Ultimately, the answer depends upon what admissions system a department uses.

• If you need to obtain admission to a group at the same time as (or instead of) the department as a whole, then it is of course absolutely critical that you make contacts ahead of time! In general, I'd even start contacting people well before the time of application. However, make sure that your contact is substantial. When it becomes time to apply, you will have a better shot.

• In many departments, you apply for admission to the department as a whole. After your admission, you select an advisor to work for. In such cases, it's not really critical for you to have a direct contact within the department, since it's not necessarily a given that the professor you want to work for sits on the admissions committee. In such cases, you'll have an indirect connection at best. That said, it's still a good idea to have a contact in a department during the admissions process. You don't lose anything by it—unless the professor has a bad reputation amongst his colleagues (and then you might not want to work for him or her, anyways!).

Now to answer some of the other issues raised.

Email or other forms of contact. Face-to-face or phone interactions rank above email interactions. There's no doubt about that. However, an email interaction—if actually substantial—can also be viable. However, a quick emil telling someone you're applying and interested in working for them won't really get you anywhere.

Number of people to contact. There is of course no limit to the number of potential advisors you can contact. To some extent, they're competing for you just as much as you're competing for them!

Can I apply on credentials alone? At most top departments (where application is done at the department level), I think it's entirely possible to apply on credentials and recommendations alone. Having the contacts can obviously help, but not having it won't ruin your chances for admissions, either.

Ultimately, the answer depends upon what admissions system a department uses.

• If you need to obtain admission to a group at the same time as (or instead of) the department as a whole, then it is of course absolutely critical that you make contacts ahead of time! In general, I'd even start contacting people well before the time of application. However, make sure that your contact is substantial. When it becomes time to apply, you will have a better shot.

• In many departments, you apply for admission to the department as a whole. After your admission, you select an advisor to work for. In such cases, it's not really critical for you to have a direct contact within the department, since it's not necessarily a given that the professor you want to work for sits on the admissions committee. In such cases, you'll have an indirect connection at best. That said, it's still a good idea to have a contact in a department during the admissions process. You don't lose anything by it—unless the professor has a bad reputation amongst his colleagues (and then you might not want to work for him or her, anyways!). By the way, I would caution strongly against going to such a school if there's only one professor at the department you'd be interested in working for. You're taking a very large risk under such circumstances.

Now to answer some of the other issues raised.

Email or other forms of contact. Face-to-face or phone interactions rank above email interactions. There's no doubt about that. However, an email interaction—if actually substantial—can also be viable. However, a quick emil telling someone you're applying and interested in working for them won't really get you anywhere.

Number of people to contact. There is of course no limit to the number of potential advisors you can contact. To some extent, they're competing for you just as much as you're competing for them!

Can I apply on credentials alone? At most top departments (where application is done at the department level), I think it's entirely possible to apply on credentials and recommendations alone. Having the contacts can obviously help, but not having it won't ruin your chances for admissions, either.