What should I do while in an undergraduate engineering course to keep a proof of the math knowledge I gain by self-study?

Question

I'm an undergraduate electronics and communication engineering student (sophomore) at an Indian university. I joined this engineering course because I found the areas of quantum computing/information and quantum engineering quite interesting, and I felt an EE background would help me later on to pursue higher studies in these areas.

Lately, although I still have great interest in those areas (quantum computing/information/engineering), I am finding myself very much interested in certain topics in mathematics and mathematical physics. Namely, mathematical foundations of quantum mechanics and quantum information (which involves learning a lot of extra math topics like functional analysis), differential geometry and topology (and their application in theoretical physics), statistical learning (I'm finding the application of statistics in machine learning quite interesting and have been reading quite a few books related to that) and discrete mathematics (graph theory and combinatorics).

The natural thing to do in such a case would be pursue a minor in mathematics. But, unfortunately, our university does not offer any minor degrees or dual major degrees. So, it's not possible for me to formally take extra classes in mathematics. Upon pondering a bit I realize that I might want to pursue my higher studies in some interdisciplinary area which involves knowing things from electronics engineering as well as from the rigorous mathematical physics and statistical learning (machine learning/data science/AI). I'm not sure if such an interdisciplinary area of study even exists at the graduate level (?). But I'm really enjoying learning the new things in mathematics and I don't want my spending time on learning these things go in vain.

So, in short, what would be the correct way to keep proof that I'm actually learning these extra things (so that I can show that I actually know these extra subjects/topics while applying for grad school) ? Should I participate in some research projects in these areas? (But then again professors don't seem to accept people who haven't taken formal courses into their research projects). If it were computer science, I could have taken online courses on sites like EdX and Coursera, for certificates. But for mathematics, no such site exists which gives out certificates based on completion of certain courses.

Answer 1

I feel sufficiently compelled to answer this as a recent math and physics major who completed his degree requirement.

In my undergraduate class, I was taking various combination of 4 math and physics units each semester -- 4 unit is the heaviest load one can take in a semester. At the top end they were partial differential equations, real analysis, Abstract Algebra, Topology and geometry, mathematical physics, Advanced topics in Quantum mechanics and atomic physics such as quantum mechanics operators, measurement theory, spin and orbital momentum, LS coupling, jj coupling, helium atom, fine structure, hyperfine interaction, atoms in magnetic fields, electron spin resonance, transition probabilities, astrophysics and condensed matter physics, dynamical systems, applied and computational modelling of physical systems. On top of these, I took various quantitative units in economics and did very well even in absence of the prerequisites.

, denotes a limiters of the different subjects/ units/ course/ modules

This is a 3 years standard bachelor but I completed it in 4 years. If you are prepared for the sheer amount of hard work, extreme burn out and a lower than average GPA, do it. Mine dipped slightly below a 3.0/4.0 but I had strong references and computational skills.

The natural thing to do in such a case would be pursue a minor in mathematics. But, unfortunately, our university does not offer any minor degrees or dual major degrees. So, it's not possible for me to formally take extra classes in mathematics.

Everything on your transcript is just ceremonial. What is more important is you knowing the subjects and actually being able to demonstrate it on a technical test/ chalkboard when called.

I'm speaking this as someone who has recently started collaborate on a research project with a professor while looking around for industrial opportunities.

If you're looking to work alongside research members, academics or professors, you may be asked to provide some insight as to what you have already covered in your time; in this case, you are free to draw upon what you have taken in your undergraduate curriculum and free time. People effectively wants to know what you know, not what you have taken.

In my limited experiences, professors are actually impressed with students who challenge themselves by taking advanced units, even if they consider you foolhardy.

Upon pondering a bit I realize that I might want to pursue my higher studies in some interdisciplinary area which involves knowing things from electronics engineering as well as from the rigorous mathematical physics and statistical learning (machine learning/data science/AI). I'm not sure if such an interdisciplinary area of study even exists at the graduate level (?).

While I am equally unfamiliar with such combination in academia, you may circumvent the lack of such opportunities in universities through independent study. Most research these days are cross functional across seemingly unrelated fields so much is dependent on the candidates to independently learn. In my short experiences with looking for opportunities in areas of data analysis and machine learning, the common litmus test is a request for candidates to undergo a technical test much like how developers are subject to technical tests despite their year of experiences.

In STEM, hardly anyone will give you the luxury of knowing something just because something came up on your transcript.

But I'm really enjoying learning the new things in mathematics and I don't want my spending time on learning these things go in vain.

If this is not a contradiction then there is insufficient data.

Answer 2

I'm really enjoying learning the new things in mathematics and I don't want my spending time on learning these things go in vain.

If you're enjoying it, it's not in vain. Also, mathematics is used in pretty much all areas of science, so no knowledge in maths will be useless, that I can promise you. For example, as an undergraduate student in mathematics I thought learning algebra and group theory was just a waste of time for me, that I just needed to pass (it was an obligatory course) and forget. However, now I'm doing a PhD in statistics and stochastic analysis, and I stumbled upon groups many, many times. You never know.

If it were computer science, I could have taken online courses on sites like EdX and Coursera, for certificates. But for mathematics, no such site exists which gives out certificates based on completion of certain courses.

There are in fact multiple online courses in mathematical subjects, especially statistics and machine learning. For example, Imperial College London offers a specialisation in Mathematics for Machine Learning (https://www.coursera.org/specializations/mathematics-machine-learning). Just browse for "mathematics" or a specific subject that interests you. Moreover, many lecturers put their notes and exercises online, it's an excellent way to learn. Just search for what is most interesting to you. You can also get a relevant textbook and do some self-study. Options are countless?
How to you prove you've done it? You don't need certificates, really! If you decide to apply for some academic programme or a job, what matters are your skills and knowledge, not certificates. I personally sometimes feel that I remember most from courses I didn't pursue certificates in. Why? Because I focus on what actually is relevant or difficult for me, not on solving many exercises that just give me "easy points".
Just pick a subject and a source, and start learning. You can even contact a lecturer for some advice (just e-mail them), they probably would be happy that someone finds their notes interesting. Whatever you do, good luck!

Answer 3

Another idea is to use MOOCs. Places like edx.org and coursera.org offer advanced coursework that might suit your needs (even if you already studied the materials and just breeze through for the certificate). You can either pay the fees to get the real certificates which you can include in your resume / applications, or you can do what I do, which is to take the course for free (audit) then screenshot your final progress report showing that you did the assignments and passed the tests, which should be nearly as good. Keeping a copy of the syllabus will also help show the relevance of the materials.

P.S. edx.org is a little easier on free users since you get a final grade on all coursework at the end, where-as coursera.org doesn't let you take some of the tests and doesn't show a final grade the same way.

Answer 4

1. Don't completely take your eye off the ball on the EE degree. That is a very portable degree with applications from microelectronics to building construction to power. Even if you like some subfields better (or neighboring fields), getting that EE degree gives you a lot of option value. Make sure you don't mess it up.

2. EE is actually a rather mathematical topic. Within EE, try to emphasize areas that are at least mathematical or connected to your neighbor fields. Examples are spectral analysis or computer circuitry.

3. Specific authors/books are failing me now, but I have come across some rather advanced math books that were written by EE profs. Maybe those are good to read. Ones is something like Irresistible Integrals (there are two books like it, can't recall which was from the EE prof). Another is the Leanord Lewin text on polylogs.

4. Don't immediately try to jump to the fanciest, most advanced area. For example, functional analysis is certainly not required for the standard junior year quantum course in physics (which is probably already more advanced than you are getting in EE courses). And you actually get a lot of benefit learning things the simple way first. Look at how physics does classical E&M three times! Freshman year, junior year and then teh Jackson in grad.