In the last program in the series, Marcus du Sautoy looks at some of the great unsolved problems that confronts mathematics in the twentieth century and tells the stories of the mathematicians who would try to crack them.

Mathematicians like Georg Cantor, who investigated a subject that many of the finest mathematical minds had avoided – infinity. Cantor discovered that there were different kinds of infinity - and that some were bigger than others. Henri Poincaré was trying to solve one mathematical problem when he accidentally stumbled on chaos theory, which has led to a range of ‘smart’ technologies, including machines which control the regularity of heart beats. But in the middle of the twentieth century, mathematics was itself thrown into chaos.

Kurt Gödel, an active member of the famous 'Vienna Circle’ of philosophers, detonated a 'logic bomb’ under 3,000 years of mathematics when he showed that it was impossible for mathematics to prove its own consistency - and that the unknowable is itself an integral part of mathematics.

In this program, Marcus looks at the startling discoveries of the American mathematician Paul Cohen, who established that there were several different sorts of mathematics in which conflicting answers to the same question were possible. He also examines the work of André Weil and his colleagues, who developed algebraic geometry, a field of study which helped to solve many of mathematics' toughest equations, including Fermat’s Last Theorem.

He also reflects on the contributions of Alexander Grothendieck, whose ideas have had a major influence on current mathematical thinking about the hidden structures behind all mathematics. Marcus concludes his journey by considering the great unsolved problems of mathematics today, including the Riemann Hypothesis - a conjecture about the distribution of prime numbers – which are the atoms of the mathematical universe. There is now $1 million prize and a place in the history books for anyone who can prove Riemann’s theorem.

http://www.open2.net/storyofmaths/infinityandbeyond.html

Other episodes:
Part 1: Ancient Mathematics - The language of the Universe
http://www.videosift.com/video/The-Story-of-Maths-1-The-Language-of-the-Universe
Part2: The Genius of the East
http://www.videosift.com/video/The-History-of-Mathematics-2-The-Genius-of-the-East
Part 3: The Frontiers of Space
http://www.videosift.com/video/The-History-of-Mathematics-3-The-Frontiers-of-Space