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.