Okay. My questions are: How do some people do doctorates in mathematics and spend so much time like three to six years trying to answer one or two open problems? How do they have the patience, psychological strength, perseverance to solve those problems for so much time?
Other questions: People like Perelman, Wiles who spend seven years each one of them for the Poincare conjecture, Fermat's last theorem to prove them respectively, how do they do it? Do they know beforehand some of the possible ways to solve those problems? What should someone study to ultimately solve problems like those? How will he know if he follows the correct paths to prove them?
Should i try to solve problems that do not take so many years or months to solve at the beginning at least? Perhaps they take like two weeks or a month to solve or less than that? How could i find those open problems?
I am an undergraduate student in math at university. I try to read some math and physics that are not included in the material covered in my undergraduate studies.