Melvyn Bragg and guests discuss the problem of P versus NP, which has a bearing on online security. There is a $1,000,000 prize on offer from the Clay Mathematical Institute for the first person to come up with a complete solution. At its heart is the question "are there problems for which the answers can be checked by computers, but not found in a reasonable time?" If the answer to that is yes, then P does not equal NP. However, if all answers can be found easily as well as checked, if only we knew how, then P equals NP. The area has intrigued mathematicians and computer scientists since Alan Turing, in 1936, found that it's impossible to decide in general whether an algorithm will run forever on some problems. Resting on P versus NP is the security of all online transactions which are currently encrypted: if it transpires that P=NP, if answers could be found as easily as checked, computers could crack passwords in moments.

With

Colva Roney-Dougal

Timothy Gowers

And

Leslie Ann Goldberg

Producer: Simon Tillotson.

Melvyn Bragg and guests discuss the rise of the idea of perpetual motion and its decline, in the 19th Century, with the Laws of Thermodynamics. For hundreds of years, some of the greatest names in science thought there might be machines that could power themselves endlessly. Leonardo Da Vinci tested the idea of a constantly-spinning wheel and Robert Boyle tried to recirculate water from a draining flask. Gottfried Leibniz supported a friend, Orffyreus, who claimed he had built an ever-rotating wheel. An increasing number of scientists voiced their doubts about perpetual motion, from the time of Galileo, but none could prove it was impossible. For scientists, the designs were a way of exploring the laws of nature. Others claimed their inventions actually worked, and promised a limitless supply of energy. It was not until the 19th Century that the picture became clearer, with the experiments of James Joule and Robert Mayer on the links between heat and work, and the establishment of the First and Second Laws of Thermodynamics.

With

Ruth Gregory

Frank Close

and

Steven Bramwell

Producer: Simon Tillotson.

In 1977, scientists in the submersible "Alvin" were exploring the deep ocean bed off the Galapagos Islands. In the dark, they discovered hydrothermal vents, like chimneys, from which superheated water flowed. Around the vents there was an extraordinary variety of life, feeding on microbes which were thriving in the acidity and extreme temperature of the vents. While it was already known that some microbes are extremophiles, thriving in extreme conditions, such as the springs and geysers of Yellowstone Park (pictured), that had not prepared scientists for what they now found. Since the "Alvin" discovery, the increased study of extremophile microbes has revealed much about what is and is not needed to sustain life on Earth and given rise to new theories about how and where life began. It has also suggested forms and places in which life might be found elsewhere in the Universe.

With

Monica Grady

Ian Crawford

And

Nick Lane

Producer: Simon Tillotson.

While glass items have been made for at least 5,000 years, scientists are yet to explain, conclusively, what happens when the substance it's made from moves from a molten state to its hard, transparent phase. It is said to be one of the great unsolved problems in physics. While apparently solid, the glass retains certain properties of a liquid. At times, ways of making glass have been highly confidential; in Venice in the Middle Ages, disclosure of manufacturing techniques was a capital offence. Despite the complexity and mystery of the science of glass, glass technology has continued to advance from sheet glass to crystal glass, optical glass and prisms, to float glasses, chemical glassware, fibre optics and metal glasses.

With:

Dame Athene Donald

Jim Bennett

Paul McMillan

Producer: Simon Tillotson.

Melvyn Bragg and his guests discuss the Earth's Core. The inner core is an extremely dense, solid ball of iron and nickel, the size of the Moon, while the outer core is a flowing liquid, the size of Mars. Thanks to the magnetic fields produced within the core, life on Earth is possible. The magnetosphere protects the Earth from much of the Sun's radiation and the flow of particles which would otherwise strip away the atmosphere. The precise structure of the core and its properties have been fascinating scientists from the Renaissance. Recent seismographs show the picture is even more complex than we might have imagined, with suggestions that the core is spinning at a different speed and on a different axis from the surface.

With

Stephen Blundell

Arwen Deuss

and

Simon Redfern

Producer: Simon Tillotson.

Melvyn Bragg and his guests discuss the scientific achievements of the Curie family. In 1903 Marie and Pierre Curie shared a Nobel Prize in Physics with Henri Becquerel for their work on radioactivity, a term which Marie coined. Marie went on to win a Nobel in Chemistry eight years later; remarkably, her daughter Irène Joliot-Curie would later share a Nobel with her husband Frédéric Joliot-Curie for their discovery that it was possible to create radioactive materials in the laboratory. The work of the Curies added immensely to our knowledge of fundamental physics and paved the way for modern treatments for cancer and other illnesses.

With:

Patricia Fara

Robert Fox

Steven T Bramwell

Producer: Simon Tillotson.

Melvyn Bragg and his guests discuss dark matter, the mysterious and invisible substance which is believed to make up most of the Universe. In 1932 the Dutch astronomer Jan Oort noticed that the speed at which galaxies moved was at odds with the amount of material they appeared to contain. He hypothesized that much of this 'missing' matter was simply invisible to telescopes. Today astronomers and particle physicists are still fascinated by the search for dark matter and the question of what it is.

With

Carolin Crawford

Carlos Frenk

Anne Green

Producer: Simon Tillotson.

Melvyn Bragg and his guests discuss the photon, one of the most enigmatic objects in the Universe. Generations of scientists have struggled to understand the nature of light. In the late nineteenth century it seemed clear that light was an electromagnetic wave. But the work of physicists including Planck and Einstein shed doubt on this theory. Today scientists accept that light can behave both as a wave and a particle, the latter known as the photon. Understanding light in terms of photons has enabled the development of some of the most important technology of the last fifty years.

With:

Frank Close

Wendy Flavell

Susan Cartwright

Producer: Thomas Morris.

Melvyn Bragg and guests discuss Behavioural Ecology, the scientific study of animal behaviour.

What factors influence where and what an animal chooses to eat? Why do some animals mate for life whilst others are promiscuous? Behavioural ecologists approach questions like these using Darwin's theory of natural selection, along with ideas drawn from game theory and the economics of consumer choice.

Scientists had always been interested in why animals behave as they do, but before behavioural ecology this area of zoology never got much beyond a collection of interesting anecdotes. Behavioural ecology gave researchers techniques for constructing rigorous mathematical models of how animals act under different circumstances, and for predicting how they will react if circumstances change. Behavioural ecology emerged as a branch of zoology in the second half of the 20th century and proponents say it revolutionized our understanding of animals in their environments.

GUESTS

Steve Jones, Emeritus Professor of Genetics at University College London

Rebecca Kilner, Professor of Evolutionary Biology at the University of Cambridge

John Krebs, Principal of Jesus College at the University of Oxford

Producer: Luke Mulhall.

Melvyn Bragg and guests discuss Isambard Kingdom Brunel, the Victorian engineer responsible for bridges, tunnels and railways still in use today more than 150 years after they were built. Brunel represented the cutting edge of technological innovation in Victorian Britain, and his life gives us a window onto the social changes that accompanied the Industrial Revolution. Yet his work was not always successful, and his innovative approach to engineering projects was often greeted with suspicion from investors.

Guests:

Julia Elton, former President of the Newcomen Society for the History of Engineering and Technology

Ben Marsden, Senior Lecturer in the School of Divinity, History and Philosophy at the University of Aberdeen

Crosbie Smith, Professor of the History of Science at the University of Kent

Producer: Luke Mulhall.