We study a system of interacting urns where balls of different colour/type compete for their survival, and annihilate upon contact. We shall consider the finite setting, i.e. when the underlying graph is finite and connected. In this case it is known that coexistence is not possible between two types. However, for competition between three or more types, the possibility of coexistence depends on the underlying graph. We prove a conjecture stating that when the underlying graph is a cycle, then the competition between three or more types has a single survivor almost surely. As part of the proof we give a detailed description of an auto-annihilative process on the cycle, which can be perceived as an expression of the geometry of a Möbius strip in a discrete setting. (Joint work with Carolina Fransson.)

**Date**: 20 November 2023, 14:00 (Monday, 7th week, Michaelmas 2023)**Venue**: Mathematical Institute

Woodstock Road OX2 6GGSee location on maps.ox**Details**: L5**Speaker**: Daniel Ahlberg (Stockholm University)**Organising department**: Department of Statistics**Organisers**: Julien Berestycki (University of Oxford), Christina Goldschmidt (Oxford), James Martin (Department of Statistics, University of Oxford)**Part of**: Probability seminar**Booking required?**: Not required**Audience**: Members of the University only- Editor: Julien Berestycki