What is the toughest math class offered in college?

Hey everyone, so I'm thinking about majoring in Math once I get to college and I'm wondering what's the most difficult math course commonly offered at universities? I've seen some discussions about Real Analysis and Abstract Algebra and stuff, but wanted to get more perspectives. Maybe those of you who've already taken some advanced math courses could give some insight? Thanks!

Different universities might offer different high-level math courses, and the difficulty can also vary based on the curriculum, the professor, and your individual strengths and weaknesses. That being said, the courses often considered among the most challenging in many programs include Abstract Algebra, Real Analysis, and Topology.

Abstract Algebra takes you into the realm of sets and operations, group theory, and field theory. It's known for its heavy emphasis on proofs and logical thinking. It's a shift from traditional calculus and algebra where you deal with numbers and now you're dealing with structures.

Real Analysis - often considered one of the big challenges in a math major - is a formal treatment of calculus and requires a strong understanding of limits, sequences, continuity, and series. It is centered on constructing proofs, which can be a new and demanding skill for many students.

Topology, which can be thought of as a sophisticated form of geometry, is another challenging course with abstract concepts and a heavy reliance on proofs. It includes topics like open sets, closed sets, continuity, compactness, connectedness, convergence, etc.

However, I'd argue that the specifics of which courses are hardest can often be individual-specific. For example, if you're more oriented toward applied mathematics and concrete solutions, you might find the abstract concepts and proofs in courses like Abstract Algebra or Topology more challenging compared to someone who thrives in abstract thinking. It's also worth thinking about the broader context and what other classes you'll be taking alongside these, as all of those factors can contribute to how rigorous a particular course feels.

Also keep in mind that being "toughest" doesn't mean you won't enjoy it – many math majors often find these courses among the most rewarding due to their complexity and abstract nature – they provide an opportunity for deeper understanding and intellectual exploration. The key is to approach each of these rigorous classes with an open mind and a willingness to put in the necessary work. It's a challenging journey but can be incredibly fulfilling for those who love mathematics.

Remember, don't be intimidated by these "tough" courses. You'll build up to them throughout your major, and you also generally have the freedom to choose the sequence in which you tackle these advanced courses. Make use of all resources available to you such as professors' office hours, tutoring centers, study groups, etc. You're not expected to understand everything immediately in these classes, but with hard work and perseverance, you surely can!