What is the hardest type of math?

 There's no single answer to that question, as "difficulty" in math is subjective and depends on individual strengths, interests, and prior experience. What one person finds challenging, another might find fascinating and logical. However, some branches of mathematics are frequently cited as being particularly challenging due to their abstract nature, complex concepts, and demanding proofs. Here are a few examples:

Highly Abstract Fields:

• Category theory: This field investigates the underlying structure of mathematics itself, focusing on relationships between mathematical objects rather than the objects themselves. Its abstract nature and reliance on formal languages can be quite challenging to grasp.
• Set theory: This branch deals with the foundations of mathematics, exploring the very concept of sets and their properties. Axiomatic foundations and transfinite sets can be difficult to comprehend for those unfamiliar with the field.
• Non-commutative geometry: This area combines differential geometry with abstract algebra, leading to complex geometric structures and intricate calculations. Its interdisciplinary nature and advanced concepts can be daunting.

Complex and Demanding Topics:

• Partial differential equations (PDEs): These equations involve multiple unknown functions and their derivatives, leading to complex calculations and challenging theoretical aspects. They are used in various fields like physics, engineering, and finance.
• Functional analysis: This branch studies infinite-dimensional spaces and functionals, leading to abstract concepts and demanding proofs. It has applications in physics, quantum mechanics, and optimization problems.
• Algebraic topology: This area combines advanced algebra with topology, studying the shapes and properties of spaces through algebraic tools. Its high level of abstraction and reliance on sophisticated tools can be difficult to master.

Additional Factors:

• Level of formalization: Some areas of mathematics rely heavily on formal proofs and rigorous logic, which can be challenging for those more comfortable with intuition and visualization.
• Unfamiliarity with prerequisites: Many advanced topics require a strong foundation in previous mathematical concepts, and encountering them without sufficient background can feel overwhelming.
• Individual learning styles: Different people learn and grasp concepts in different ways. What one person finds challenging, another might find stimulating and enjoyable.

Ultimately, the "hardest" type of math depends on the individual and their perspective. What might be a struggle for one person could be a fascinating intellectual challenge for another. The key is to approach any area of mathematics with an open mind, a willingness to learn, and a sense of curiosity.