I haven't finished applying to one of the grad schools I'm interested in. But, one of the professors emailed me asking if I was still applying. He said I had strong letters of recommendations and told me a little bit about a project he is working on. Says he has student positions to fill. I assume he's partially interested in me because of my degree in mathematics where other applicants probably have degrees in other fields.
I want to know a few things. One, if I havent actually sent in my application can he see it? Or can he only see the letter of recommendation? Two, is it common for professors to be the ones to reach out to a student? I know most people encourage contacting professors you potentially want to work with but was surprised when a professor contacted me. Three, would this increase my odds of getting into the grad school? I.e. if this professor decides he wants to work with me could he have influence over my admission? Lastly, he said if there's enough interest on both of out parts he'd be willing to "support a trip to his lab". How should I interpret that?