Hey all! I'm new here to prepare for the graduate school and need some suggestions. Currently, I'm an international student studying pure math as an undergrad at a big public school in Midwest. Right now, my GPA is 3.90/4, major GPA is 3.94/4 and I have a minor in CS. I haven't taken GRE nor GRE math sub.

So far, I took Linear Algebra I (A+), Probability (A-), Ordinary Differential Equation (A+), Foundation Analysis (A+) and I'm currently taking Vector Calculus, Linear algebra II, and a graduate level real analysis.(I began to like math at a very late time lol) I just started a research with a prof which probably won't yield any paper(lol). After taking an Abstract Algebra course next semester (probably the honor one ), I'm done with all the requirement.

Here is my question.

(1) Is there anything I can do before the end of next semester to improve the chances of being admitted to a satisfying graduate program? I'm aware some of my weakness like: few grade courses, no academic activities (talks, posts, etc..), no papers...

(2) My advisor recommended me to prepare Putnam exam next semester (I didn't take it before) which means I'll take 2 undergrad proof based class, prepare for the exam and do the research I mentioned before. Or I can take 2 undergrad math classes and 1 graduate class. Which seems better just in terms of improving my overall math ability? My math prof didn't recommend me take 3 math courses and prepare the exam since he believed it would be overwhelming.

(3)How did you guys find the area you want to do research? Right now I'm very confused. For example, I really like Real Analysis which I'm taking now. But since the math courses I've taken are limited, I may like some other math subjects like topology, complex analysis or the combination of CS and math in future...

Any response will be appreciated! Wish all you guys get wonderful offers!

Thank you!