Paul Erdős (1913 – 1996) is one of the most celebrated mathematicians of the 20th century. During his long career, he made a number of impressive advances in our understanding of maths and developed whole new fields in the subject. He was born into a Jewish family in Hungary just before the outbreak of World War I, and his life was shaped by the rise of fascism in Europe, anti-Semitism and the Cold War. His reputation for mathematical problem solving is unrivalled and he was extraordinarily prolific. He produced more than 1,500 papers and collaborated with around 500 other academics. He also had an unconventional lifestyle. Instead of having a long-term post at one university, he spent much of his life travelling around visiting other mathematicians, often staying for just a few days. With Colva Roney-Dougal Professor of Pure Mathematics at the University of St AndrewsTimothy Gowers Professor of Mathematics at the College de France in Paris and Fellow of Trinity College, CambridgeandAndrew Treglown Associate Professor in Mathematics at the University of BirminghamThe image above shows a graph occurring in Ramsey Theory. It was created by Dr Katherine Staden, lecturer in the School of Mathematics at the Open University.

Melvyn Bragg and guests discuss the pioneering Danish astronomer Tycho Brahe (1546 – 1601) whose charts offered an unprecedented level of accuracy.In 1572 Brahe's observations of a new star challenged the idea, inherited from Aristotle, that the heavens were unchanging. He went on to create his own observatory complex on the Danish island of Hven, and there, working before the invention of the telescope, he developed innovative instruments and gathered a team of assistants, taking a highly systematic approach to observation. A second, smaller source of renown was his metal prosthetic nose, which he needed after a serious injury sustained in a duel. The image above shows Brahe aged 40, from the Atlas Major by Johann Blaeu. With Ole Grell Emeritus Professor in Early Modern History at the Open University Adam Mosley Associate Professor of History at Swansea University and Emma Perkins Affiliate Scholar in the Department of History and Philosophy of Science at the University of Cambridge.

Melvyn Bragg and guests discuss the discovery made in 1911 by the Dutch physicist Heike Kamerlingh Onnes (1853-1926). He came to call it Superconductivity and it is a set of physical properties that nobody predicted and that none, since, have fully explained. When he lowered the temperature of mercury close to absolute zero and ran an electrical current through it, Kamerlingh Onnes found not that it had low resistance but that it had no resistance. Later, in addition, it was noticed that a superconductor expels its magnetic field. In the century or more that has followed, superconductors have already been used to make MRI scanners and to speed particles through the Large Hadron Collider and they may perhaps bring nuclear fusion a little closer (a step that could be world changing).The image above is from a photograph taken by Stephen Blundell of a piece of superconductor levitating above a magnet.With Nigel Hussey Professor of Experimental Condensed Matter Physics at the University of Bristol and Radbout UniversitySuchitra Sebastian Professor of Physics at the Cavendish Laboratory at the University of CambridgeAndStephen Blundell Professor of Physics at the University of Oxford and Fellow of Mansfield CollegeProducer: Simon Tillotson

Melvyn Bragg and guests discuss the voyage of HMS Challenger which set out from Portsmouth in 1872 with a mission a to explore the ocean depths around the world and search for new life. The scale of the enterprise was breath taking and, for its ambition, it has since been compared to the Apollo missions. The team onboard found thousands of new species, proved there was life on the deepest seabeds and plumbed the Mariana Trench five miles below the surface. Thanks to telegraphy and mailboats, its vast discoveries were shared around the world even while Challenger was at sea, and they are still being studied today, offering insights into the ever-changing oceans that cover so much of the globe and into the health of our planet.The image above is from the journal of Pelham Aldrich R.N. who served on the Challenger Surveying Expedition from 1872-5.WithErika Jones Curator of Navigation and Oceanography at Royal Museums GreenwichSam Robinson Southampton Marine and Maritime Institute Research Fellow at the University of SouthamptonAndGiles Miller Principal Curator of Micropalaeontology at the Natural History Museum LondonProducer: Simon Tillotson

Melvyn Bragg and guests discuss one of the greatest changes in the history of life on Earth. Around 400 million years ago some of our ancestors, the fish, started to become a little more like humans. At the swampy margins between land and water, some fish were turning their fins into limbs, their swim bladders into lungs and developed necks and eventually they became tetrapods, the group to which we and all animals with backbones and limbs belong. After millions of years of this transition, these tetrapod descendants of fish were now ready to leave the water for a new life of walking on land, and with that came an explosion in the diversity of life on Earth.The image above is a representation of Tiktaalik Roseae, a fish with some features of a tetrapod but not one yet, based on a fossil collected in the Canadian Arctic.WithEmily Rayfield Professor of Palaeobiology at the University of BristolMichael Coates Chair and Professor of Organismal Biology and Anatomy at the University of ChicagoAnd Steve Brusatte Professor of Palaeontology and Evolution at the University of EdinburghProducer: Simon Tillotson

Melvyn Bragg and guests discuss an atomic particle that's become inseparable from modernity. JJ Thomson discovered the electron 125 years ago, so revealing that atoms, supposedly the smallest things, were made of even smaller things. He pictured them inside an atomic ball like a plum pudding, with others later identifying their place outside the nucleus - and it is their location on the outer limit that has helped scientists learn so much about electrons and with electrons. We can use electrons to reveal the secrets of other particles and, while electricity exists whether we understand electrons or not, the applications of electricity and electrons grow as our knowledge grows. Many questions, though, remain unanswered.With Victoria Martin Professor of Collider Physics at the University of EdinburghHarry Cliff Research Fellow in Particle Physics at the University of CambridgeAndFrank Close Professor Emeritus of Theoretical Physics and Fellow Emeritus at Exeter College at the University of OxfordProducer: Simon Tillotson

Melvyn Bragg and guests discuss the abrupt transformation of stars after shining brightly for millions or billions of years, once they lack the fuel to counter the force of gravity. Those like our own star, the Sun, become red giants, expanding outwards and consuming nearby planets, only to collapse into dense white dwarves. The massive stars, up to fifty times the mass of the Sun, burst into supernovas, visible from Earth in daytime, and become incredibly dense neutron stars or black holes. In these moments of collapse, the intense heat and pressure can create all the known elements to form gases and dust which may eventually combine to form new stars, new planets and, as on Earth, new life.The image above is of the supernova remnant Cassiopeia A, approximately 10,000 light years away, from a once massive star that died in a supernova explosion that was first seen from Earth in 1690WithMartin Rees Astronomer Royal, Fellow of Trinity College, CambridgeCarolin Crawford Emeritus Member of the Institute of Astronomy and Emeritus Fellow of Emmanuel College, University of CambridgeAndMark Sullivan Professor of Astrophysics at the University of SouthamptonProducer: Simon Tillotson

Melvyn Bragg and guests discuss one of our ancestors, Homo erectus, who thrived on Earth for around two million years whereas we, Homo sapiens, emerged only in the last three hundred thousand years. Homo erectus, or Upright Man, spread from Africa to Asia and it was on the Island of Java that fossilised remains were found in 1891 in an expedition led by Dutch scientist Eugène Dubois. Homo erectus people adapted to different habitats, ate varied food, lived in groups, had stamina to outrun their prey; and discoveries have prompted many theories on the relationship between their diet and the size of their brains, on their ability as seafarers, on their creativity and on their ability to speak and otherwise communicate.The image above is from a diorama at the Moesgaard Museum in Denmark, depicting the Turkana Boy referred to in the programme. With Peter Kjærgaard Director of the Natural History Museum of Denmark and Professor of Evolutionary History at the University of CopenhagenJosé Joordens Senior Researcher in Human Evolution at Naturalis Biodiversity Centre and Professor of Human Evolution at Maastricht UniversityAndMark Maslin Professor of Earth System Science at University College LondonProducer: Simon Tillotson

Melvyn Bragg and guests discuss the study of earthquakes. A massive earthquake in 1755 devastated Lisbon, and this disaster helped inspire a new science of seismology which intensified after San Francisco in 1906 and advanced even further with the need to monitor nuclear tests around the world from 1945 onwards. While we now know so much more about what lies beneath the surface of the Earth, and how rocks move and crack, it remains impossible to predict when earthquakes will happen. Thanks to seismology, though, we have a clearer idea of where earthquakes will happen and how to make some of them less hazardous to lives and homes.WithRebecca Bell Senior lecturer in Geology and Geophysics at Imperial College LondonZoe Mildon Lecturer in Earth Sciences and Future Leaders Fellow at the University of PlymouthAnd James Hammond Reader in Geophysics at Birkbeck, University of LondonProducer: Simon Tillotson

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