Mathematical disciplines with high thresholds

Question

Are there mathematical disciplines that are extremely inaccessible, with very high thresholds even for those who have the necessary prerequisites?

Number theory and graph theory, for example, are very complicated in themselves, but everyone can still form an idea of ​​what it's all about. Also, category theory is relatively easy to understand, although the definitions and methods are very abstract.

General topology is more abstract from scratch and many get lost (just unnecessary). Those who practice the methods of analysis understand the importance of open sets, but it is not easy to develop a proper intuition.

Many people have problems with homological algebra, in my opinion because they misunderstand the diagrams: large charts do not mean more difficult problems, but more information; the diagram is just a very efficient way to display information that facilitates problem solving with a few standard methods.

However, I ask for very inaccessible disciplines which are fundamentally difficult to understand even for mathematicians with prior knowledge.

Answer 1

Here are two examples by changing our underlying logic (perhaps someone else can comment on "Inter-universal Teichmüller theory" since I know nothing about it except that it is hard to get into):

• constructive and intuitonistic mathematics: is probably not easy to get into for the average mathematician, since there are many possible pitfalls. Just look at the notion of "finite set". There are 5 different notions given on the nLab that are all equivalent in $\mathsf{ZFC}$ but generally not equivalent in constructive mathematics. Parts of measure theory and topology get extra complicated since the classical notions start to "fall apart" (c.f. Cheng spaces and locales). The situation gets even spicier if we start to contradict the law of excluded middle (c.f. smooth infinitesimal analysis).
• inconsistent / paraconsistent mathematics: this even much more obscure than constructive mathematics. The basic idea is the if we get rid of ex falso quodlibet then we are allowed to talk about contradictory statements without our logic becoming trivial (note that this usually stops the disjunctive syllogism from working). You can try to do mathematics in this setting. If I'm not mistaken: one motivation is to talk about "real numbers" $\varepsilon$ with $\varepsilon > 0$ and $\varepsilon = 0$.