# Abstracts

**05 October 2018: Agelos Georgakopoulos (Warwick) From mafia expansion to analytic functions in percolation theory**Abstract:

**I will present a (finite) random graph model that admits various definitions, one of which is via a percolation model on an infinite group. This will lead us to an excursion into classical results and open problems in percolation theory.**

**12 October 2018: Robert Gilman (Stevens Institute) Generating the symmetric group**

Abstract:

It is well known that two randomly chosen permutations generate the alternating or symmetric group of degree n with probability tending to 1 as n goes to infinity. This is bad news if you would like to sample 2-generator permutation groups. We study a sampling method which provably avoids this problem, and we discuss a practical heuristic variation.

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19 October 2018: Joel Hamkins (Oxford) The rearrangement number: how many rearrangements of a series suffice to validate absolute convergence?
**Abstract:

**The Riemann rearrangement theorem asserts that a series $\sum_n a_n$ is absolutely convergent if and only if every rearrangement $\sum_n a_{p(n)}$ of it is convergent, and furthermore, any conditionally convergent series can be rearranged so as to converge to any desired extended real value. How many rearrangements $p$ suffice to test for absolute convergence in this way? The rearrangement number, a new cardinal characteristic of the continuum, is the smallest size of a family of permutations, such that whenever the convergence and value of a convergent series is invariant by all these permutations, then it is absolutely convergent. The exact value of the rearrangement number turns out to be independent of the axioms of set theory. In this talk, I shall place the rearrangement number into a discussion of cardinal characteristics of the continuum, including an elementary introduction to the continuum hypothesis and an account of Freiling's axiom of symmetry.**

**26 October 2018: Andrzej Zuk (Paris 7 and Imperial)**

*Abstract:*

**From PDEs to groups**We present a construction which associates to a KdV equation the lamplighter group. In order to establish this relation we use automata and random walks on ultra discrete limits. It is also related to the L2 Betti numbers introduced by Atiyah which are homotopy invariants of closed manifolds.