I am looking to find a forum in the Internet intended for math enthusiasts to discuss math ideas outside the questionanswer format but at high level. Can anyone advise such sites?

4$\begingroup$ On Stack Exchange, apart from Q&A site there are also chatrooms. You can look at some rooms associated with Mathematics: List of chatrooms. Of the rooms associated with MO only Homotopy Theory seems to be rather active. See also: Specialized chat rooms, You can also check out some of the links mentioned here: Useful mathematical fora. $\endgroup$ – Martin Sleziak May 14 at 17:04

$\begingroup$ I agree with Martin. Lots of suggestions at math.stackexchange.com/q/95787/442 $\endgroup$ – Gerald Edgar May 15 at 10:03

$\begingroup$ @GeraldEdgar chat rooms here are limited by time. In sci.math I found only garbage. Some other links lead to either lowpopulated sites or school help sites... $\endgroup$ – Anixx May 15 at 23:18

$\begingroup$ I am not sure what you mean by "chat rooms are limit by time". It is true that if a room is completely silent, it gets frozen after 14 days. (7 days if only a single user used the room.) However, if you talk in the room once a week, you can keep it indefinitely. Maybe you meant something else. Anyway, if needed we can discuss it further in chat  so that we do not digress too much from the original question. $\endgroup$ – Martin Sleziak May 16 at 4:51

$\begingroup$ This also reminded me of an older question here on Meta MathOverflow: Any other site besides mathoverflow and mathematics stackexchange? $\endgroup$ – Martin Sleziak May 16 at 5:34

1$\begingroup$ What is "discussing math ideas" for you? Reading recent articles on a very specific topic and discussing research trends? Commenting expositions on math blogs such as Terence Tao's, or 3blue1brown's Youtube videos? Recreational math articles such as the ones in the Mathematical Intelligencer? Math Olympiad problems and puzzles? Discussing methods to square the circle and comparing each other's proof of the Riemann Hypothesis? $\endgroup$ – Federico Poloni May 18 at 10:32