Does he want something from me?

Hi everyone, I need some help regarding a situation I'm currently in.

So there's this guy I met 2 weeks ago and he's beginning to really make an impression on me.

How we met? He came and sat in front of me out of the blue in the study room, when the whole room was pretty damn empty, so why did he choose to sit in front of me instead of anywhere else?

Moments later, he spoke to me using the excuse of asking for info regarding coffee in the common room, and from there we started to talk about all kinds of things.

It was a friendly social type of chat, but we really clicked in that we talked about things we were commonly interested in, we're both deep and reflective and we shared our opinions and thoughts on various matters.

I notice that he stares at me a lot accompanied with a smile whenever I talk. He will listen to me attentively, and address whatever I say and add on to it further. If I look away or look down, I can see from the corner of my eye that he is staring at me.

I also notice him observing me, as in he follows my every move and listens attentively to what I'm saying.

So when we finished our chat, he asked for my email, but I had given him my phone number instead. He also gave me his number.

He text me at some point that if I'm free the following week, to let him know so we can meet up again. He left the ball in my court, so it was up to me to decide when we can meet up. I text him in return to let him know what days and times I'm exactly free so that he can choose whenever we can hang out, so I left the ball in his court this time.

A week later, he contacted me via phone call to ask if I was on campus, but I wasn't, so we spoke for a while and I asked him when would he like to meet in the rest of the week. He said he doesn't prefer planning ahead and we can meet up whenever. I don't know why he said it like this- is it because he wasn't very bothered about meeting up, or did he want me to initiate it?

So I text him later that day and arranged for us to meet, and he was up for it.

We met again, but this time I was very tired and a bit slow in my interactions, not to forget kind of nervous. I didn't sleep the whole night, is why I was very tired. He did the same things- stare and smile, and follow my every moves. But this time he was showing more disagreement to my points/opinions whenever we discussed anything, did he do this because he was comfortable in the conversation?

He would also advice me a lot about a particular situation I'm in, and he told me that he was helping me or being a catalyst for me to improve my situation. However, I felt that he might have been kind of bored in this particular meet-up, because I was pretty nervous, and shy because of being tired and out-of-tune.

Before he left, he vaguely and casually said "if you're free just let me know", but he didn't say it assertively or clearly. Does this mean he isn't bothered anymore about seeing me or meeting me? Or does this mean he is trying to put the ball in my court so I can make the first moves, since it's the only way he can tell if I'm interested in him the way he's insofar shown interest in me?

Also, what could he want?

Thank you!

It sounds like he is interested in you but is not very good at asking girls out. You might have to be more proactive if you want this to go anywhere, but not too forward since guys like a little challenge.