名失己, 非士也 [One who pursues fame at the risk of losing one's self, is not a scholar]. (莊子 [Zhuangzi], “大宗師 [The Grandmaster]“)

Going into a field or department simply because it is glamorous is not a good idea, nor is focusing on the most famous problems (or mathematicians) within a field, solely because they are famous – honestly, there isn’t that much fame or glamour in mathematics overall, and it is not worth chasing these things as your primary goal. Anything glamorous is likely to be highly competitive, and only those with the most solid of backgrounds (in particular, lots of experience with less glamorous aspects of the field) are likely to get anywhere.

A famous unsolved problem is almost never solved ab nihilo. One has to first spend much time and effort working on simpler (and much less famous) model problems, acquiring techniques, intuition, partial results, context, and literature, thus enabling fruitful approaches to the problem and ruling out fruitless ones, before having any real chance of solving any really big problem in the area. (Occasionally, one of these problems falls relatively easily, simply because the right group of people with the right set of tools hadn’t had a chance to look at the problem before, but this is usually not the case for the very intensively studied problems – particularly those which already have a substantial body of “no go” theorems and counterexamples which rule out entire strategies of attack.)

For similar reasons, one should never make prizes or recognition a primary reason for pursuing mathematics; it is a better strategy in the long-term to just produce good mathematics and contribute to your field, and the prizes and recognition will take care of themselves (and be well-earned when they eventually do appear).

On the other hand, it can be worth researching why a problem or mathematician is famous, or how an institution or department earnt its prestige; such specific information can help you decide whether this problem, mathematician, or department would be of interest to you. See also “Which universities should I apply to?