When the mathematician AK Erlang first used probability theory to model telephone networks in the early twentieth century he could hardly have imagined that the science he founded would one day help solve a most pressing global problem: how to wean ourselves off fossil fuels and switch to renewable energy sources.

Progress in pure mathematics has its own tempo. Major questions may remain open for decades, even centuries, and once an answer has been found, it can take a collaborative effort of many mathematicians in the field to check that it is correct. The New Contexts for Stable Homotopy Theory programme, held at the Institute in 2002, is a prime example of how its research programmes can benefit researchers and its lead to landmark results.

Artificial neural networks grew out of researchers' attempts to mimick the human brain. In 1997 the Isaac Newton Institute hosted a landmark research programme in the area. Today, neural networks are able to learn how to perform complex tasks and are crucial in many areas of life, from medicine to the Xbox.

Will we ever be able to make computers that think and feel? If not, why not? And what has all this got to do with tiles? Plus talks to Sir Roger Penrose about all this and more.

Astronomer Royal Sir Martin Rees gives Plus a whistlestop tour of some of the more extraordinary features of our cosmos, and explains how lucky we are that the universe is the way it is.

What happens when one black hole meets another? Professor Kip Thorne shows us how to eavesdrop on these cosmic events by watching for telltale gravitational waves.