Maths Areas Ranked by Coolness (most to least)

Georg Cantor (Wikipedia)

Maths areas ranked by coolness: the most cool at the top: my subjective judgement.

1. Transfinite Set Theory: Bigger than infinity is still a mind-blowing concept.
2. Complex Analysis: The beauty and power of the imaginary unit and its applications make it a strong contender for the top spot. Holomorphic functions in complex analysis have surprising properties, like being infinitely differentiable.
3. Abstract Algebra: Exploring elegant structures and operations like groups, rings, and fields, feels like entering a new mathematical universe. Its generalisation to Universal Algebra leads to foundational computer science concepts.
4. Real Analysis: Taking calculus to the next level, dealing with abstract concepts like continuity and convergence. Leads to the wonderful Calculus of Variations, a foundation of theoretical physics.
5. Topology: Point-set topology, also called general topology, is the foundation of most branches of topology. It studies the basic properties of shapes and spaces by focusing on how close points are without relying on specific distances. It defines concepts like open sets, continuity, and connectedness, forming the essential toolkit for exploring the geometrical nature of mathematical objects..
6. Differential Geometry: Bending space and time with the power of calculus? Sounds pretty cool for those who enjoy the physics connection.
7. Number Theory: The timeless elegance of prime numbers and their mysteries remain fascinating. The extraordinary conceptual depth you get starting with the simple notions of the object 0 and the unary operator s, combining thus: (0, s(0), s(s(0)),...).
8. Linear Algebra: Essential but the practicality might overshadow the coolness factor. The link with Quantum Theory might boost the coolness a little - Hilbert Spaces.

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This list - very subjective - came from a dialogue between Gemini Pro and myself. I studied most of these at undergraduate level excepting Number Theory (an option I didn't take) and Differential Geometry (not offered). I have forgotten almost all of the maths I studied at university...

I would like to add formal logic to the list: predicate calculus; modal logic, lambda calculus. I'm not sure the mathematicians would allow it entry, but if they did it would be very, very cool.