James Parkinson (University of Sydney) Friday 20 May, 12-1pm, Place: Carslaw 273 Title: Cone types and automata, and regular partitions in Coxeter groups. Abstract: In 1993 Brink and Howlett showed that finitely generated Coxeter groups are automatic. One ingredient was the construction of a finite state automaton recognising the language of reduced words in the Coxeter group using the remarkable (finite!) set of roots of "elementary roots" of the associated root system. Recently Dehornoy, Dyer, and Hohlweg introduced the notion of a Garside shadow in a Coxeter group, resulting in further constructions of automata recognising the language of reduced words, and various conjectures. In this talk we outline recent joint work with Y. Yau towards resolving these conjectures. This work centres around the notion of a "regular partition" of a Coxeter group, and we show that such partitions are essentially equivalent to the class of automata recognising the language of reduced words. Moreover, we use this framework to prove some fundamental facts about the minimal automata recognising the language of reduced words, and study the associated "cone types" in Coxeter groups.