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What is math research like day-to-day/curious about your life experiences

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I read papers and preprints

Try to have ideas of new interesting problems or solutions to known problems - usually generalizations of known results, gaps on the current knowledge, applications of ideas in different contexts etc

Talk to colleagues about my and their ideas.

Try to find the proof for some of my/our ideas, combining known results and new calculations. This involves a lot of wrong ideas, wasted ink on the board and piles of paper sheets angrily torn apart.

When I/we find a new relevant proof, I/we write a paper!

Essentially our products are new proofs of new relevant results. Finding that requires a lot of "experiments".
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First thing to point out is that most research jobs are not 100% research. Second thing to point out is that many research jobs are light on research and heavy on managing others to do your research for you.

For example, I'm a statistician and my job is only about 20% research time. I would say at least 70% of that time is spent writing the manuscript. 25% of the time is spent developing the computer code that runs the analysis and generates the tables/figures for the manuscript,. The remaining 5% is spent on other odds and ends, like reading papers.

To be more specific, research projects take years to complete. It takes a good year or two for projects gestate in my head. When I am ready to start, I spend about a month writing notes to prepare to write the computer code. I usually spend 2--3 months writing the computer code and waiting for results. When results are ready, it usually takes another 6--12 months or so to finish writing and submit the manuscript.
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I frequent math.stackexchange.com (Math.SE) and just read the various questions and answers there.

I review the current question(s) I am asking about the problem I'm working on and try to push the boundaries of what I know, whether by asking new questions, or by writing out the expanded form of an expression I'm considering, or even just push a bunch of data from one of my expressions into a spreadsheet and see if there are any manipulations that are fruitful.

If I come up with new questions that I don't immediately have answers for, I search for resources on them, including Math.SE (as above) or mathoverflow.com, generic web searches, or any sources with papers that might be related.

I spend a lot of time driving, so a good deal of my mathematics "research" revolves around asking questions within the problem space and seeing if I can come up with answers without needing to stop and write things down or look something up.
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@OP, how did you even get a research oppo in an organic chemistry lab without the background?!!!
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For context, I’m an applied mathematics Ph.D student working in numerical analysis with a focus on probably convergent methods

At the start of my first project, I did a lot of reading of specific papers chosen by my advisor. I worked to build a codebase which could reproduce the results of the current state of the art methods and could be built upon for developing improved methods. For about a month or two my days consistent of writing code, testing the code for known benchmarks, and trying to find and read relevant ideas. Somewhere along the way I got lucky and stumbled upon a well known result from a different field which could be modified to fit our problem.

This idea was not the original project my advisor has set for me to do, so it became a bit disorganized. For about a year my days followed some form of this process: do some math on paper trying to figure out how to implement this new idea, implement what I had come up with and see what works and what didn’t, make reports for my advisor on what I’ve done and the results, have her explain why the problems are probably happening and offer some potential solutions.

For myself personally, I found that there are three types of work modes I had while doing research. There’s solving problems I know how to solve already. This is usually in the form of coding or doing some simple asymptotic analysis. Then there is solving problems I don’t know how to solve. This can be aided with computers to get an idea of what the answer should be, but often it’s just banging your head against a wall. This can also involve talking to colleagues, mentors, reading papers, etc. The final mode is trying to solve problems that you don’t know that you’re trying to solve. This is often the most fun part of research, but I’ve found I really have to limit myself here because it’s easy to “lose yourself” (being a grad student usually leads to having lots of time constraints and not a lot of time to waste). This process is usually sparked by conversation, talks, or reading (though sometimes a random idea just pops into your head), and often involves rapidly writing or coding an idea for preliminary testing without much of an idea of where the results will take you. For example, I may come across a result that allowed someone to compute a specific quantity in a “fast” way. This quantity may be related to a quantity I need to compute, so I’d try to modify what they’ve done to fit what I need to do.

Now that I’m further into my program, my days are a bit more organized. I have some material I’m trying to learn, so I have daily time set aside for working through that. A lot of my days go to writing up the work I’ve done for my thesis and publication. The same sort of iterative process I described above still happens, though now sometimes my advisor comes to me with problems.

Hope this gives some insight!

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