How do I interact with a person at a bare minimum level without seeming excessively rude?
Background:
I had an incident a couple months ago where a friend of mine told me that a person Yanny from our university had accused me of doing some rather horrible things to my ex from two years ago.
This person is quite close with my ex and they might be having some sort of relationship. After some probing of my friend for what was said, I found out that Yanny hadn't said much, but implied that I had done terrible things (terrible enough that my friend assumed physical abuse or worse).
Our break up was not pretty, and we did some stupid things, the both of us. My worst mistake was not breaking it up sooner and getting to the point where I rummaged through her things and read through her texts at some point when my trust in her was completely gone (yeah really dumb I know definitely a low point in my life). Under no circumstances did I ever though, in any way, threaten her physically or anything of the sort.
Anyhow, over the following three days I had an extended talk over text with this Yanny. He initially wanted me to profusely apologise for what I had done to my ex in the previous relationship (the invasion of her privacy). I explained to him that this is between me and her and none of his business (in a more polite way).
When he wouldn't listen I told him that we had already talked about all these things with my ex and that I thought that I had definitely crossed a line and apologised for it and stated that if she ever wanted to talk about this I'd be open to that. I also still added that this was none of his business and it was a two year old matter anyhow, him opening old wounds would not bring anything good to anyone's life. I also stated that it is a really sucky thing to do, to imply that I'd have abused my ex or something worse.
Finally he did come around apologising himself. I asked if he had said similar things to other people during these two years and he denied it at least (I don't know whether to believe him though, it's not an easy thing to admit).
My friend who brought this to my attention is now more cautious around me (we weren't super close to begin with but he's an honest guy), maybe looking for signs of me to be some sort of covert abuser. My friend also contacted another mutual friend of ours on this and she advised him to have a little faith me and talk to me (and good that he did). I feel like this put some strain on my relationship with that other friend as well.
I have had a couple of months from this incident, but university is starting soon. I'll have to inevitably spend time with this Yanny person but I find myself still feeling angry and hurt. I feel like this person has crossed boundaries and I have trouble forgiving him.
I don't know how I can handle being with him in the same space and social circles (he's well enough liked). I will see him quite a lot in different extracurricular activities since we share a lot of friends in university. For example we go running together for an hour or so each week and have movie nights together. Also in the student commons area we see nearly every day.
I'd really prefer to just not interact with him at all and forget about him and this nonsense, but it's hard since we have so much overlap in university life. How do I interact with Yanny so that I can seem as little rude as possible and still minimize any contact with him, especially in events that last longer periods of time?