  The unprecedented prowess of measurement techniques provides a detailed,
multi-scale look into the depths of living systems. But understanding these
avalanches of high-dimensional data -- by distilling underlying principles of
more general nature -- necessitates dimensional reduction. We will explain how
some geometric insights by mathematicians allow one to identify genuine
hallmarks of biological systems and distinguish them from universal properties
of generic data sets. In particular, life involves many layers of information
processing and internal representations that give rise to extreme dimensional
reduction. We illustrate these general principles in a concrete example of a
model of protein evolution.
