My Epic Battle Against That One Snarky Classmate

Hey everyone! So I have this ongoing battle with this classmate in my grad studies course. His name is Alex and, oh boy, we've been at each other's throats for months now. It started when he made a snide comment about the research topic I was presenting on, which, by the way, got an A+ from our professor! Ever since then, it's been nothing but passive-aggressive notes in the margins of group assignments and some pretty heated discussions during seminars. You know how they say you can't judge a book by its cover? Well, I decided to apply that philosophy here. I mean, sure Alex seems like this super sharp-witted academic type who probably wears tweed and sips tea all day long while reciting Shakespearean sonnets in perfect iambic pentameter. But deep down inside, under the layers of his scholarly exterior, there's a guy who can't take criticism, or is it just me? Maybe I'm overthinking this whole thing. Anyway, last week was particularly bad when we were working on our project together for the semester-long seminar. Alex had some really strange ideas about how to present our findings and kept pushing back against every single one of my suggestions. By the end, I was so frustrated that I wrote an entire page of notes detailing why his approach wouldn't work, just in case anyone else asked me what I thought. Here's where it gets juicy though: after class yesterday, Alex came up to me with a smug grin and said he'd found a critical flaw in my research methods. My initial reaction was to blow him off because I've heard enough of his critiques for a lifetime already. But then something clicked, maybe if we could actually talk about this stuff without all the drama, there might be room for us both in our project. So here's my question: How do you deal with classmate conflicts like this? Should I continue trying to work things out or should I just accept that sometimes people are jerks and move on? Your thoughts would mean so much to me right now!