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In the corner of the garden between the Centre of Mathematical Sciences and the Isaac Newton Institute in Cambridge, sits a reminder of our ongoing quest to understand gravity: an apple tree that was taken as a cutting from the tree at Newton's birthplace, the tree that is said to have inspired his theory of gravity. Newton's theory was extended to the cosmological scales by Einstein's theory of general relativity – but can supergravity explain how gravity works in the quantum world?

This is an excerpt from Stephen Hawking's address to his 70th birthday symposium which took place on 8th January 2011 in Cambridge.

"Astronomers are used to large numbers, but few are as large as the odds I'd have given this celebration today," is how Astronomer Royal Martin Rees started his presentation at Stephen Hawking's birthday symposium yesterday. He was talking about the 1960s when he first met Hawking who was then already suffering motor neurone disease. But Rees' prediction has been proved wrong. Hawking turned 70 yesterday and since the time of their first meeting he has made enormous contributions to cosmology and physics.

Human reasoning is biased and illogical. At least that's what a huge body of psychological research seems to show. But now a psychological scientist from the University of Toulouse in France has come up with a new theory: that logical and probabilistic thinking is an intuitive part of decision making, only its conclusions often lose out to heuristic considerations.

In this, the second part of our interview, John Conway explains how the Kochen-Specker Theorem from 1965 not only seemed to explain the EPR Paradox, it also provided the first hint of Conway and Kochen's Free Will Theorem.

On August 19, 2004, John Conway was standing with his friend Simon Kochen at the blackboard in Kochen’s office in Princeton. They had been trying to understand a thought experiment involving quantum physics and relativity. What they discovered, and how they described it, created one of the most controversial theorems of their careers: The Free Will Theorem.

In this, the third part of our interview, John Conway continues to explain the Free Will Theorem and how it has changed his perception of the Universe.

Researchers in Germany have created a rare example of a weird phenomenon predicted by quantum mechanics: quantum entanglement, or as Einstein called it, "spooky action at a distance". The idea, loosely speaking, is that particles which have once interacted physically remain linked to each other even when they're moved apart and seem to affect each other instantaneously.

Whenever you smell the lovely smell of fresh coffee or drop a tea bag into hot water you're benefiting from diffusion: the fact that particles moving at random under the influence of thermal energy spread themselves around. It's this process that wafts coffee particles towards your nose and allows the tea to spread around the water. Diffusion underlies a huge number of processes and it has been studied intensively for over 150 years. Yet it wasn't until very recently that one of the most important assumptions of the underlying theory was confirmed in an experiment.

In 1982 Dan Shechtman discovered a crystal that would revolutionise chemistry. He has just been awarded the 2011 Nobel Prize in Chemistry for his discovery — but has the Nobel committee missed out a chance to honour a mathematician for his role in this revolution as well?

This year's Nobel Prize in Physics was awarded for a discovery that proved Einstein wrong and right at the same time.