The holy grail for 21st century physics is to produce a unified theory of everything that can describe the world at every level, from the tiniest particles to the largest galaxies. Currently the strongest contender for such a theory is something called M-theory. So what is this supposed mother of all theories all about?

The men's and women's 100 metre freestyle swimming races are set to be two of the most glamorous events of the London 2012 Olympic Games. Much has been made of the swimming events for London 2012 because the previous 2008 Beijing Olympics saw an unprecedented number of new world records, due to the use of controversial swimsuits. So how do these suits improve performance?

A recent study from Harvard reported that eating red meat is associated with a 13% increased risk of death. But what does this mean? Surely our risk of death is already 100%, and a risk of 113% does not seem very sensible? To really interpret this number we need to use some maths.

If you look at the pattern of rowers in a racing four or eight rowing boat then you expect to find them positioned in a symmetrical fashion, alternately right-left, right-left as you go from one end of the boat to the other. However, the regularity of the rower's positions hides a significant asymmetry that affects the way the boat will move through the water.

Recent discoveries have made it possible to control computer games by thought alone, or work out what kind of item someone is thinking about from their brain signals. And that's not all. Researchers were able to use brain scans to reconstruct what someone was looking at. In these experiments the scientists were literally able to see what people were thinking. A worrying thought, perhaps. But how did they do it?

Some things are so familiar to us that they are simply expected, and we may forget to wonder why they should be that way in the first place. Sex ratios are a good example of this: the number of men and women in the world is roughly equal, but why should this be the case? A simple mathematical argument provides an answer.

The Jerusalem Chords Bridge, Israel, was built to make way for the city's light rail train system. Its design took into consideration more than just utility — it is a work of art, designed as a monument. Its beauty rests not only in the visual appearance of its criss-cross cables, but also in the mathematics that lies behind it. So let's take a deeper look at 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?

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.