I am an undergraduate student from Brazil. I am not a mathematician yet because I've never published an article. I am very close to starting my master's degree and would like to have a personal page to talk about the areas I'm interested in — General Topology and Algebraic Topology — and post the lists of exercises and LaTeX solutions that I typed. I am afraid of seeming arrogant and pretentious because I am not even in my doctorate.
Yes, it is fine, but if you intend to maintain it into the future, keep it professional in all respects.
It isn't likely to be found by many people, however, unless you also make it useful in some way. That is, useful to mathematicians in your case. Book reviews might work, or alternate problems/exercises in your field.
But stay away from things like solutions to book exercises, which might be used by others to cheat in courses. And pop-culture lessens the professional nature of the site.
IIRC, some universities provide web space for students, so it isn't unheard of. Those might disappear at graduation, however.
I had a webpage as a grad student where I had a lot of tutorials on how to use the various tools that we used in circuits classes. I also had some of my lectures, and tools. It got about 100k hits a year by the time I left, and then it went away a year later. It wasn't until it went away that I found out how much it was used as a resource. The Internet is actually written in pencil.
I would suggest having a github or something similar as it's free (currently) and free of the requirements of the department. This will let you keep the work long after you are finished with school and it can follow you when you change affiliations.
It used to even be expected. When I started university in 2001 every student got allocated some space for a homepage on the university servers. That might vary by country, and be less popular post social-networking, but it's completely normal.
For Computer Science students graduating, having a personal Github page with most of you major projects is almost a must, currently.
I have learned some very difficult mathematics from blog posts written by some (admittedly, genius) mathematicians when they were in early years of college or even still in high school. So having a web page with your personal thoughts on mathematics should not shed any negative light in you.
But if you are afraid of coming off as arrogant, then don't be arrogant! Having something along the lines of "Hey, I am a second-year student, and here I will teach all of you how you should think about mirror symmetry" is something that could only be justified by immense knowledge of mathematics, while "Hey, I am a second-year student, I am interested in mirror symmetry and this is my understanding so far" sounds completely fine, even if the actual content following this line turns out to be completely wrong.
As others have suggested, you should probably stay away from posting exercise solutions. Even though there is precedent for that too, you really should consult with a senior colleague before doing so. The most important thing to consider is whether the result of the exercise is useful on its own (there are books in which critical results of the subject are hidden away in the exercises), or is the point of the exercise is for a student to practice doing something.
You should have a web page listing your name, the university where you study, and a way to contact you. If your name is common, you might add something else that would help identify you. No matter what you do in your future career, this is a helpful thing to have.
Including any math in your site is not particularly important. Anyone searching for your page will be looking for your contact information.
No it's not. From what you describe, it looks like you want to have a web page like it was in the good old days of the Internet of Ideas (also called Internet 1.0 ... T&C apply, these names are made up by me).
If you have an apple and you share it with me, we both have half an apple. If you have an idea and you share it with me, we both have an idea.
If your page is pretentious, it will look like pretentious. If your page is full of claims like "I will demonstrate Fermat's theorem", it will look like it is a crank page If your page is full of exercises and their solutions in Latex ... it may be interesting both for you and for who will find it.
I do not have an opinion on the general issue with cheaters and future versions of ChatGPT using it, you may reflect on this aspect.
Yes, it's fine. Moreover, this will help you to showcase your accomplishments get more opportunities in each step of your life. Just gradually add it over time. In addition, if you add blog section and put stuff there, you will be able to improve your English, your writing skills, shich is helpful in both, in academia and in industry; moreover, in life.