Absolutely. There are a few ways to do this, depending on your needs.
You may make a Resource with alias "user" and write a Snippet that checks if the current logged in user is a valid user (sanitizing MODx). Then you may get that user's profile.
For further exploration: you may use @sottwell's suggestion and create a Plugin that fires on OnPageNotFound and accepts a parameter. (Such that http.//domain.tld/user/5). That parameter could grab the profile of that user ID when present, but when not present is the current user's profile. This is fairly simple to implement and I use such techniques in
this tutorial. This tutorial does not directly apply to profiles, but is pretty easy to apply for just profiles.
Both suggestions would require a single initial Resource named "user", and the Snippet to get the Profile information.