Are there common "phrases" in complex Mathematical notation? More precisely, are there complex mathematical-notation-patterns that convey more meaning to somebody who's seen and understood them before, than they do to somebody who's just seen them? (Even if the person just seeing them understands all the notation.)
To give an example of a simple phrase, in probability, $|$ means "given", and $P(.)$ means the probability of whatever takes the place of the dot. Taken together $P(X|Y)$ means "The probability of $X$ given $Y$". This could be considered a simple mathematical phrase.
Personally, I don't have to think about what it means because I've seen it before. But it might take somebody who hasn't seen it a non-trivial amount of time to understand it.
At a post graduate level, are there a greater number of far more complex phrases that you have to memorise to be fluent in maths?
 A: Practitioners in any branch of mathematics  build a vocabulary of terms that encapsulate abstractions. Those terms might be natural language or more formal mathematical expressions. Then they communicate using those terms. Newcomers to the field gradually learn  the local language. That isn't primarily memorizing complex phrases, it's becoming fluent.
Augmenting language this way is not limited to mathematics, or even to technical fields. Every sport or religion or government or corporation does it.
Edit in response to this comment from the OP:

I'm a material science Post-Doc, so perhaps not as far removed as my
comments are suggesting. At the very least, I'm definitely not a PhD
program away from understanding the papers. My question was prompted
by some papers in game theory, which I'm interested in understanding
for practical reasons - Explainable AI. In one paper there was an
innocent, short mathematical phrase that I didn't think much of. Then,
looking deeper, it seemed that the phrase was actually a short way of
saying something complex. My question is intended to discover how
common this is.

I think you may have asked an xy question. Rather than wondering how common it is for mathematicians to pack a lot into a single formal statement you can ask about the particular statements that puzzle you. one at a time on this site. Show us your best effort and the background you bring and why the meaning matters and we may be able to help.
