Tomorrow morning (9am GMT) I am giving a seminar in Novosibirsk (virtually of course). The abstract is below. If you are interested in attending let me know I will give you the Zoom link. Title: Nonschurian primitive association schemes with many automorphisms Abstract: This talk is about primitive coherent configurations of degree with more than automorphisms, for constant . Babai conjectured that all such are so-called Cameron schemes, the orbital configurations of large primitive groups for . We will describe several families of examples that show Babai’s conjecture is actually wro ..read more

The pancake sorting problem is to determine the diameter of the pancake graph, the Cayley graph of with respect to the generating set , where is defined (in two-line notation) by (i.e., flips the top pancakes). It is not hard to see that the diameter is between and . Both lower and upper bound have been sharpened by a constant factor, and a natural conjecture is that converges to some intermediate constant (to be determined, ideally). Here is a natural model problem: estimate for each fixed as . Clearly , and we should be able to detect some fall-off from this trivial upper bound. Pro ..read more

Yesterday Urban Jezernik and I uploaded to the arxiv our preprint Babai’s conjecture for high-rank classical groups with random generators. I want to use this space to explain what our work is about and where it fits in the literature. To try to cater for a possibly wide range of mathematical backgrounds, I will start friendly and informal and get progressively more precise and technical. Thanks in advance for your attention! The subject matter is diameters of groups. A familiar example is Rubik’s cube, which has ( billion billion) possible configurations. The set of all configurations of th ..read more

Problem 1 Let be a group of order . Is there a bijection such that the map is also a bijection? For example, if has odd order, you can just take . Then , and because every element has odd order this defines a bijection. If , then it suffices to find a linear map without an eigenvalue. Cyclic groups of even order never have complete mappings (option 1: stop reading and try to prove this as a diverting puzzle; option 2: read on, and it will be explained and become essential). In general, a solution to Problem 1 is called a complete mapping. The definition of complete mapping is a little s ..read more

Jeff Kahn, Bhargav Narayanan, Sophie Spirkl, and I have just uploaded to the arxiv our note On symmetric intersecting families of vectors. We are interested in problems in extremal set theory with symmetry constraints. Often in extremal set theory it is useful to “compress”, which you can think of as trying to reduce your arbitrary system to a simpler system, one that may be easier to understand. With symmetry constraints, the added challenge is that your compression method should also satisfy those symmetry constraints. Our particular focus is symmetric intersecting families. A family of sub ..read more