university - How to comfortably interact with famous professors?

I know that they are "just as human as we are", but I feel somewhat uncomfortable to talk nonchalantly with the top-notch math researchers and professors in my department because of their social position, their extremely superior knowledge and talent, and also because of their age.

Still, I feel that it is like a waste to have such interesting mind-expanding people around and not to interact properly and talk about maths when I meet them in the department (outside the lecture hall).

So my question is: according to your experience, what is the best (where best means: most polite, appropriate, acceptable, but also profitable) way to interact and make the most of the presence of such awesome professors? Can I discuss "lower" [mathematical and non-mathematical] subjects with them even though their actual (research) interests lie in much more abstruse topics?

Any suggestion (even in the form of a comment) is really appreciated.

Answer

From a student's point of view, here is how I tackle the situation. You can break it down by the circumstances in which you meet, and thus guesstimate how much time the professor has for you.

Asking "may I talk to / email you later about...?" can be useful if it's not the right time and place for a useful discussion.

DTK's answer has good points about how you should approach a conversation.

Of course there will be some variations depending on the culture and environment of your department, and how well you know the prof you're talking to.

Hallway / elevator encounters

If they look busy or rushed, perhaps best save it till later. Otherwise, keep questions very brief, concisely answerable, and related to something you know they're interested in (i.e. their own research field).

Discussions at semi-social occasions, such as at a gathering after a seminar, or breaks at a conference

Here, the professor probably has no immediate obligations or things urgently pressing on his or her mind (or else they'd have disappeared by now). This is a better opportunity to ask broader or more complicated questions, or if they work in a related field you could ask for an opinion on a problem you've been thinking about.

Social occasions such as at lunchtime, conference dinner, etc.

As above, but they may even prefer to talk about something other than their own field of expertise -- sports, politics, an interesting paper from a completely unrelated discipline that you read whilst procrastinating, etc. If you don't know the professor very well, you might want to wait for him or her to initiate steering the conversation in that direction.