Learning to do independent research without a PhD program

[mathiverse]

Hi all,

I know there are a bunch of research scientists on the forums, so I thought I might get insight here.

I'm very interested in learning to do independent research. My goals are:

• to be the first to discover something interesting,
• to make it up the CCCCCC ladder to become an expert in some topic,
• to enhance my abstract thinking skills,
• and to do something where I can have a lot of personal growth in general.

I'm mostly interested in mathematics research or maybe mathematical physics. I have a Bachelor's in Math. I don't have any more specifics than that at this point since it's pretty early in my exploration phase. I've been learning about different areas of math and open problems in the field. Doing math also means I have no need for institutional resources like huge experimental apparatuses, so let's assume I don't need that type of financial/tool support for the research I'd like to do.

I can obviously do this all of this via a PhD program. However, I want to explore whether there are alternatives that can be done without enrollment in an educational institution. I recognize the final answer might be that you can't or it's extremely unlikely, but let's not start there. I get that it's unlikely and unusual. Still, how can I approach this?

Here are a few questions to start things off.

• To enroll in a PhD program, one often needs to pass quals, is the ability to pass quals necessary beyond institutional hazing? Basically, I'm wondering to what extent it's really necessary to learn a broad subsection of the field in such a way that one is fluent in the types of problems on qualification exams if you aren't in a PhD program that requires them?
• To what extent can someone learn up to the point of unknown problems outside of a PhD program? How do I know what to learn? I know one thing I could do is to learn a grad school curriculum given by some school, but to some extent that curriculum is aimed at getting students to pass quals rather than aiming to get them to do independent research, so I'd appreciate any insights to this question beyond that.
• What do you expect the largest challenges to doing independent research at home?
• To what extent is being a part of a research group or research collaboration necessary?

I may ultimately enroll in a PhD program, but I'd like to learn about and to do all I can do to start the real work of independent research before doing so.

[ertyu]

Doing some independent work will probably increase your chances of getting in, as in, "i have already surveyed abc toping and achieved D results, and I am now at a point where it makes sense to continue working on this in a more structured environment"

idk to what extent the broad curriculum is aimed to get students to pass quals. it can be argued that you need to be aware of the general state of a field before you know what there is to research in a field. the academic goalpost keepers are also unlikely to give credence to the abilities of someone who says "naaah i don't want to do the dumb general bullshit, just show me the cool stuff"

my experience in a phd program (which i dropped out of) was that the classes aimed to present the current state of a sub-field of the larger discipline - providing the background knowledge, if you will - and then research was by and large self-taught. it was up to the individual student to locate an area of interest within the subject, to read up on the papers of others who are working in that area, and to write up what one thinks one can add to the current state of the "conversation" in the appropriate academic format (a properly structured academic paper). You learn how to produce such papers by aping other papers in the field. None of my professors explicitly taught how to write an academic paper. One was also responsible for teaching oneself any quantitative/programming/data crunching skills that might be required.

So in my experience, the process is supposed to go, 1. I am interested in this, 2. Because I am interested in this, I read up on the current state of knowledge about this, 2. because i am an intelligent, independent thinker, i have something to say about this current state -- i can identify errors others have made, ways in which their arguments are incomplete -- or further directions in which i can take this subject that others haven't thought of. 3. I formulate my thoughts and findings according to the conventions of my field and i present them for the edification of others (i produce a peer reviewd publication)

I am not sure how necessary it is to be affiliated with a university. From my current vantage point, if you can't get funding, it might be a good move to try and be accepted somewhere in a developing country with cheaper tuition and i assume more lax expectations of grad students (in order to keep said tuition coming). Combines adventure with cheaper col with a structured environment anyway

[jacob]

Here are a few questions to start things off.

• To enroll in a PhD program, one often needs to pass quals, is the ability to pass quals necessary beyond institutional hazing? Basically, I'm wondering to what extent it's really necessary to learn a broad subsection of the field in such a way that one is fluent in the types of problems on qualification exams if you aren't in a PhD program that requires them?

In my direct and indirect experience, quals usually comprise 80-90% of the advanced courses the given department offers out of which only 10-20% are actually useful for your research. It's mainly about checking off the box on a certain amount of credit hours. Other departmental strategies includes having to take so-called seminars of the kind where attendance is mandatory, little work is involved, and everybody gets an "A", ... all so that the university can underpay research assistants with MSc's by calling them [grad] students.

A phd is really a master-apprenticeship program. How it turns out and what it turns into very much depends on the two people involved and their relationship. Quals also serve as a test in the sense of "you must be this tall to ride this ride". This does not eliminate the Peter Principle. While research requires a certain amount of intelligence, it more importantly requires creativity and a high frustration tolerance. More than one A+ student has failed out because while good at textbook problems and oral regurgitation, they couldn't stand the open-ended nature of research.

• To what extent can someone learn up to the point of unknown problems outside of a PhD program? How do I know what to learn? I know one thing I could do is to learn a grad school curriculum given by some school, but to some extent that curriculum is aimed at getting students to pass quals rather than aiming to get them to do independent research, so I'd appreciate any insights to this question beyond that.

Quals have nothing to do with the actual research. In physics, the rule of thumb is that undergraduate work is about 3 decades behind the bleeding edge. It takes 30 years to translate new discoveries into textbooks. Grad school courses are maybe 10 years behind. The thing is that research doesn't have a curriculum, because research is fundamentally about making up the curriculum as you go along.

The risk of making it up as an independent researcher is unknown-knowns. Lacking structure, there's a high likelihood of entering crankdom. In physics, departments often receive very excited letters from independent researchers who have been working for years to discover something. The usual problem is that they missed something very fundamental that basically invalidates their entire theory. (You see this a lot in climate denialism.) This is where either a solid intellectual foundation or collaborating with/working for a professor is valuable.

• What do you expect the largest challenges to doing independent research at home?

Lack of infrastructure such as tools, software, data, and journals. After retiring from physics, I spent ~4 years as an independent researcher in a quant research group. Finance is interesting in that if some strategy is profitable, it doesn't get published, so in that sense you're automatically at the bleeding edge of research ... or are you. My adventures in that field may be illustrative. My background in finance is entirely self-taught. As such it has holes. And because of the practitioner secrecy, there are other holes. As a result, this means there are lots of independent discoveries to be made. As I was researching, that is, making up theories and testing them on data, I would continually believe that I had discovered "something new" only to later realize that someone else had already figured out in 1983 or 1995. My indicator for progress was that the publication dates of various articles would increase until I couldn't find a match anymore. This told me that I had entered the "bleeding edge" where there was potential profit. OTOH, I wasted a lot of time researching "unknown-knowns". One may argue that it's part of the learning process to "Redo it yourself", but it's not very efficient.

I had access to great infrastructure and financial data. I would not have been able to do this at home or even from a university since financial data is too expensive for individuals or universities to buy. Secondly, in terms of software, it helps not to have to spend time developing ALL of your own code---I'm sure mathematicians use computers too---but unless you know someone who knows someone, you may have to. Research is not like software engineering. People won't automatically post their code since that is not their primary product.

• To what extent is being a part of a research group or research collaboration necessary?

Aside from that ... some people need to bounce ideas off on other people. Others don't. As far as I understand, mathematics is usually single-authored work or at most has two authors. In physics, this is rare because there are too many complications. If one person has to solve all of them, they are not effective enough. These days an experiment in high energy physics may have more than 100 people involved. You're not going to compete with that with your one-man garage-cyclotron. Finance is somewhere in between but getting increasingly closer to physics.

• to be the first to discover something interesting,

What is interesting to you? The first thing, where I was the first human in the world to discover something was a few more decimals on the half life on a certain isotope of nickel. See https://arxiv.org/abs/nucl-th/9902005 At the time I found that extremely exciting. I even went and got the preprint framed on my wall, which in retrospect is very dorky and slightly embarrassing. After about a dozen of such "discoveries", it gets a bit more ho-hum. Scientists talk a good game. The more irrelevant the field is to the world at large, the more "selling" becomes important. "Astrophysicists detect large amounts of gold in colliding neutron stars" sound exciting, except if you're nerdy enough to know that the process also creates large amounts of 500 other isotopes and not a MOP.

There's a risk that the definition of "interesting" will change over time. For example, I eventually found it more "interesting" to solve mundane personal finance issues here on planet Earth via the blog and this is a big reason why I quit physics. Likewise, my quant perspective changed from "solving systemic second-order issues of the world's economic nervous system" to "staring at numbers on a screen".

I'm not sure that my process of changing perspective is universal. It doesn't seem to happen for those who stay in research but perhaps that's just survivor bias.

• to make it up the CCCCCC ladder to become an expert in some topic,
• and to do something where I can have a lot of personal growth in general.

In terms of phd programs, I distinguish between researchers and "glorified technicians". The latter only make it up the first four CCCC as their supervisor does all the coordination and creativity for the student only telling them to calculate this and compute that. It is, therefore, not a given that the top two CCs will appear. Personal growth may or may not appear. This goes back to why the fit between the apprentice and the master is important. Keep in mind that just because someone is a professor it does not mean that they are explicitly aware of this [Kegan] like dynamics. Like most people, they'll presume that others are like themselves. If you're looking for experience in coordinating your own research but your boss is micromanager, who believes in frequent check-ins and daily milestones, you will not find the droids, you're looking for.

• to enhance my abstract thinking skills,

I'm not sure about this one. The quals mainly exist to see if you have [enough of] them.

[7Wannabe5]

the rule of thumb is that undergraduate work is about 3 decades behind the bleeding edge. It takes 30 years to translate new discoveries into textbooks. Grad school courses are maybe 10 years behind.

Sometimes books written for lay reader are published more quickly in the cycle than the undergraduate texts are updated. It seems like it is becoming increasingly common for researchers who have the ability to write an engaging book for the lay reader to "popularize" their work and/or that of their mentor(s) prior to it making its way into a more general text. Generally, it's much quicker to make your way towards the bleeding edge topic-wise if you don't have to do all the math exercises along the way. Although, of course, if you want to do some actual work in the field (as opposed to simply indulging curiosity) you will need to go back and read/do the math.

Also, I have found that online courses offered by the sort of universities that are known for cutting edge research are more likely to be of interest than any generalized textbooks meant to be taught at any general-purpose university.

Textbooks meant for classes on technology topics are so quickly made out of date, the one assigned for my course on operating systems contained unedited expletives.

Actually, I'm interested in this topic too, because the grad courses I am taking are kind of too easy for me, because more towards writing executive summaries and bullet-point reports on technological topics rather than learning more math. So, I am trying to find the time/mental-energy to create my own simultaneous independent course of study. However, since I am an irredeemable generalist, it is tending more towards reading/doing some of "every possible math topic that might overlap with data science" vs. any sort of focused higher-level PhD type research. For instance, this week I spent 4 hours trying to figure out the math behind the "wasted' surplus created by adherence to the 4% SWR Rule. Last week, I spent 8 hours fiddling around with Scott Page's "Model Thinking" and "Doing Bayesian Data Analysis." The week before that I was reading Polya. And these are all digressions from the giant tomes of "Concrete Mathematics: A Foundation for Computer Science" and "Probability Theory: The Logic of Science" by Jaynes, both of which, IM(likely less than well-informed)O are kind of towards cutting edge for "real world" applications of math.

[mathiverse]

Thanks for the reply, ertyu! It was very helpful.

Doing some independent work will probably increase your chances of getting in, as in, "i have already surveyed abc toping and achieved D results, and I am now at a point where it makes sense to continue working on this in a more structured environment"

Yeah. Plus I want to validate I want to do the work before I start jumping through hoops.

idk to what extent the broad curriculum is aimed to get students to pass quals. it can be argued that you need to be aware of the general state of a field before you know what there is to research in a field. the academic goalpost keepers are also unlikely to give credence to the abilities of someone who says "naaah i don't want to do the dumb general bullshit, just show me the cool stuff"

If I enroll in a PhD program, then I will jump through the hoops I need to. Don't worry about that.

The quals question is more to figure out whether I was missing some motivation to go broad before going deep when I'm doing this on my own rather than as a part of a program. From the answers so far, it sounds like the answer is no. (But I'll see if anyone else has input there!) It was also wondering if mastering quals-type questions (which are not research questions) to quals-level standard has some purpose that matters when doing independent research on my own. It sounds like jacob suggests that being able to solve those types of questions (not broadly, but at least in the field you intend to research in) is basically necessary, but not sufficient to do research. As in, if you do original research in a field, you will pass the milestone of being able to do those types of problems in the field one way or another.

my experience in a phd program (which i dropped out of) was that the classes aimed to present the current state of a sub-field of the larger discipline - providing the background knowledge, if you will - and then research was by and large self-taught. it was up to the individual student to locate an area of interest within the subject, to read up on the papers of others who are working in that area, and to write up what one thinks one can add to the current state of the "conversation" in the appropriate academic format (a properly structured academic paper). You learn how to produce such papers by aping other papers in the field. None of my professors explicitly taught how to write an academic paper. One was also responsible for teaching oneself any quantitative/programming/data crunching skills that might be required.

So in my experience, the process is supposed to go, 1. I am interested in this, 2. Because I am interested in this, I read up on the current state of knowledge about this, 2. because i am an intelligent, independent thinker, i have something to say about this current state -- i can identify errors others have made, ways in which their arguments are incomplete -- or further directions in which i can take this subject that others haven't thought of. 3. I formulate my thoughts and findings according to the conventions of my field and i present them for the edification of others (i produce a peer reviewd publication)

Oh, I can see how a class like you describe would be useful for getting started in your subfield. Also thanks for that insight in to what you do and do not learn from classes/profs in the program. That paragraph with the steps was also pretty interesting.

So in your program, what was the point of being under an advisor or in a research group? What's the point of the PhD program anyway?

More replies to come...

[ertyu]

So in your program, what was the point of being under an advisor or in a research group? What's the point of the PhD program anyway?

Mind that I didn't finish - I just did the "schoolwork" and dropped out before writing the dissertation. In my program, many did not complete the dissertation if they'd found a consulting gig that resulted in a satisfying employment position. So, you have one round of drop-outs that really shouldn't have been in a PhD program in the first place but went because they weren't able to find a "real job" (me), one round of drop-outs that, after completing 2-3 years of classes and getting an MA awarded in the process, managed to secure various entry level gigs at the IMF/WB/various gvt departments etc and the dissertation/research fell on the backburner and never got done, and finally, there were those that finished.

The benefit to being in my program was mostly networking. You were in classes with the older grad students, the older grad students had secured some of the aforementioned jobs and gigs, and there was institutional knowledge and a pipeline that made getting accepted easier. Or, thinking of this one particular classmate, she'd been a couple of years working full time at the UN in some capacity, had a child, etc., and needed the PhD to further advance her career. At the same time, as you might imagine, research and scholarship weren't at the forefront for her. At that stage, her advisor who already had connections in various developing countries suggested a dissertation topic to her (because the advisor knew what there was to research that was feasible to complete with the local data available) and connected her with local researchers, whose connections they could use to locate populations to study for various focus groups, to locate a local grad student fluent in English that could serve as an interpreter on the project, etc.

There were definitely a couple of cases of older profs whose quantitative skills had fallen behind who used grad student work to get their data crunched and get the boring part of paper-writing done in exchange for co-authorship for the grad student. This other prof I remember really insisted that her advisees master clear academic writing - it was a pet peeve of hers that often, scholarly papers in the discipline were unclear and atrociously written, and she demanded a certain standard from her advisees - they bonded over how rigorous she was about it but I'm certain it served them well in the long run. So I guess the tl;dr is, in addition to the benefit you get from being affiliated with an institution (subscriptions to journals, someone to fund your attendance at conferences so you can travel for free and network, etc), the benefits to working with an advisor very much depended on the advisor.

[mathiverse]

Thanks for the reply, Jacob! Lots of good info there.

Lack of infrastructure such as tools, software, data, and journals. After retiring from physics, I spent ~4 years as an independent researcher in a quant research group. Finance is interesting in that if some strategy is profitable, it doesn't get published, so in that sense you're automatically at the bleeding edge of research ... or are you. My adventures in that field may be illustrative. My background in finance is entirely self-taught. As such it has holes. And because of the practitioner secrecy, there are other holes. As a result, this means there are lots of independent discoveries to be made. As I was researching, that is, making up theories and testing them on data, I would continually believe that I had discovered "something new" only to later realize that someone else had already figured out in 1983 or 1995. My indicator for progress was that the publication dates of various articles would increase until I couldn't find a match anymore. This told me that I had entered the "bleeding edge" where there was potential profit. OTOH, I wasted a lot of time researching "known-unknowns". One may argue that it's part of the learning process to "Redo it yourself", but it's not very efficient.

That was a very useful illustration for me! Thanks for sharing that.

Quals have nothing to do with the actual research. In physics, the rule of thumb is that undergraduate work is about 3 decades behind the bleeding edge. It takes 30 years to translate new discoveries into textbooks. Grad school courses are maybe 10 years behind. The thing is that research doesn't have a curriculum, because research is fundamentally about making up the curriculum as you go along.

Do you have any other advice for getting to the bleeding edge at home alone? Is it enough to orient your "curriculum" toward whatever theory you have or problem you're trying to answer?

I'm not sure about [improving abstract thinking skills through research]. The quals mainly exist to see if you have [enough of] them.

Hm, that's unexpected. Thanks for the input.

[guitarplayer]

I first wrote an answer to OP but figured I likely have little to contribute compared to some other forumites so never posted. I have some observations about this in social sciences but will keep these to myself since you are interested in mathematics and / or mathematics physics.

I will echo this last post of @jacob that having a mentor works well. As of recently I am in the same boat as you @mathiverse as I just got my BSc in Mathematics and Statistics. In the course of the degree I twice applied for a summer research bursary which I never got. However, by the second time one professor got in touch asking if I would be interested in casually collaborating on a research problem. The problem is promising (i.e. the questions are relatively well defined and at least some of them NP-complete) and the bits suggested by the professor to look at have occurred to him as 'relatively well grasped by people new to the topic'. I find this interaction valuable, always grateful for his time. It is clearly working on unanswered problems with potential applications in the world.

Otherwise, this forum deals with independent research and is not a PhD programme.

[mathiverse]

Neat set up! Also I am happy to hear input from everyone.

[mountainFrugal]

I think that @jacob's post nails most of it. In the end it is up to you to do the work and being open to both being slightly wrong or completely wrong or it not working out at all. Sounds fun doesn't it? .

The main challenge would be to make sure you are getting critical feedback of the ideas from other experts.
A few ideas around that:
1) write a blog - this could also focus mainly on whatever the grad level curriculum is. This could show you understand the content and perhaps provide a deeper understanding of it by writing about it or pretending you are teaching it to undergrad you.
2) With a blog (or similar in place) you could reach out to expert mathematicians pointing to your writings/idea development
3) Regardless if you go to grad school or not, I think that many profs would be interested in discussing ideas with a self-taught expert if they had put in the work to show they were not falling for the "research" fallacies similar to that @jacob talks about for climate deniers.
4) Eventually post your work as pre-prints for others to comment on in a more formal way than a blog
5) Post reviews of other pre-prints within the sub-field you are trying to work in. If the reviews show deep nuance to understanding, you will gain credibility and deeper understanding of the field.
6) Others?

[jacob]

Another thing to mention is that grad school is much closer to a "real job" than the undergraduate experience of "taking classes and passing exams". Unlike undergraduate "projects", one rarely gets to choose what to research. Each research group has a [very narrow] specialty and they only do that and nothing else. To define narrow, it corresponds to maybe a paragraph of a regular textbook (recall https://matt.might.net/articles/phd-school-in-pictures/ ). Insofar one collaborates with academia as an amateur, it will be similarly constrained.

I believe that "match" is much more important than "subject". One can always become interested in something else, but it's nearly impossible to change the working habits/expectations of yourself/professor/research group. Fortunately, "real job" experience means having a lot more awareness of potential issues than the random undergraduate.

Since the goal is not to join a program, the trick is to find a professor (or a postdoc) who is willing to entertain such a match. Most people at that level will be able to come up with 10-20 different "summer projects" that they could finish themselves in a couple of days but is usually reserved for summer students. If I was in an academic position, I'd personally be willing to offer a box of tools and a project, but I wouldn't want to spend much of any time "explaining" or "bouncing off ideas" with amateur students until they demonstrated a lot of invested time in the box I gave them. However, other "masters" may be more excited about this kind of outreach. I think from the perspective of the professional researcher, the expectation of them getting anything useful (=publications) out of the random grad student or amateur is low if not negative until the "apprentice" has a few thousand hours under their belt (on top of a bachelor) so a lot depends on how much effort someone is willing to put it. There's a big difference between "I'm going to dedicate my life and only eat and breathe my research" and "A fun thing to spend my evenings on when I have the time".

PS: These insights may vary a lot between fields. For all I know, mathematics is very different. The best thing would be to ask a math professor.

[mountainFrugal]

I think this is what I am getting at. In the end if you want to do novel research, you have to put in the work. The academic system has a structure for that which is very pragmatic. And so while the DIY version is possible it is also harder in many ways because all of that work falls on the individual instead of some of the burden being taken up by the structure of the program. Maybe that does not come through in my previous post.

FWIW, my PhD program was research first. You were only let in if you had a professor to work with already rather than spending a few years on grad classes and then working on a project. You only took classes that were directly relevant to your research. Our qualifying exam was research based. You could not even do the exam *until* you had at least 1 "publishable unit" of research completed. You then wrote up the results and a full research proposal for what the next 2-3 chapters of your dissertation were going to be given what you knew at the time. I think the "general knowledge" hazing section of my quals was only 45 minutes where I was asked to derive a biophysics model that I was using from first principles on the whiteboard. When I got stuck and had to admit where I was unclear on a few things, I would get hints and move on. It was as much about how I was thinking as it was about what I already knew. If you failed, I knew a few who did, they had to come back a year later or write up their publishable unit as a masters thesis and leave with a masters.

[guitarplayer]

In our last call, at the end of it the post doc (it is a post doc rather than a prof as I looked up just now, I don't know if tenures work the same way in the UK as they do in the US) said something along the lines of 'I must tell you, in research mathematics, it is impossible to know upfront if the problem you are working on has a solution at all, if the solution can be found and how difficult or easy it is to find it. Most of the time you end up going nowhere, this is the way it is and you don't have to feel bad about it'.

I certainly get a sense of freedom that comes with the fact of not being bound by reality Bound by logic yes, but I think perhaps due to me being limited in appreciation of various kinds of logic.

[Britz]

I'm not sure that I can add much beyond the excellent answers given by everyone above but, as a maths lecturer, my comments below might possibly add more detail about the PhD/research circumstances in maths.

First of all, you can definitely do maths research yourself and fulfill those four goals that you state without doing a PhD or working within any institution. Grab a good maths book like A Course in Combinatorics by Van Lint and Wilson (sorry for my bias! ) or any other maths book that might appeal to you, browse through it until you find some topic of interest, and enjoy the content and the challenges of filling in the intentional left-to-the-reader gaps and the problems, while at the same time wondering how you could possibly generalise or modify mathematical results to become something larger or new. Webpages like www.cut-the-knot.org and ProjectEuler.net are awesome sources of fun and inspiring problems and potential research ideas and projects, and there are journals such as Parabola (www.parabola.unsw.edu.au) that publish papers by non-professional mathematicians. If you see any articles that you like, then you could contact the authors and see whether any happy correspondence, or even collaboration, might happen. That happens more often than one might think.

If you would like to do more "professional" maths, then the suggestions above of asking professors for "summer project" type research projects are very good. Many professors are able to provide you with these, and many professors are eager to help you, and are grateful for help in their research. As mentioned above, it's a bit of a gamble for a professor to take you on in any supervisory capacity, but if you tell them that you already have a maths degree and are very independent, not asking time or effort from them, then most professors would be happy to help you. That is, if they have projects lying around and if they aren't too busy to answer their emails(!). Unfortunately, I (Thomas Britz, UNSW Sydney) don't have any good research projects to offer myself right now; otherwise, you'd be more than welcome to look at them! (You're also more than welcome to email me about these issues or anything else!)

Most pure maths doesn't require equipment. I sometimes do large-scale computational searches using university computer clusters but if you are good with computers and can figure out how to parallel program GPUs, then you can do these sort of computations with a $200 graphics card. A friend of mine does that.

A bigger obstacle to maths research is literature, at least if you want to work with cutting edge research (which you might not! Cutting edge doesn't necessarily mean fun.) Pure mathematicians were the first researchers to introduce and embrace open access journals but most maths research papers are still inaccessible to people without institutional access (unless you're willing to pay a fortune). There are ways around that, and sites like arXiv.org help, as do the increasing number of open access journals. You're also more than welcome to ask me for any articles too!

As for the quals, they are mostly to vet you: it's a gamble to take on the PhD studies for the student, the supervisor and the institution, and if a student's maths skills or motivation aren't excellent, then it's a lot of pain for everyone, especially the poor student.

With respect to background knowledge, it usually helps a lot to have some, but this depends a lot on the area of mathematical research. You might need 4-5 years of uni studies in order to understand questions about cohomologies, say, whereas many problems in graph theory can be explained to school kids. On the other hand, the answers to those cohomology questions might possibly be a lot easier to find, given that the questions are very specific and that far fewer people are looking at them. The importance of background studies - and those quals - is maybe more that you've trained yourself to top mathematical "fitness", and already have a diverse array of general tools to use. It might be a struggle to return to studies and to learn from scratch, but it's definitely possible and I've supervised thesis students who were old and retired from their careers.

As a PhD student working independently from at home, the biggest challenges are usually lack of motivation, inspiration and loneliness (funny enough, I was mostly in that situation and loved it, finding it to be free, creative and happy - but this is probably atypical).

If you're doing independent maths research for fun and personal development, then this is not that much of a problem, and you can find likeminded people to chat and perhaps collaborate with.

However, if you want to publish research journal articles, then it might be hard to find projects to work on, to get good ideas and feedback on how to find new results, to be able to look through the literature to see whether your new results are actually new, and to know how to write up and present your results. StackOverflow is a good source of good research ideas and feedback for non-professional mathematicians, and zbmath.org is a free and excellent database for searching maths articles. Still, the PhD studies provide a lot of cultural training, including how to write a maths paper clearly and presentably, that is hard to get elsewhere.

I mostly worked on my own during my PhD project and had little supervision and next to no feedback from supervisors. That made me work hard and independently but I missed out on collaboration and support that you get from a group and its networks. In the other extreme, some people are so supported by their group that they don't learn to be independent. Most maths is usually collaborative, though often in parallel or turn-based. Some people work more directly together; it depends on people's dynamics and the work at hand, and the modes of collaboration can be quite diverse. It's usually constructive and collegial in maths, though, often socially boring but also without unnecessary frictions or politics.

I hope that these comments added something to the discussion and that you might find them useful!
If you're interested, I've written more about related topics here:
https://www.quora.com/profile/Thomas-Britz

Oh, and good luck and enjoy your maths research, in whatever form it might take!

[guitarplayer]

Brilliant @Britz, huge thanks for adding the comment!

Also, just want to state here that Combinatorics and Graph Theory is much of my interest at the moment as I am looking at the topic of and around general position problem in graphs :]

[jacob]

I'm not sure that the "learning to do independent research" part of the question was completely answered. There are two aspects to this. The first one is "independent" and the other is "research".

Independence develops once one realizes that "the buck stops here" and that there's literally no other human in the world who knows the answer. This is a huge shift in mindset and attitude compared to the undergraduate years where one was essentially solving problems that the professor or author would always know the answer to or at least be able to find out in short order. With actual research, this is no longer the case. You're even guaranteed to find an answer. The Socratic quote about wisdom being about knowing what you don't know is often frivolously used to justify ignorance, but research is when it's fully appreciated for what it is. As Britz alluded to, there's also a risk of not developing this independence insofar one becomes dependent or interdependent on other researchers.

Approaches to research are probably quite idiosyncratic. Personally I've never been explicitly taught how to do research. In general, there's not a locked-in approach. The reason is that insofar you have one [such approach or method], the kind of questions and answers you can find will become constrained by that(*). Indeed, good research often comes about from finding a novel approach to a known problem resulting in a new perspective which in turn reveals previously unseen/unconsidered aspects of the problem. Also see viewtopic.php?t=12873 ...
In my case, my approaches tend to be 1) exhaustive literature search. I tend to trace most references back to the original discovery(**). For my thesis, I had read and made notes for 600+ papers and could cite them ad libitum. I think that helped for the summa cum laude. This came naturally to me because I'm a reader, but that much reading is unusual; 2) parameter studies. As you might have noticed I still frame almost every question in terms of 2 or 3 variables and then try to understand all 4 or 8 quadrants. This is possibly due to spending most of the time developing computational tools. Once you have them, why not use them for the full range. I credit the financial part of ERE with the 80% savings rate (see chapter 7) to that. Apparently nobody in professional finance had ever considered looking at that part of the parameter space even if the equations (in chap 7) were well-known to probably millions of people; and 3) visualization ... making fancy color plots. This may be trite now, but 20 years ago, color plots and animations were still new and rarely used(***), so I could see/do things that my competitors couldn't.

(*) In some fields, like maybe solid state physics, every project and paper is almost formulaic in form and content.
(**) For example, the first X-ray astronomy was done in 1949 using a captured V-2 rocket to get above the atmosphere.
(***) I wrote an f90 subroutine to output raw postscript files which required learning postscript.

In that regard, you'll probably make the most significant contribution based on your personal "edge(s)". Maybe that's programming GPUs? Maybe something else. The thing is, you might not be aware of what your edge is (compared to other researchers). The good news is that you'll use them anyway insofar your independence is free to run. In the best case, you'll collaborate with someone whose own edges complement your own. For example, in my case, I'm very much a "leave me alone and I'll figure it out eventually, do or die" whereas my phd supervisor was a great networker/connector constantly hooking me up with the right people and sending me all over the place---I would never have naturally gone myself---instead of acting like the more common boss asking for a slide or two in order to present my latest results at some conference he'd go to himself. The importance of "the whole >> sum of the parts" can not be overemphasized, because its opposite is also a genuine possibility.

[mathiverse]

I am thankful for all of the wonderful replies so far! Thanks mountainFrugal and Britz for the excellent additions! I also appreciate the additional posts from gp and Jacob!

I have a lot to think about. I'm glad to hear that there is a lot I can do on my own. I'm excited to try some of the suggestions everyone has made here!