Date:	Tue, December 1, 2015
Time:	14:15
Place:	Research I Seminar Room

Abstract: Entire holomorphic functions can be iterated, yielding a complex dynamical system. One then looks at the behaviour of points under such iteration and tries to classify maps by their dynamical behaviour. Combinatorial objects dependent on this dynamics only provide an attractive means of classification. Post-critically finite polynomials and also some classes of rational functions have been fully classified, hence the focus is on trying to extend this towards transcendental functions. The appropriate class are post-singularly transcendental functions now. For rational functions, all such classifications crucially depend on a deep theorem by Bill Thurston, proven in the 1980s already. The lack of such a theorem in the general transcendental case has prevented further progress here - with the only exception being post-singularly finite exponential maps, for which an extension of Thurston's theorem by Hubbard, Schleicher and Shishikura in 2009 enabled a combinatorial classification. In this talk, I will review and motivate the underlying basic concepts,introduce the main tools to be used and then describe several ideas for further progress towards a combinatorial classification, including classifying new large families of maps as well as better understanding the combinatorial tools at hand. This is based on current in-progress work by Hazemach and Schleicher towards proving an analogue of Thurston's theorem in full generality.