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The -omic revolution in biology, and parallel developments in microscopy and imaging, have opened up fascinating new opportunities for analysing biological data using tools from the mathematical sciences. However, the kind of data we have and the way we interpret them are determined by the conceptual landscape through which experimentalists reason about biology. In this talk, I will consider how mathematics can help to shape that conceptual landscape and thereby suggest new experimental strategies. I will describe some of our recent work on how eukaryotic genes are regulated, which tries to update conventional thinking in this field, which is largely derived from bacterial studies, and I will point out how this exercise gives rise to mathematical conjectures for which we currently have no solutions.