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I completed my undergraduate in Mathematics in 2018 and masters degree in 2020. After that, I taught in a college for two years. I was not satisfied with my masters degree, and also was getting rejected from a lot of institutes for Ph.D. due to my masters college. So, 2022, I joined a good reputed institute to do Master's again. I completed my SEM I, with good marks. I am way more serious than I was before, regarding my career and my studies. Although I have good teachers here, there is a lack of guidance in some other things. I want to do a research project in during this course. I have applied to some good institutes for that, but they ask for research experience in the form, and since I have none, I get rejected. Also, maybe because I am doing my master's for the second time. I don't know what to do. How do I convince people I am focused and passionate about research in pure mathematics? So far, I have only studied and taught college courses, so I get empty headed when asked to write the statement of purpose. I have no one to ask to, friends or family. My teachers are not particularly from the field I wish to apply. They only suggest to read more books. Can anyone please guide me as to how to overcome this block? How can I get a summer research project without having any experience? What should I specifically mention in my statement of purpose?

P.S. My aim is to do research in cosmic topology. The subjects I want to do a project is Abstract Algebra, Topology, and Geometry.

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    Do I understand it correctly that your current Masters degree is not finished yet (I don't know what a SEM is)? Do I understand correctly that you are looking for research internships at the moment, before having finished the degree, rather than for doing a PhD once more? It may help to say in which country you are, and where you look. Commented Mar 18, 2023 at 11:30
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    Your teachers’ advice is good — you should read more books. Mathematics research is not something you typically jump into early. There is a lot of background knowledge to learn before you are at the stage where you can contribute to active research. You can do this no matter what program / institute you’re at. So it’s generally good advice, especially from someone who otherwise isn’t involved in the fields you’re interested in so can’t give more specifics.
    – Eric
    Commented Mar 18, 2023 at 16:50
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    "I have applied to some good institutes for that..." Could you clarify what you mean? It is somewhat unusual for a masters student to have their first research project be outside their home university.
    – user137975
    Commented Mar 18, 2023 at 19:45
  • @ChristianHennig Yes, my current master's degree is ongoing. SEM is semester...so as of now my second semester is going on and after this we have a 2.5 month break. During that break, I wish to work on a research project. Many institutes offer such programmes, and I have applied for those. But rejections are making me question my decisions as what exactly am I doing wrong and where can I improve so I can get selected. I am from India, btw
    – user519535
    Commented Mar 19, 2023 at 13:29
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    @user519535 Please add relevant information to your question rather than just posting it as comment. Commented Mar 19, 2023 at 13:36

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Coming from Physics, my impression is that you need a lot more training before you can begin to do meaningful research into mathematics than you have to do for physics. When I did my honours (master's equivalent) me and all the other physics students did some basis research, while the maths honours students basically did a lit review. So maybe you should change your goal from doing research to doing a guided review of a topic/review a paper and learn how everything in the paper works and how it fits together (which is what I think the maths honours students did).

I also noticed a master's thesis from university of Helsinki on your topic of interest: https://arxiv.org/pdf/1612.04157.pdf which I noticed described itself as a comprehensive review.

The other thing to keep in mind is if you go to a mathematician and say you want to do cosmic topology, then they are going to wonder why you are coming to them to do a physics research project (its really mathematical physics but I've noticed pure mathematicians call it physics, and physicists call it mathematics). So that might also be leading to rejections.

Another thing to do would be to find out if there are any weekly mathematics seminars or reading groups on a topic you might be interested in at your university (or possibly if there are any that run virtually/hybrid) and ask if you can join. You probably won't understand most of what's going on but you may be able to build relations with other mathematicians and perhaps then they'd be more likely to take you on as a research student since they already know you.

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