Toughest Math Courses?

Hey everyone, I'm a bit of a math enthusiast and I'm looking to challenge myself. What might be some of the most difficult mathematics coursework at the undergraduate level?

Absolutely, I can suggest a few! These aren't universally considered the toughest as it will also depend on individual interests and strengths, but in general undergraduate students tend to rank these among the most challenging:

1. Abstract Algebra: This is a course that typically explores topics such as group theory, ring theory, and field theory. It takes you out of the familiar number systems and into abstract algebraic structures. To do well in this class, you need to get used to dealing with non-intuitive and abstract concepts.

2. Real Analysis: This is a notoriously hard course among math students. It rigorously lays the foundations of calculus through an exploration of real numbers, sequences, series, and functions. Proofs are the big focus here.

3. Differential Geometry: It’s a course that explores calculus in the context of curves and surfaces in three dimensions. It tends to be abstract and the visualization needed can be tricky.

4. Topology: Sometimes called "rubber sheet geometry", topology is a branch of mathematics looking at the properties of space preserved under continuous transformations. This course often involves a different method of reasoning through problems compared to most other math courses.

5. Partial Differential Equations: PDEs involve multivariable calculus and differential equations, and can be quite challenging. Solutions often involve advanced techniques and a deep understanding of multiple areas of mathematics.

6. Number Theory: While its concepts might seem simple and it doesn’t require a significant number of prerequisites, number theory is highly abstract, and can involve complex proofs.

Remember that what is "difficult" can be very subjective and depends on where your strengths in mathematics lie. It's also worth pointing out that the above courses provide a strong foundation for many fields in mathematics and physics, so they're very worthwhile if you're passionate about the subject!