What are the most academically rigorous math classes in college?

I'm a junior in high school and absolutely love math. I hope to major in it when I get to college. I'm curious, what are considered some of the most challenging math classes in undergrad? I just want to push myself and take the most rewarding courses. Thanks in advance!

You undoubtedly have a strong passion for mathematics, and your enthusiasm is commendable. An interest like yours can certainly lead to an exciting and challenging academic journey in college. The specific math courses that are deemed the "most challenging" can vary between universities due to course content and teaching style. However, there are some advanced math courses that are typically viewed as difficult in most programs:

1. Abstract Algebra: This course introduces abstract algebraic structures, such as groups, rings, and fields. This often involves sophisticated and abstract reasoning.

2. Real Analysis: Also known as advanced calculus, this course delves into the rigorous proof and principles behind calculus. It often starts with the foundations of the real number system and works up to sequences, series, continuity, differentiation, and integration.

3. Topology: This subject fundamentally deals with the concept of 'closeness' without necessarily relying on numbers. It's an abstract part of geometry and is tricky because it can defy your intuition.

4. Linear Algebra: Although many encounter this as a freshman, a more advanced study in this area can be quite challenging. It includes topics like vector spaces, linear transformations, determinants, eigenvectors and eigenvalues.

5. Partial Differential Equations (PDE): This course revolves around equations that involve an unknown function and its partial derivatives. PDE is generally regarded as challenging due to its theoretical and practical complexity.

6. Mathematical Proofs: This can be a standalone course or woven into all advanced mathematics courses. Here students learn to construct rigorous proofs, which can be a steep learning curve.

Remember, while these options are traditionally challenging, the difficulty level will depend on your abilities, how the course is taught, and the support you receive. Be sure to balance your course load and make the most of professors, study groups, and tutoring resources. Go at a pace comfortable for you, and enjoy the journey. Remember, mathematical inquiry isn't just about final answers, but also about the process of exploring, conjecturing, and discovering!