## Interactions between group theory, number theory, combinatorics and geometry

**Monday 16th March 2020 to Friday 20th March 2020**

09:30 to 09:50 | Registration | ||

09:50 to 10:00 | Welcome from David Abrahams (INI Director) | ||

10:00 to 11:00 |
William Timothy Gowers (University of Cambridge)Partial associativity and rough approximate groups
Let X be a finite set and let o be a binary operation on X that is injective in each variable separately and has the property that x o (y o z) = (x o y) o z for a positive proportion of triples (x,y,z) with x,y,z in X. What can we say about this operation? In particular, must there be some underlying group structure that causes the partial associativity? The answer turns out to be yes ? up to a point. I shall explain what that point is and give some indication of the ideas that go into the proof, which is joint work with Jason Long. I shall also report on a natural strengthening that one might hope for. We identified a likely counterexample, which was recently proved to be a counterexample by Ben Green, so in a certain sense our result cannot be improved. (However, there are still some interesting questions one can ask.) |
INI 1 | |

11:30 to 12:30 |
Michael Giudici (University of Western Australia)Simple groups and graph symmetry |
INI 1 | |

14:30 to 15:30 |
Emmanuel Breuillard (University of Cambridge)Expanders and word maps.
An understanding of word maps on semisimple Lie
groups helps for establishing spectral gap bounds for finite simple groups of
Lie type. In this talk I will discuss how this can also be reversed. Joint work
with P. Varj\'u and O. Becker |
INI 1 |

10:00 to 11:00 |
Tim Burness (University of Bristol)The length and depth of a group |
INI 1 | |

11:00 to 11:10 | Group Photo | ||

11:30 to 12:30 |
Aner Shalev (Hebrew University of Jerusalem)Subset products and applications
In the past two decades there has been intense interest in products of subsets in finite groups. Two important examples are Gowers' theory of Quasi Random Groups and its applications by Nikolov, Pyber, Babai and others, and the theory of Approximate Subgroups and the Product Theorem of Breuillard-Green-Tao and Pyber-Szabo on growth in finite simple groups of Lie type of bounded rank, extending Helfgott's work. These deep theories yield strong results on products of three subsets (covering, growth). What can be said about products of two subsets? I will discuss a recent joint work with Michael Larsen and Pham Tiep on this challenging problem, focusing on products of two normal subsets of finite simple groups, and deriving some applications. The proofs involve algebraic geometry, representation theory and combinatorics. |
INI 1 | |

14:30 to 15:30 |
Nikolay Nikolov (University of Oxford)Rank gradient in profinite groups
In this talk I will focus on the growth of generators in open subgroups of profinite groups. In the case of abstract groups this topic has received a lot of attention but in the profinite situation there are not many results and they tend to be different from those for abstract groups. For example a pro-p group of positive rank gradient must have a dense free subgroup. As far as I know the analogous question for profinite groups is still open. |
INI 1 | |

16:00 to 17:00 |
Cheryl Praeger (University of Western Australia)Kirk Lecture: The mathematics of Shuffling
The crux of a card trick performed with a deck of cards usually depends on understanding how shuffles of the deck change the order of the cards. By understanding which permutations are possible, one knows if a given card may be brought into a certain position. The mathematics of shuffling a deck of 2n cards with two ``perfect shuffles'' was studied thoroughly by Diaconis, Graham and Kantor in 1983. I will report on our efforts to understand a generalisation of this problem, with a so-called "many handed dealer'' shuffling kn cards by cutting into k piles with n cards in each pile and using k! possible shuffles. A conjecture of Medvedoff and Morrison suggests that all possible permutations of the deck of cards are achieved, as long as k is not 4 and n is not a power of k. We confirm this conjecture for three doubly infinite families of integers, including all (k, n) with k > n. We initiate a more general study of shuffle groups, which admit an arbitrary subgroup of shuffles. This is joint work with Carmen Amarra and Luke Morgan. |
INI 1 |

10:00 to 11:00 |
Martin Kassabov (Cornell University)Presentation of simple groups of Ree type |
INI 1 | |

11:30 to 12:30 |
Martin Liebeck (Imperial College London); (Imperial College London)Binary groups |
INI 1 | |

13:30 to 17:00 | Free Afternoon |