Towards the Complexity of Riemann Mappings (Extended Abstract)

We show that under reasonable assumptions there exist Riemann mappings which are as hard as tally $sharp$-P even in the non-uniform case. More precisely, we show that under a widely accepted conjecture from numerical mathematics there exist single domains with simple, i.e. polynomial time computable, smooth boundary whose Riemann mapping is polynomial time computable if and only if tally $sharp$-P equals P. Additionally, we give similar results without any assumptions using tally $UP$ instead of $sharp$-P and show that Riemann mappings of domains with polynomial time computable analytic boundaries are polynomial time computable