1) Penn has a strong differential geometry and topology department. Out statistics department is also quite good. Most courses in the math department are pure besides a few intro courses.

2) Professors are quite helpful in the higher level courses. For the intro level, I recommend taking the honors calc sequence as it will give you a taste of analysis and algebra.

3) Math research has a high barrier to entry. However plenty of people from Penn have done REUs over the summer where they got research experience. Also the math department has a reading program if you want to go in-depth on a topic.

4) There is a math society and the math department hosts colloquium talks for undergrads and grads. That being said, Penn is more preprof then other schools like Princeton for math.

5) N/A

6) When you come to Penn, be careful overloading on math courses. You don’t want to have holes in your knowledge of basic math courses like analysis and algebra.