Struggling with that new year's resolution to lose a few pounds? Weight not dropping off as fast as you'd expected? A new mathematical model has some good news and some bad news for you. Which would you like to hear first?

How many people died? It's one of the first questions asked in a war or violent conflict, but it's one of the hardest to answer. In the chaos of war many deaths go unrecorded and all sides have an interest in distorting the figures. The best we can do is come up with estimates, but the trouble is that different statistical methods for doing this can produce vastly different results . So how do we know how different methods compare?

Plants are amazingly good at something that is still flummoxing us humans in our quest for sustainable energy sources: turning sunlight into energy in an efficient way. Around 100 bilions tons of biomass are produced annually through photosynthesis. The question is, how exacty do plants do it?

The number 1 can be written as a sum of unit fractions, that is fractions with 1 in the numerator. But how long can we make such a sum?

Yesterday's refusal by the UK government to posthumously pardon Alan Turing makes sad news for maths, computer science and the fight against discrimination. But even if symbolic gestures are, symbolically, being rebuffed, at least Turing's most important legacy — the scientific one — is going stronger than ever. An example is this week's announcement that scientists have devised a biological computer, based on an idea first described by Turing in the 1930s.

Some of the things I overheard at Stephen Hawking's 70th birthday conference did make me wonder whether I hadn't got the wrong building and stumbled in on a sci-fi convention. "The state of the multiverse". "The Universe is simple but strange". "The future for intelligent life is potentially infinite". And — excuse me — "the Big Bang was just the decay of our parent vacuum"?!

This is the first part of the lecture given by Astronomer Royal Martin Rees at Stephen Hawking's birthday symposium.

This is the second part of the lecture given by Astronomer Royal Martin Rees at Stephen Hawking's birthday symposium.

A traditional view of science holds that every system — including ourselves — is no more than the sum of its parts. To understand it, all you have to do is take it apart and see what's happening to the smallest constituents. But the mathematician and cosmologist George Ellis disagrees. He believes that complexity can arise from simple components and physical effects can have non-physical causes, opening a door for our free will to make a difference in a physical world.

Most of us think that we have the capacity to act freely. Our sense of morality, our legal system, our whole culture is based on the idea that there is such a thing as free will. It's embarrassing then that classical physics seems to tell a different story. And what does quantum theory have to say about free will?

Many risks we take don't kill us straight away: think of all the lifestyle frailties we get warned about, such as smoking, drinking, eating badly, not exercising and so on. The microlife aims to make all these chronic risks comparable by showing how much life we lose on average when we're exposed to them.

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?