Flexible and user-adaptive statistical inference In this proposal we want to develop mathematical theory for testing many hypotheses simultaneously. Our new theory brings a threefold flexibility into the research: in terms of sample size, analysis and model choice. The researcher may look at the data and the intermediate results, and decide on that basis to add more data, to add more hypotheses – or focus on the most promising ones – and she does not have to restrict herself to a specific pre-specified model. In the mean time our methods retain strong statistical error guarantees.