Please help me select my undergrad subjects [maths]

[AutumnWater]

Hi all, I’m about to begin yet another freshman year this year majoring in mathematical economics (I know I wanted to major in mathematics but this is the closest I can get when taking into account prior learning credits being efficiently fitted into my new degree.)

A little about myself:

I did one semester of university maths, consisting half of linear algebra and half of single variable calculus a few years ago, the rest of my studies were across business/finance/economics and I have since taken gap years to work and play so I haven’t finished my undergrad yet still.

During my initial exposure to university level maths, I found limit problems and convergence/divergence topics most interesting, whilst numerical approximations, and hand computing matrices were rather boring albeit exploring techniques in integration was also fun. Linear algebra seemed more fun during analytical stages rather than computational stages. And I’m pretty weak at hand drawing 3D conics for multivariable calculus (contour maps or anything requiring 3D visualization are slow for me to catch onto, not sure if this also means I’ll be weak at visualizing linear transformations.)

I also didn’t do much about sigma notation on sequence/series based maths in high school so in stats it was at first a shock to me, but this increased my interest to self study for them and now I’m comfortable with them, I also liked how maclaurin and taylor series were explained.

I’ve read subject reviews from student forums about how the alumnis found their maths subjects, and the majority of the opinions were that vector(multivariable) calculus was an exceedingly “applied” unit rather than pure, and was therefore boring for them. I don’t know anything about how I might feel towards that opinion nor about whether I’m going to be able to have a stronger focus in Applied or Pure maths.

Given my current situation, and that I plan to major in stochastic calculus for both undergrad and possibly Msc in maths later on down the track with measure theory in mind as well, what should be my sequence of taking the following subjects in order to maximize understanding/ease of learning curve, and what might be some good co-requisites to take together? And which subjects can more easily be self-learned so I don’t have to enroll in them? And from non-stats related math branches, what subjects am I missing that are of relevance?

I can fit a total of 10 maths subjects from the following list into my course before I graduate:

Multivariable(vector) calculus 2nd year

Real analysis 2nd year

Probability 2nd year

Mathematical statistics 2nd year

DE with modelling 2nd year

Linear algebra with applications 2nd year

PDE 3rd year

Intro to computational mathematics (numerical analysis) 3rd year

Advanced ODEs 3rd year

Complex analysis 3rd year

Random processes in the sciences and engineering 3rd year

Time series and random processes in linear systems 3rd year

Financial mathematics 3rd year

Are there anything redundant from these 3 units that need not be repeated:

Financial mathematics

Random processes in the sciences and engineering

Time series and random processes in linear systems

And for the following subjects, which ones are relevant to Measure theory and/or have a probabilistic emphasis, when should I take them after the initial list of subjects:

Algebra and number theory

Algebra and number theory II

Differential Geometry

Functional analysis

Applied mathematical modelling

Fluid dynamics

Anything I may have missed from the above list of subjects?

As for coding, I plan to learn in the next 10 years or so, in this following order of priority:

Matlab, C, R, C++, Python. (Should I change the priority in any ways?)

Thanks in advance!

 

[jambaugh]

Study Python before C++, it's a better language for learning OOP. I suggest learning C and Python together, also learning how to interface (calling C routines from python).

As to the math courses, hmmm... hard to narrow down to 10.
I think you'll definitely need the multivariable calculus and linear algebra. And given your field of course the probability and statistics. The real analysis courses should give you an idea of measure theory. I don't know what the various "random processes" courses entail so cannot advise you there. The differential equations courses give you lots of tools for understanding models.

I would also suggest you build up a tree of prerequisites from this list. For example you will need multivariable calc before you can take PDE's.

 

[AutumnWater]

Thanks for the tips on coding. I will endeavor to do so as suggested.

I have just met up with some university course advisors and decided I will do a double degree in Economics and Science, this way I will be able to do 17+ maths subjects alongside of the econometrics units shared by both degrees.

If I can get into the double degree, then I can do all of the stats/applied units offered by my university, and still have 3 units available for pure maths besides real analysis which I will already take.

Whilst unsure which 3, do these two look appropriate:

*Complex analysis
*Algebra and Number theory 1 (the first entry level of number theory course contains group theory, is that relevant to measure theory? I recall group theory and advanced linear algebra being the pre-req of a unit called "metric and hilbert spaces" at a different institute, which is their pre-req for measure theory)

*And 1 more to choose from:
Algebra and number theory II
Differential Geometry
Functional analysis

The random process subjects are both from the stats major:
Random processes in the sciences and engineering is about poisson processes, markov and hidden markov chains, martingales, random walks and other physical biological applications;
Time series and random processes in linear systems is about ARMA/ARIMA/kalman filter/spectral analysis

As for pre-reqs, pretty much everything in 3rd year are requiring either 2nd year multivariable calculus or both that and 2nd year linear algebra, plus the 2 random process and financial math courses require 2nd year probability and 2nd year mathematical statistics as pre-reqs.

But there are no other requirements horizontally across 2nd or 3rd year units from each other, between say, multivariable calculus, DE with modelling, Linear algebra, Probability, mathematical statistics; I do plan to take multivariable calculus and linear algebra this semester along with a unit called 'discrete maths for computer science' which is a first year unit and has a bayesian/logic and proofs oriented flavor; and next semester I will take DE with modelling, Mathematical statistics, and Probability. If however, things change, I'm unsure as to which one of mathematical statistics and probability should I take first, if not together.

Then at last for the 3rd year subjects,
I don't know which ones to take first.
Is it better to study the pure then applied then stats last, or in some other sequence?

 

[jambaugh]

Complex analysis is a beautiful subject with some amazingly useful calculation techniques. I'd be hard pressed to chose between the 2nd algebra, the differential geometry, and the functional analysis. It depends on your application and I'm not the best advisor away from physics. What science courses will you plan to take?

With regard to sequencing, although at the lower levels typically statistics precedes probability in its introduction I think at the higher level one should begin with probability theory then the statistics. You may also want to check the timing on course offerings. Depending on the size of the department, not all courses may be offered every semester.

But I don't know what else to tell you. Your major advisor likely has better perspective than I.

 

[AutumnWater]

At this stage, my major area of science is still mathematics. With a little bit of computational science involved as a sort of minor kind of thing.
My interest is in quantitative finance and is complex analysis related to signals processing? That's something that I want to look into as well that's why I picked it.

As for other applications, I'm trying to keep options open for data scientist/econometrics/actuarial careers.

 

[ttyu6]

I’ve read subject reviews from student forums about how the alumnis found their maths subjects, and the majority of the opinions were that vector(multivariable) calculus was an exceedingly “applied” unit rather than pure, and was therefore boring for them.

They are correct. There is an emphasis on computation rather than theory. You could also try to enrol in the advanced version of the course which is supposed to be more theoretical.

Multivariable(vector) calculus 2nd year
Real analysis 2nd year
Probability 2nd year
Mathematical statistics 2nd year
DE with modelling 2nd year
Linear algebra with applications 2nd year
Random processes in the sciences and engineering 3rd year
Time series and random processes in linear systems 3rd year
Financial mathematics 3rd year*

I'd say that these would all be core units. They are either required for a major or prerequisites for the third year subjects. Though I would suggest completing MTH3140 rather than MTH2140. The three third year probability courses have a tiny bit of overlap. As I recall they covered martingales in all of them but there wasn't significant overlap between them. Though I think most people take MTH3251 and MTH3241 at the same time(I did) so you may see some stuff covered in one subject earlier in the semester before seeing it later.

*If you don't manage to transfer into comm/sci (perhaps even if you do) I'd suggest taking ETC3510 instead of MTH3251. They are the exact same unit(you'll be in the same tutes/lectures as the maths students, do the same assessments). The only difference is the code, so this frees you up to take an additional maths unit.

PDE 3rd year
Intro to computational mathematics (numerical analysis) 3rd year
Advanced ODEs 3rd year
Complex analysis 3rd year

PDEs has quite a heavy workload. You'll be doing either tests or assignments every week. If you do MTH3051, learn Matlab before the start of the course. Otherwise you have to learn to program and learn the numerical analysis stuff at the same time. I've never taken MTH3060, however MTH3051 used to be a prerequisite so it might be a good idea to take it first and I've been told it is a very challenging subject. Complex analysis is fairly straight forward, though it helps a lot to have done real analysis. They tend to gloss over the theoretical stuff and some things weren't very well explained, such as open and closed sets. Of course if you've done real analysis it's much easier to pick up. I'd wish I'd taken it before Time Series though. Time series had a few integral transforms and I had no idea what I was doing until I did complex analysis the following year.

And for the following subjects, which ones are relevant to Measure theory and/or have a probabilistic emphasis, when should I take them after the initial list of subjects:

Algebra and number theory
Algebra and number theory II
Differential Geometry
Functional analysis
Applied mathematical modelling
Fluid dynamics

Perhaps the only two that would be useful are Functional analysis and Applied mathematical modelling. Functional Analysis replaced "Analysis and Topology" which I took and is typically a prerequisite for Measure theory. As for Applied mathematical modelling, I was told there was a stochastic modelling competent in the course.

I have just met up with some university course advisors and decided I will do a double degree in Economics and Science, this way I will be able to do 17+ maths subjects alongside of the econometrics units shared by both degrees. If I can get into the double degree, then I can do all of the stats/applied units offered by my university, and still have 3 units available for pure maths besides real analysis which I will already take.

Great choice, that's what I did :). Most of the econometrics course I did were great, but avoid the ones which overlap with mathematics, etc2520 and etc2440. I'd recommend etc2450 and etc3250 which require you learn to program in R. Other interesting ones I took were ETC3500,ETC3400,ETC3460 and ETC3410(though I hated the lecturer in this subject).

Whilst unsure which 3, do these two look appropriate:

*Complex analysis
*Algebra and Number theory 1 (the first entry level of number theory course contains group theory, is that relevant to measure theory? I recall group theory and advanced linear algebra being the pre-req of a unit called "metric and hilbert spaces" at a different institute, which is their pre-req for measure theory).

Complex analysis would be more useful if only for the integral transforms. I suspect the reason that "Group theory and linear algebra" is a prerequisite for "metric and hilbert spaces" is because of the linear algebra component and not the group theory component, since after all hilbert spaces are vector spaces.

I do plan to take multivariable calculus and linear algebra this semester along with a unit called 'discrete maths for computer science' which is a first year unit and has a bayesian/logic and proofs oriented flavor

Don't waste your time and money on the discrete maths unit. It's an easy HD but it isn't a prerequisite for anything and doesn't contribute to a major/minor. When I took it, lecturer called it "mathematics for vegetables" and was complaining about what a waste of time it was to the class in MTH3121 which he taught the following semester. If you want to do an easy unit, I'd suggest BIO2060. I took it last year, when it was offered for the first time. You learn stats stuff in there but it's very hands on. You have a one hour lecture, one hour tute and three hour lab where you will basically do stuff in R. The subject is focused on application to biological sciences but they teach the statistics as intuitively as possible in contrast to the mathematical nature in econometrics/stats units, which might be helpful if you take it before the more advanced econometrics/stats units.

next semester I will take DE with modelling, Mathematical statistics, and Probability. If however, things change, I'm unsure as to which one of mathematical statistics and probability should I take first, if not together.

I suggest taking all three if possible since they are prerequisites for further study. If you have to drop one I'd honestly drop DEs but of the other two I'd drop mth2222. MTH2232 can be used as a prerequisite(for ETC3400 and personally I suggest doing MTH2232 over ETC2520 which is poorly taught) and it has a crash course in probability before you start the stats stuff(they don't expect you to have done probability before). The reason I suggest dropping DEs is that the probability and stats course will be very helpful for your econometrics courses.

The best advice I can give you though is to talk to the advisors. Not the admin staff in the faculty but the ones in charge of subjects. For statistics I believe it's Kais who is one of the probability lecturers(and one of the best lecturers you'll have). They'll be the ones who can give you the best advice.

 

[AutumnWater]

Wow what do you know, it's a small world isn't it ;) really appreciate such elaborate advice.

PDEs has quite a heavy workload. You'll be doing either tests or assignments every week. If you do MTH3051, learn Matlab before the start of the course. Otherwise you have to learn to program and learn the numerical analysis stuff at the same time. I've never taken MTH3060, however MTH3051 used to be a prerequisite so it might be a good idea to take it first and I've been told it is a very challenging subject. Complex analysis is fairly straight forward, though it helps a lot to have done real analysis. They tend to gloss over the theoretical stuff and some things weren't very well explained, such as open and closed sets. Of course if you've done real analysis it's much easier to pick up. I'd wish I'd taken it before Time Series though. Time series had a few integral transforms and I had no idea what I was doing until I did complex analysis the following year.

So MTH3051 before MTH3060, but should MTH3060 precede PDEs? And Complex analysis before Time series gotchya.

Perhaps the only two that would be useful are Functional analysis and Applied mathematical modelling. Functional Analysis replaced "Analysis and Topology" which I took and is typically a prerequisite for Measure theory. As for Applied mathematical modelling, I was told there was a stochastic modelling competent in the course.

So should the three 3rd year probability courses be done before or after Applied Mathematical Modelling and Functional Analysis?

Don't waste your time and money on the discrete maths unit. It's an easy HD but it isn't a prerequisite for anything and doesn't contribute to a major/minor. When I took it, lecturer called it "mathematics for vegetables" and was complaining about what a waste of time it was to the class in MTH3121 which he taught the following semester. If you want to do an easy unit, I'd suggest BIO2060. I took it last year, when it was offered for the first time. You learn stats stuff in there but it's very hands on. You have a one hour lecture, one hour tute and three hour lab where you will basically do stuff in R. The subject is focused on application to biological sciences but they teach the statistics as intuitively as possible in contrast to the mathematical nature in econometrics/stats units, which might be helpful if you take it before the more advanced econometrics/stats units.

The reason I have the discrete math subject on my course plan is because I thought the science faculty requires me to do a 6th (5th if excluding ETC1000 counting as STA1010) 1st year science subjects as course requirement. If that requirement can be bent in a double degree then I will definitely not do that. Although I was hoping it'll help me with real analysis with the set theory and logic stuff involved.

Great choice, that's what I did :). Most of the econometrics course I did were great, but avoid the ones which overlap with mathematics, etc2520 and etc2440. I'd recommend etc2450 and etc3250 which require you learn to program in R. Other interesting ones I took were ETC3500,ETC3400,ETC3460 and ETC3410(though I hated the lecturer in this subject).

If I was to pick 5 out of ETC2450,ETC3250,ETC3400(compulsory),ETC3460, ETC3500, ETC3410, I'm not sure between ETC3410 and ETC3500, but seeing as ETC3410 will build foundations for Applied econometrics 2 in econometrics honours, I might be inclined towards that, and if I can't get rid of Discrete math from science course requirement then I might have to drop 1 more ETC unit from here...

 

[AutumnWater]

Wow what do you know, it's a small world isn't it ;) really appreciate such elaborate advice.

The reason I have the discrete math subject on my course plan is because I thought the science faculty requires me to do a 6th (5th if excluding ETC1000 counting as STA1010) 1st year science subjects as course requirement. If that requirement can be bent in a double degree then I will definitely not do that. Although I was hoping it'll help me with real analysis with the set theory and logic stuff involved.

Actually I didn't see the phrase "Note 8: Students in a double degree course can replace one level 1 science listed unit with a level 2 or level 3 science listed unit.".
So that settles that, I just withdrew from MAT1830 and am enrolling in either ETC2410 or ETC2450, which should I do first? is ETC2410 better done after MTH2232?

 

[ttyu6]

Wow what do you know, it's a small world isn't it ;) really appreciate such elaborate advice.

It's really no problem at all.

So MTH3051 before MTH3060, but should MTH3060 precede PDEs? And Complex analysis before Time series gotchya.

I don't think it matters if you do MTH3060 or MTH3011 first since they don't have much in common. All you really need is MTH2032.

So should the three 3rd year probability courses be done before or after Applied Mathematical Modelling and Functional Analysis?

I'd take the third year first semester probability courses before Applied Mathematical Modelling since some of the material may come up but I'm not sure how much help they'd be. It doesn't really matter when you take functional analysis since you won't really be using any of the material in any of the probability courses. Occasionally some stuff from analysis comes up but it's not significant enough that you'd need to have seen it before hand.

If I was to pick 5 out of ETC2450,ETC3250,ETC3400(compulsory),ETC3460, ETC3500, ETC3410, I'm not sure between ETC3410 and ETC3500, but seeing as ETC3410 will build foundations for Applied econometrics 2 in econometrics honours, I might be inclined towards that, and if I can't get rid of Discrete math from science course requirement then I might have to drop 1 more ETC unit from here...

Definitely take ETC3410 over ETC3500(and as I recall it is a prerequisite for econometrics honours). The material in ETC3410 is much more challenging than ETC3500 which is a lot more hands on(I suspect because it's co-taught with the marketing department). In terms of difficulty I'd say ETC3400>ETC3410>ETC3460>ETC3500.

ETC3400 uses Casella and Berger's "Statistical Inference" and Greene's "Principles of Econometrics" which are fairly challenging.

ETC3410 uses Wooldridge's Intro econometrics text(or did when I took it), though I recommend his more advanced "Econometric Analysis of Cross Section and Panel Data". The intro text doesn't use matrices, but matrices are used in the course and the advanced text. I prefer the advanced version(since the intro text is verbose) which you can tackle if when you've done mth2232,etc2410 and mth2021, sadly I found this out after I had finished the course. Solutions for both texts are online and I suggest doing plenty of problems from the texts since when I did the course there were no practice questions. Instead, the labs primarily consisted of a short introduction of the material and typing commands in statistical software. I do highly recommend getting in a group to solve the assignments so talk to people in your tute(or wander around the computer labs after 3...true for most of the 3rd year econometrics courses). Lastly, you don't get past exam solutions and may not get your assignments back in time for the exams but the lecturer is supposed to be very approachable so if you're stuck go see him in office hours. Some of the material in this course appears in ETC3500 and ETC3460 so it might be worthwhile taking it first.

ETC3500 has a group assignment starting mid semester so do get to know classmates so you don't pick people who are deadweight.

ETC3460 last year when I took it had removed an introductory finance unit as a prerequisite. I honestly found the first half of the course which was finance oriented quite challenging since I hadn't done finance before, but the lecturer is very engaging and knowledgeable. The tutors made a website for the course http://www.etc3460fineco.com .

So that settles that, I just withdrew from MAT1830 and am enrolling in either ETC2410 or ETC2450, which should I do first? is ETC2410 better done after MTH2232?

It won't really matter when you do ETC2450, but I'd do ETC2410 after MTH2222 and MTH2232. It's not completely necessary but I didn't initially understand some of the things they glossed over in the course such as conditional expectations and unbiased estimators(and log correction), which are covered in depth within the maths courses.

If you aren't already familiar with proofs, then I'd suggest teaching yourself. Something like Vellemen and Spivak's Calculus text with Pete Clark's supplementary notes would be a good idea.

 

[AutumnWater]

If you aren't already familiar with proofs, then I'd suggest teaching yourself. Something like Vellemen and Spivak's Calculus text with Pete Clark's supplementary notes would be a good idea.

Thanks, I'll have a look at those too.
I'm currently reading Victor Bryant's "yet another introduction to analysis" in preparation for Rudin's text. When reading those kinds of texts, I sometimes feel guilty about looking at the solution too soon for their proof exercises, but other times feel stupid for not doing so. Is it generally bad to look at a solution of a proof and try to replicate the methodology onto other proof problems? Is it worth the wait of a couple hours or even days or weeks to figure something out myself rather than looking at the solutions?

And I think my taste in the type of proof questions are a bit biased too, which can affect my drive to think about things myself.
For instance, the proof of square root of 2 as an irrational number by way of contradiction was way less interesting and hence harder to replicate than if I was to try out an Epsilon Delta Proof of a limit. It just felt like I'm getting owned by elementary arithmetic when failing to do such proofs on the spot.

 

[ttyu6]

I'm currently reading Victor Bryant's "yet another introduction to analysis" in preparation for Rudin's text. When reading those kinds of texts, I sometimes feel guilty about looking at the solution too soon for their proof exercises, but other times feel stupid for not doing so. Is it generally bad to look at a solution of a proof and try to replicate the methodology onto other proof problems? Is it worth the wait of a couple hours or even days or weeks to figure something out myself rather than looking at the solutions?

I don't necessarily think it's a bad thing but obviously I try to refrain from doing so. Rather than look at the whole solution I'll often just read the first line or two, if I'm drawing a blank. That way I'm attempting to complete as much of the proof myself. As for replicating the methodology onto other proofs, that's something you should be doing for all proofs. For instance try to replicate the proof of the irrationality of the square root of 2 for the square root of 3. I usually wait a few days of thinking about tutorial problems, for instance, before I go off for hints/solutions. I do think that you should at least sleep on a problem before looking for help. If you aren't already aware of the Maths Learning Centre, it's located on the bottom floor of the maths building where your tutorials are held. You can get help from the tutors there for most maths subjects. As for Rudin, expect to slowly work through his proofs, some of them may take a while to digest. Here's a helpful guide I used to study. While it's focused on the algebra side, a lot of the stuff is applicable to analysis as well.

And I think my taste in the type of proof questions are a bit biased too, which can affect my drive to think about things myself. For instance, the proof of square root of 2 as an irrational number by way of contradiction was way less interesting and hence harder to replicate than if I was to try out an Epsilon Delta Proof of a limit. It just felt like I'm getting owned by elementary arithmetic when failing to do such proofs on the spot.

Could this be lack of experience? It took me a while to get into proofs for each of the different units and to develop the different way of thinking. Doing more proofs and understanding their structure rather than rote learning helps a lot. As for failing to do proofs on the spot don't worry it happens to every one. Happened to one of my tutors. He forgot how to do an epsilon delta proof took and it him more than 10 minutes to figure it out.