Today was the first day of Integers 2025. I haven't been to Integers since 2013, so this is a very exciting trip for me. Alfie is here and I've already met two new gamesters!
But I'm getting ahead of myself. Quick explanation: Integers is a Number Theory and Combinatorics conference, with lots of number theorists and (as I've experienced in my previous visits) a small bunch of gamesters. This year it looks like there's four of us and we all talked today. (I hope I'm wrong about that!)
I went to a lot of number theory talks, but I didn't write things up about them, mostly because I just don't know a lot about that field. The plenaries were great. Number theorists are apparently hilarious. Mel Nathanson had some great quotes, including "If I had to explain to grown-ups why I was interested in this, that'd be hard!" and "... one of the rare times doing mathematics I felt like a real scientist, ..." I followed that talk a little bit, because I could correctly discern the words when "sumset" and "some set" were used in the same sentence spoken aloud.
This was also the first time I ever introduced speakers at someone else's conference. (I think this counts as "chairing" though they weren't sessions.) I will freely admit that I was 1 minute late to one of the talks I was introducing. (We found lunch too far away.) People were very nice about it, probably because I showed up quite sweaty.
You're likely here to read my summaries of talks, though. Enough delaying, here they are:
Francesca Yu: "Structure-Biased Maker-Breaker Games"
Francesca talked about maker-breaker games on complete graphs where both players choose unclaimed edges and the Maker is trying to create some kind of structure (e.g. a triangle). In these games, however, the Breaker gets to claim b edges per turn instead of just one. The threshold bias for a given game is the lowest possible b such that the Breaker still wins. Francesca looked at a cool variant of this: the Breaker instead has to claim a specific structure in the graph instead of just any b edges. She found bounds for the threshold sizes for structures of stars and disjoint edges when the Maker is trying to build a triangle. A lot of these thresholds are in O(n). It is very worth mentioning that Francesca just graduated with her bachelor's degree!
Caroline Cashman: "Black Hole Zeckendorf Games"
Caroline reminded us that any positive integer can be written as the sum of non-adjacent Fibonacci numbers, known as the Zeckendorf representation. She then talked about the Zeckendorf game on weighted sums of Fibonacci numbers where moves include changing the representation via some small adjustments. Each of these moves changes the weighted sum towards the Zeckendorf decomposition, so that the player that finally moves to the correct representation wins. She looked at a "Black Hole" variant so that terms for the fourth fibonacci number (3) and above just fall off, meaning each state can be written as a triple (a, b, c). She showed that (a, 0, 0) is always a P position and that (1, 0, c) is in P when c is not equal to 3. She had other such classifications as well. It is very worth mentioning here that Caroline is a rising senior in college!
Alfie Davies: "Comparisons in the Misère Blocking Universe"
Alfie talked about the misère play convention and the definition of the <= operator. He did a really good job of explaining the difficulties of misère analysis as compared to normal play using Aaron Siegel's recent work on the numbers of unique values. He explained the idea behind Thane Plambeck's work on restricting to specific universes. he gave the definition of universes and talked about how only finite steps are needed to prove their feasibility in many cases. He then explained the details of those tests in the blocking universe. He really did an excellent job of explaining the important parts to the audience of mostly number theorists.
What a great day of talks! I'm especially impressed by these two undergrads for their excellent work. I hope they continue to do work in combinatorial games!
I don't expect there will be other CGT talks here, but if I'm wrong, I'll post again!
