"Hope we can see each other soon."
I can answer you briefly about this. Similar to English the concept about possibility and ability, and sometimes requests, is not entirely isolated. If we ask "Can you do this?” it can be interpreted as whether the listener is able to do it, or whether the listener will do it. There are also several ways in Chinese to express such related concept. 可以 by its radical meaning is about a possibility - whether it is possible that we see each other; 能 is more about an inherent capacity or ability acquired through training - whether we are able to see each other. In this sentence these 2 words can be used interchangeably.
In some other sentences they cannot. However, it's going to need a lot of pages to iterate all the possible variations. If you have a particular sentence that puzzles you, I can point out which words are acceptable.
It's not very comprehensive, but I hope it helps.
This doesn't quite agree with what i've read, which is that 可以 basically is about permission, 能 about possibility and 會/会 about knowledge. See e.g. http://www.chinese-grammar.com/beginner/hui-neng-keyi.html or https://resources.allsetlearning.com/chinese/grammar/Comparing_%22hui,%22_%22neng,%22_%22keyi%22.
I would break down the syntax like this:
希望 (I) hope (that)
= Main clause
--> 我们 We
= Subject of sub-clause
-->> 可以 + 很快 --> 再见 can + quickly --> see each other again
= adverbial phrases and the verb of sub-clause
The term "adverbial phrases" may not be very well used, but anyway those are phrases that elaborate on the verb. Time can also be used like this.
That’s a great video, thanks!
Summary for those who can’t view it:
能 = to be able to (think possible/impossible)
可以 = to be allowed to (a rule is involved. Could someone tell you that you can’t do x?)
会 = (a learned skill, something you have to practise to be able to do) or (something that’s very probable or likely to happen)
But you really should see the video, it’s great for getting into the nuances of usage.
I love the fact that they have a mandarin speaker naturally explaining things and then the English speaker rehashes it (makes it excellent practise for all levels of learning). Don’t worry if you don’t get everything that’s said, they go over everything multiple times and give multiple examples.
Thanks @JanZeze !
Yes 再见 is basically 'meet again'. They use it to say 'goodbye', just like 你好 (you good) is used to say hello.
I don't know where you got 'part' from. The literal translation would be 'hope (希望）we (我们) soon (很快) can (可以) see you again later (再见)'. See Keith_APP's explanation under Muanui's question.