Can a Facebook application or anyone track that information? Some applications advertise this feature. I'm not sure whether I can trust them.
Facebook does not allow users to track profile views or see statistics on how often a particular piece of content has been viewed and by whom without your consent.
If an Application claims to provide this functionality, you can report an application for abuse by going to the application’s About page and clicking "Report Application" at the bottom of the page, or by clicking "Report" at the bottom of any canvas page within the application.
Well, You can use Google Analytics. There are many apps available on Facebook. I have added some of them to my profle page and added Google Analytics code in that. Now I can view the number of views of the profile.