Undecided on the last unit of my math major

[Floatzel98]

Hi all,

I'm a bit undecided what to pick as my last unit for my math major. I'm about to the the 3rd year of my degree. I'm a physics and math major, although I'll be continuing with physics into honours (and beyond). I'm all set with my physics units. There isn't much leeway in accelerating any of those units anyway. I've currently done all available physics units, The only physics units I'm not doing this year is a Photon, Synchrotron and Optics unit and a Condensed Matter unit, as I'm just not interested in those topics anymore. I've also completed all the math prereqs for all physics units (calc 2, calc 2 and intro DEs).

The math units I have completed so far are:
Calc 2
Calc 3 (Vector Calc)
Intro DEs (ODEs and PDEs)
Linear Algebra
Abstract Algebra (Intro to group theory + number theory)
Complex Analysis and Integral Transforms
Advanced ODEs (BVPs and Dynamical Systems)

And so far this semester I know I am going to take Differential Geometry. With all that I'm left with choosing one unit. Basically the choices I have are between:

Topology
Real Analysis
Advanced PDEs
Fluid Dynamics
Comp Maths

I am interested in Topology, but it is (obviously) a rigourous, proof based course. Although Real Analysis is not a direct prerequisite for it, I'm hesitant to take it without having done it. I have ended up disliking the other 'pure' math units I have done so far mainly due to the fact that I'm so bad at the proof work. Most of the time I can follow proofs, but have never been able to do them myself. That's the reason I haven't done Real Analysis yet. It's treated as the hardest math unit at uni and I'm too scared to take it haha. Also I don't really want to go on to Topology without having done Real.

PDEs seems to be the most logical choice to go alongside Physics, however I'm just not fully interested in doing it. With all the other DE units I've already done, I just feel bored. Content-wise, the only reason I would do it is because it is more easily applicable to physics than the other units. It seems to just go over everything we did in intro DEs (PDEs) again but with a proper explanation for everything. It is quite an easy unit apparently, which is something I am taking into consideration because I am looking to increase my GPA and WAM a bit this year (being my final year).

Ideally my goal is to go into some sort of theoretical physics. At the very least I want to be able to study QFT and GR at some point. What would you recommend based on that? What would you recommend I take? I have some people telling me that I won't ever understand Diff Geo and GR properly without Topology. I know the content in Real won't be directly applicable in the future, but I feel the skills from it will be useful. Most of my friends following the same path as me have done Real already, which is another reason I feel I need to take it. However the reality is I don't want to take it because I'm sacred of how hard it will be. But I also don't want to take Topology without Real because it will be just as hard. PDEs is the easy unit, but I'm not very interested in doing it. Technically I've already gone over the content in the unit before anyway. Are PDEs that important that it would be useful to do an advanced course on them?

Thanks for reading. Any help or advice would be great!

 

[lekh2003]

At the very least I want to be able to study QFT and GR at some point. What would you recommend based on that? What would you recommend I take? I have some people telling me that I won't ever understand Diff Geo and GR without Topology.

I've never been to university. I've never looked to deep into university, but this seems like a clear-cut problem here. You are afraid of taking something difficult and ultimately lowering what could have been a higher GPA if you took PDE.

I can't say I know for sure what all these subjects involve, but the harder subject will definitely help you considering that people have advised you that Topology will aid you in GR. If you want to do GR, don't be afraid to take what is necessary for it. Don't worry about your GPA as much, but more about what will be a better skill when actually doing Physics as a career.

However the reality is I don't want to take it because I'm sacred of how hard it will be. But I also don't want to take Topology without Real because it will be just as hard.

Try.

PDEs is the easy unit, but I'm not very interested in doing it.

Then don't take it.

I might be terribly incorrect about all this, so please listen to some more experienced members as well. But here are my two cents.

 

[Stephen Tashi]

You mentioned your outlook on every course except "comp math". What's the deal on that course? Is it taught by the physics dept, the math dept, or the computer science department? How do you like computer programming?

 

[Floatzel98]

I've never been to university. I've never looked to deep into university, but this seems like a clear-cut problem here. You are afraid of taking something difficult and ultimately lowering what could have been a higher GPA if you took PDE.

I can't say I know for sure what all these subjects involve, but the harder subject will definitely help you considering that people have advised you that Topology will aid you in GR. If you want to do GR, don't be afraid to take what is necessary for it. Don't worry about your GPA as much, but more about what will be a better skill when actually doing Physics as a career.

Yeah I see what you mean. I definitely am hesitant in taking it due to the effect it could have on my scores. Another reason I'm hesitant is because back in first year I failed an elective unit I disliked for being too hard, because I just stopped doing work and study for it. I'd like to think I'm mature enough to not let that happen now (and again), but its always in the back of my mind now which makes me instinctively pick units that I won't hate. (I'm not, and I haven't, avoided hard units but the stigma behind the difficulty of Real Analysis and the fact that I am not very proficient at proofs, which is the entire unit, have made me avoid it).

Try.
Then don't take it.
I might be terribly incorrect about all this, so please listen to some more experienced members as well. But here are my two cents.

I do agree with you. At this point I'd probably just like to know if doing Real Analysis is worth doing (for a physics student / in general) / Is doing Topology without Real reasonable?

You mentioned your outlook on every course except "comp math". What's the deal on that course? Is it taught by the physics dept, the math dept, or the computer science department? How do you like computer programming?

Yeah, I listed it because it was an option, but I don't really plan to take it at all. It is a math unit though; Introduction to Computational Mathematics. Its just a bunch of numerical methods using MATLAB. I can definitely see the usefulness of the course, but my Adv ODE course basically had a small chunk of this stuff in it and I absolutely dreaded it (at least in the setting it was taught). Also the PDE course has some numerical content in it, which is another reason I don't want to take it. I did do a research project last year on numerical methods relating to the Schrodinger equation, which was actually fun. But I don't think I would want to, or need to, take an actual class on it.

I didn't talk about Fluid either, but I don't plan on taking it. It was just another option. I have had friends take it who say it was super hard and not very well run. I'm not entirely interested in the field and am not planning on going into any fields like plasma physics or any astrophysical fluid fields.

 

[Dr. Courtney]

Any inkling what employment or further education you might pursue after graduation? Different choices may be better based on different future paths.

 

[ZapperZ]

But the more obvious question here is, has the OP talked to his/her Academic Advisor on this?

Zz.

 

[Floatzel98]

Any inkling what employment or further education you might pursue after graduation? Different choices may be better based on different future paths.

At the moment its looking like I'm going to try to get into a PhD program after honours (Starting 3rd year this year and honours next year).

But the more obvious question here is, has the OP talked to his/her Academic Advisor on this?

Zz.

Yeah I've asked my previous supervisor, course coordinator and some lecturers I know about this before (all physicists, not mathematicians). With regards to PDEs, most say to do it, or at least that it's super useful for physics. I'm not sure they actually know the specifics of the unit or are just suggesting that PDEs are useful in general. One professor has said to me "We all need to solve PDEs, but it sounds too luxurious to spend a whole semester on them".

Most of the don't of them recommend to do a real/functional analysis course, or at least say that a physicist does not need to do them. But most of them don't think it matters what I do at all because most of the time the relevant math will show up in the physics classes themselves. Unfortunately that never really helped me make a decision.

But I did email the coordinator of the Topology unit, asking how I would cope without having done Real Analysis. As expected he said that the course will reflect the style and content of real analysis more than any of the other courses I have done and that more effort would be required without having previously taken it. I think based on that I'd rather do Real Analysis. Might as well start from the basics, build up the foundations. At least I will have a better chance of learning Topology in my free time after going through Real. It also sets me up to do a functional analysis course next semester (if i don't fail or end up hating math haha)

Now I got to decide PDEs vs Real Analysis

 

[bhobba]

I was in your position. I however knew I was going to work as a computer programmer and PDE's or any of the other subjects you mention was available to me, plus others. Now even though I was going into computing I knew it wasn't likely scientific computing, so Computational Fluid Dynamics wasn't much use (they combined the two - comp math and fluid dynamics at my school - we all had to do basic numerical methods in year 2 so it was mostly applying computational methods we knew to fluid dynamics). Here is how I made my choice. I wanted to do analysis 2 (I was weird - I liked analysis - most don't) for which you needed analysis 1 (it was on Hilbert spaces, Lebesgue integration and such) but you need it for analysis 2. However I knew the importance of PDE's for physics which at the time I wasn't really into - but thought - I should keep my options open. What I did was go along to the Analysis 2 teacher, who I knew fairly well from other subjects he took me for. I said could I be excused from Analysis 1 as a prerequisite for Analysis 2 if I studied over the break - he said - you are a good student - OK I know you will acquire the necessary knowledge. So I took PDE's and self studied Hilbert Spaces etc. In physics only for some advanced work like Rigged Hilbert Spaces etc do you make use of analysis, but I liked it. I personally would self study analysis as needed and do PDE's. You can also teach yourself computational methods - mostly its just hooking together programs others have written from what people tell me these days. Topology - great if you are into pure math - but for physics - again of use in advanced work but isn't as widely applicable to physics as PDE's.

I would do PDE's and self study the other stuff as required.

BTW analysis isn't hard - you just need to get used to the epsilonic way of thinking. Most don't like learning an entirely different way of thinking, so avoid it like the plague. At my school you had to do introductory analysis year 1 - everyone hated it except me - I loved it because it explained things if you actually think is a real issue like Zeno's so called paradox. They removed it after I finished my degree - it became a 3rd year option - I thought it sad. But as you come across issues you need it for you will appreciate it, and with a bit of motivation will likely find it more bearable and eventually see it's value in things like Hilbert Spaces necessary to get a teal understanding of QM. BTW you need PDE's for that as well.

I studied GR using a number of sources such as the mathematically more rigorous text Wald and never did study topology. Many years ago I got a book on it and picked up the basics but I have to say it never reared its head much in what I studied later, so much so I pretty much have forgotten it all.

IMHO for the mathematics that actually explains some rather interesting things in physics I would watch the following:


I contains stuff you won't finds elsewhere.

In analysis you define carefully convergence of sums.

He gets around it very simply.

Say you wanted to show L =.9999999999... is 1 its so simple its trivial

10L = 9.9999999999...
10L-L = 9L = 9 or L=1.

Want to show that a recurring decimal is rational

Let L=.I/10^n + I/10^2n +++ where I is the repeating integer

10^nL - L = I or L = I/(10^n -1)

How do you define L= 1 -1 +1 -1 ...

L = 1 - L or L = 1/2

Its interesting stuff.

Thanks
Bill

 

[Dr. Courtney]

At the moment its looking like I'm going to try to get into a PhD program after honours (Starting 3rd year this year and honours next year).

That's a start, but applied math or pure? Theoretical physics or experimental? If theoretical, more pencil and paper analytical or computational?

 

[romsofia]

Analysis is a waste of time for what your goals are, so is topology. I'd pick Fluid dynamics if I were in your shoes because regardless of what field you want to go into, you still have your qualifying exams. Fluid dynamics is a tough subject, but very rewarding. The concepts you can learn from it can be applied to ANY physics field you go into, and the math is tougher than GR, QFT, etc. Even in GR, you idealize matter to be something called a "perfect fluid" most of the time.

So I'd pick fluid dynamics.

 

[Floatzel98]

snip

Thanks
Bill

Wow thanks for the reply. It really helps. They way you put it, it does make sense to go with PDEs and then study analysis in my own time. At least then I can go at my own pace, without all the added pressure of a class.

That's a start, but applied math or pure? Theoretical physics or experimental? If theoretical, more pencil and paper analytical or computational?

So far I've done more applied math, than pure. I want to go into theoretical physics, not experimental. Not too sure what beyond that though. The research projects I have done have incorporated analytical and computational components. I'd be happy and willing to go into computational fields, as I understand that a lot of research requires that. However I still don't think I would want to do the comp math unit as I'm just not interested in that kind of content. Plus one of my physics units this semester is computational physics, so I don't feel the need to do the other unit.

Analysis is a waste of time for what your goals are, so is topology. I'd pick Fluid dynamics if I were in your shoes because regardless of what field you want to go into, you still have your qualifying exams. Fluid dynamics is a tough subject, but very rewarding. The concepts you can learn from it can be applied to ANY physics field you go into, and the math is tougher than GR, QFT, etc. Even in GR, you idealize matter to be something called a "perfect fluid" most of the time.

So I'd pick fluid dynamics.

It does sounds compelling. I understand your argument on how it has applications in a large range of fields. My electromag professor last semester does work on quantum vortices and used a lot of fluid mechanics analogues to aid his explanations.

I have done some more looking around in the handbook for my uni and found that an astrophysics unit on relativity (and a bit of cosmology) can be accredited to the math major. From what I've read of the unit it goes into mainly special relativity and a basic intro to GR (Tensors, Field equations and some metrics). I never had enough room to do this unit on the physics side of my major (especially because it is an astrophysics unit), but since I can use it as a unit in the math major it seems perfect. I'm pretty sure its the correct decision to do this unit instead of another math? I'm interested in the content unlike PDEs and its not super difficult unlike Real (i.e its just not analysis. I'm not expecting it to be easy, it just falls in the category of difficulty that I can deal with). One of the reasons I'm doing Diff Geo as well is to better understand GR, so I might as well start learning it if I have the chance.

Any thoughts on that?

Thanks for all the previous help as well though. I'll probably still end up trying to go through some analysis on my own some time, as I'm still interested in it.

 

[bhobba]

Any thoughts on that?

GR is actually simple - its the proofs and understanding tensors that's hard.

Here is the theory - particles move along geodesics according to the principle of maximal time ie you maximize ∫dτ = √Guv dxu dxv. Space-time curvature is Euv = Tuv. Euv is a measure of curvature called the Einstein Tensor and assuming some reasonable things follows from Lovelock's theorem.

Its very math heavy - you may not like it.

Thanks
Bill