, this requires an outright rejection of outside interests from science or anything else. If you read Hardy's "Apology", an essay from a classical "pure mathematician", you will find that the only mathematics which he cares about is pure mathematics. On the other hand, it's not that "pure mathematics" isn't interested in the outside world, it just chooses to ignore it.

If you go into applied math, you have to learn pure math because that's the foundation for all of it. If you go into pure math, however, you may never directly deal with anything relevant outside of mathematics again. Your work may still be highly relevant to other fields, but this would not be your concern.

4. Aug 5, 2009

### planethunter

I myself majored in pure math for bachelor's degree. I can tell you first hand that if you want to get into applied math with a pure math degree, do the pure math degree with applied electives if this is available. This is what I did so I was able to take number theory, analysis, abstract algebra while still taking linear algebra, fourier analysis, ODE, etc.

Some undergraduate math programs require you to take pure math courses as part of your degree, be it pure or applied. Just take more applied than pure and you will be able to get into either an applied or pure math graduate program. This way it keeps your options open.

5. Aug 5, 2009

### PieceOfPi

Thanks for your replies. It looks like the best thing I can do now is to take both pure and applied math classes as well as subjects that are outside of mathematics. Right now, I am thinking of taking upper-division computer science course, hoping that would give me a chance to see how math is applied in real world.

n!kofeyn: I really liked Robert Ghrist's website, especially the story about how he became a mathematician. Looks like he had a same issue about a department of applied math telling him that taking analysis is a waste of time. And interestingly, he got accepted to Cornell's applied math, which does recommend students to have been exposed to analysis and algebra during undergrad. I gotta check out more about Michael Atiya.

mordechai9: That's what I believe--I do believe that I need to know the foundation of math (i.e. analysis, topology, and algebra) in order to study applied math. The only thing that is concerning me is that many applied math departments don't explicitly say that I should take those pure-math courses.

planethunter: That's my plan for now--take as many analysis/algebra/topology as I can while taking courses like ODE, PDE, Fourier, stats, and numerical analysis. The math department at my school doesn't really have a tough requirement--you only need to take some core classes (Intro to DE, M.V. Calculus, linear algebra, and elementary analysis), and you only need to take 4 quarters of either pure math or applied math courses in order to satisfy the bare minimum of graduation. My plan definitely exceeds that bare minimum, though.

Here is my tentative schedule for the fall quarter:
Intro to Analysis (Text: Rudin, this is a year-long sequence)
Intro to Topology (Text: Munkres, this is a two-term sequence)
Ordinary Differential Equations (upper-div DE course, Text: Boyce/DiPrima, this is a stand-alone course.)
Computer Science III (The last term of the intro to CS sequence. I guess we'll be covering digital circuits, Boolean algebra, and basics of algorithms and data structures.)

Other options that I have this term are algebra, numerical analysis, econometrics (from what I've heard, this class is straight-up math, and no economics) and statistics (which is offered at the same time topology is offered). The school starts in September, which means I still have some time to make a change, so please let me know if you have any suggestion.

6. Aug 5, 2009

### Landau

What is the algebra class about? What text?

7. Aug 5, 2009

### PieceOfPi

Algebra is just a first course in abstract algebra. The text is Beachy/Blair's Abstract Algebra.

8. Aug 5, 2009

### mordechai9

If you want to know how math is applied in the real world, I would suggest some other alternatives instead of the CS course. Computer science is really useful/interesting/practical, but I think it should not be your first choice for this kind of outcome.

Instead, I would recommend some courses from engineering or physics, and in particular, if you could get into one of these courses, I think it would be really interesting for you. They might not be named this exact same way, but they should be entitled something similar:

1. structural dynamics or vibrations (usually in mechanical engineering, civil or aerospace engineering) -- provides lots of experience in solving ODE's and systems of ODE's that model physical systems

2. solid mechanics (usually in mechanical, civil, or aerospace engineering) -- excellent class in the basics of mathematical modeling using Newton's laws

3. statistical mechanics (usually in physics) -- extremely fundamental usage of probability theory and statistical theory

4. quantum mechanics (physics) -- solving pde's, probability theory, linear algebra, algebraic vs. analytic representations of PDE's

5. fluid dynamics (mechanical, aerospace) -- formulating PDE's from Newton's laws, kinematics, solving some basic PDE's, excellent examples of "physical assumptions" that you must make to solve any problem

6. electrodynamics (physics) -- elegant mathematical theory, with extremely challenging problems in vector calculus and integral computations

--

The first class (vibrations) is often sometimes listed as a basic "waves" type of class in the physics department, covering fairly similar material. If you could get into any of these courses, they would really give you a "close and personal" view of basic applied mathematics. These classes have more emphasis on physics/formulation/computation than what an academic would call "applied math", but still, they represent fundamental/basic motivation for things like ODE's, probability theory, PDE theory, and so on. Notice the relative lack of algebraic topics -- algebra rarely comes into play in this level of sophistication -- with the notable exception of quantum mechanics.

9. Aug 6, 2009

### PieceOfPi

That's not the only reason I'm taking CS courses though. I just took a few introductory CS courses, and found them to be pretty interesting in general (besides the mathematical aspects).

I might take a few upper-division physics courses someday (we don't have an engineering department at my school). I actually took the first-year physics class last year, and found them to be somewhat interesting, but I'm not really fond of labs, which kind of made me think I shouldn't major (or minor) in physics. But theoretical physics that you've mentioned does sound interesting, so I'll consider it if I have time.

10. Aug 6, 2009

### mrb

Just to throw this out there... This is a decidedly non-expert opinion, and my impression may be quite wrong, but: In my experience, applied mathematicians aren't actually researching math. They're using math developed by others to do research in the area of their application. Rather than "mathematician, who does traffic modelling," he is more along the lines of a "traffic researcher, who knows a lot of math."

11. Aug 6, 2009

### qspeechc

I think doing numerical analysis will give you a good idea of the kind of work applied mathematicians do. So try taking that if you can. If you like it, then check for courses in fluid dynamics or advanced numerical methods to see if applied maths is really your cup of tea. But if you think you might like applied math, I strongly recommend numerical analysis.
Oh and ODEs and PDEs are very important.

12. Aug 6, 2009

### PieceOfPi

That sounds right, especially if you take a look at the type of careers that SIAM describes in its http://www.siam.org/" [Broken]. That actually sounds interesting to me too, but then again, my current interest seems to fit somewhere between pure math and applied math. I wonder if there is anything called bridge math where I get to work in that gray area.

Thanks for your advice. I think what I will be doing in the first few days of the class is to just sit in all of the math classes that I am interested in taking (i.e. analysis, topology, ODE, algebra, and numerical analysis) to see which one seems most interesting to me. One area of applied math that I am currently interested is dynamical systems (or I guess this is sometimes called nonlinear dynamics), as it seems to have some interesting applications in fluid dynamics, biology, and economics, and the ODE course that I mentioned seems to cover this topic, but I can also see numerical analysis is also critical. But I still don't know much about this subject yet (I just took a course in discrete dynamical systems last quarter, and that's it.), as well as any other area of math, so I am still open to wide range of mathematics.

Last edited by a moderator: May 4, 2017
13. Aug 7, 2009

### mal4mac

Yes it is quite wrong :-) Applied mathematicians are .. er.. applied mathematicians, and applied mathematics courses are about applied mathematics, not traffic or whatever. Similar equations crop up in physics and traffic research. Applied mathematicians solve the equations without bothering about whether they describe cars or quarks. They are applied mathematicians if they use known techniques to solve the equations. They become pure mathematicians if they find new, proven, ways to solve the equations.

The new "solutions" may not come with adequate proofs, but seem to work in practice, so the "mathematician" uses them, but their purist mathematician colleagues disown them and call them theoretical physicists (or theoretical trafficers*, or whatever.)

*The course that Stringer Bell took at college?

14. Aug 7, 2009

### mal4mac

A quarter of my BSc degree involved taking applied mathematics courses. They were far and away the most boring courses. I liked the pure maths I did (1/8th of my course) and most of the physics (3/8). I liked pure maths because of the ..er.. purity. Seeing theorems and proofs of areas of mathematics beyond the usual popular examples (Pythagoras and all that...) was very interesting, especially in areas like group theory. I liked physics because it was about the real world! Applied mathematics seems to have neither advantage. The "equations to be solved" were not motivated by neat real world examples, and the "theorem, proof" purity was lost in the messy details. In summary, I would find out of you really like pure mathematics or physics and specialise in one or the other. If you do a physics degree and have to do courses in another subject, do pure mathematics not applied. Do anything but applied!

15. Aug 7, 2009

### PieceOfPi

Would you like to tell me what types of applied/pure math courses you took?

16. Aug 8, 2009

### mal4mac

Applied mathematics - endless ways to solve partial differential equations, calculus of variations, that kind of thing. After a couple of decades I can't remember much about it. I've blocked out the pain, but the memory of the pain still lingers.

Note, if the lecturer in applied maths had been better the course might have been interesting. She had no concept about presenting a 'larger view' of the subject, no idea about motivating people. There wasn't even a set textbook, she just put endless very hard equations on the blackboard and went full speed ahead through endless, tricky, solutions. You had to be very quick at copying down all the hieroglyphics as well 'cause she handed out no notes.

So your idea to sit in on all the classes is a good one. If the lecturer doesn't provide a good overview at the start, doesn't recommend one good textbook, avoids eye contact, starts by producing reams of complicated equations that you have to spend hours deciphering after the lecture, then look elsewhere!

Also talk to TAs, other lecturers, and people who have done the classes you are thinking of doing. Look at the pass/fail ratio. Do anything you can to avoid the rubbish courses. And be especially careful about courses you are interested in! I was really interested in astrophysics until I hit a really bad advanced course given by the worst lecturer I have ever encountered...

Check out websites that rate lecturers, like:

http://www.ratemylecturer.com/

17. Aug 12, 2009

### PieceOfPi

I see. Yeah that might be boring to take a class from a lecturer like that. At least, the math professors at my school are almost all pure mathematicians, so it's unlikely that they will just present complicated equations without explaining why that equation works and how that is relevant to the story. I've taken combinatorics and discrete dynamical systems here--both of them belong to the applied side--but both of the class included good amount of proofs, so we still get some mathematical rigors in what we're doing.

18. Aug 24, 2009

### amezcua

In the Book of Curves there was a description of the 4 dogs in a back yard ,describing the way the dogs would move if a rabbit was released.Each route would be like a spiral .as the dogs would be chasing a rabbit altering its course due to the next dog that sees it.(see book)---Thats pure maths. I turned that into Applied maths when playing Rugby.If I wanted to stop a player running along the touchline I roughly work out where he would be(near the goal line)and ran in a straight line.It was very noticeable that all the rest of the team followed a curve across the pitch. We won our last game 43 -0. Long time ago now .They had the hot shot sprinter and I`ve got short fat hairy legs.

19. Aug 24, 2009

### amezcua

In addition to previous posts I have seen International players getting this wrong as well.
Part two is an aside .If Olympic runners are so keen on personal best times ---why do they all lean back and slow down yards before they touch the tape?

20. Sep 8, 2009

### vegetto34

Does one recommend taking a pure mathematic course or two outside an engineering degree? I would like to understand that area of mathematics.