One of the most bitterly contested proofs in modern mathematics may be on the verge of being untangled. Two projects, both aiming to use a computer program to cast new light on the controversy, are now up and running - with one having operated in secret for more than two years already. The developments are a positive sign that the row might find a solution, say mathematicians. The saga began in 2012 when Shinichi Mochizuki at Kyoto University, Japan, claimed to have proved a famous idea called the ABC conjecture, posting a 500-page proof online. The conjecture is simple to state, concerning prime numbers involved in solutions to the equation a + b = c and how these numbers relate to each other.

Amateur mathematicians are closing in on an unimaginably huge number – one so large that it brushes up on the edge of what is even knowable within the framework of modern mathematics. It all stems from a seemingly simple question: how do you know if a computer program will run forever? Answering this starts with mathematician Alan Turing. In the 1930s, he showed that any computer algorithm can be mimicked by imagining a simple "Turing machine" that reads and writes 0s and 1s on an infinitely long tape by following a set of instructions called states, with more complex algorithms requiring more states. For every number of states, such as 5 or 100, there are finitely many corresponding Turing machines, but it is unclear for how long each of these machines must run.

We describe the new field of mathematical analysis of deep learning. This field emerged around a list of research questions that were not answered within the classical framework of learning theory. These questions concern: the outstanding generalization power of overparametrized neural networks, the role of depth in deep architectures, the apparent absence of the curse of dimensionality, the surprisingly successful optimization performance despite the non-convexity of the problem, understanding what features are learned, why deep architectures perform exceptionally well in physical problems, and which fine aspects of an architecture affect the behavior of a learning task in which way. We present an overview of modern approaches that yield partial answers to these questions. For selected approaches, we describe the main ideas in more detail.