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Interferometry images living cells in 3D

Tiny biological samples must normally be prepared before they can be viewed in 3D. Cells, for example, often have their inner components highlighted with fluorescent dyes. But such modifications can disrupt a cell’s normal functions, limiting the possibilities for analysis.

Feld and colleagues have done away with such preparations, and instead use the optical properties of the cell in its natural state to generate a 3D image. First, a laser beam is split into two: one beam goes through the sample while the other bypasses it. The beams are then recombined and shone onto a digital camera where they produce an interference pattern.

From this pattern the US team deduce the phase difference between the two beams, which changes according to the refractive index of the material that the sample beam passes through. By mapping this refractive index, a 2D image of the cell’s interior is generated.

To get a 3D image the researchers must place a mirror in front of the sample and rotate it incrementally with a galvanometer – a device that converts a small current into a mechanical motion. For each rotation, which alters the angle of the laser beam through the sample, they record an interference pattern.

Feld and colleagues demonstrated their technique, called tomographic phase microscopy, by imaging a cervical cancer cell (See Inside a cell). For the first time, an unaltered cell’s detailed 3D structure with elements such as the nucleus can be seen.

“Accomplishing this has been my dream, and a goal of our laboratory, for several years,” said Feld. “For the first time the functional activities of living cells can be studied in their native state.”

The resolution currently stands at about 0.5 µm, but the group says it should be able to improve it to 0.15 µm or less. They expect that it will complement electron microscopy, which can probe as small as 10 nm but requires samples to be either frozen or coated in a layer of conductive material.

Physicists minimize ‘sticky friction’ in tiny machines

Stiction is a problem in nano- and micro-electromechanical systems (NEMS and MEMS) whereby the tiny components stick together, often greatly reducing the reliability and long-term durability of these devices. It occurs when capillary, van der Waals and electrostatic forces between surfaces overpower the built-in restoring forces of the overall structure. In smaller systems the effect is more pronounced because of a larger surface-area-to-volume ratio.

Practical methods to eliminate stiction-related failures involve designing devices with high mechanical restoring forces, or by using “passivants” – special treatments for reducing surface energy. Some researchers have employed time consuming and costly molecular-dynamics simulations to see how roughness affects stiction, but so far these have not given any useful insights. Liu and co-workers, however, have performed experiments that demonstrate a correlation between surface roughness and stiction – a result that they hope could be used to minimize stiction in MEMS, and possibly NEMS, components.

The US team started with a series of silicon wafers, each with a different average roughness – that is, with different sized lumps or “asperities” on the surface. They then brought the cantilevers of an atomic force microscope with various tip radii into contact with a single asperity on the surface and measured the size of the adhesive force.

The researchers found that the adhesive force falls quickly as the average roughness is increased, but reaches a minimum beyond which it steadily rises again – in other words, there exists an optimal roughness. This value increases with the radius of the cantilever tip.

According to Liu and co-workers, this is because a tip resting on a completely smooth surface is strongly attracted by the majority of the surface’s adhesive forces. A small asperity on a slightly roughened surface acts to distance the tip from the surface, so these forces are less pronounced. But too large an asperity on a very rough surface will have its own strong adhesive forces which cancel the distancing effect.

“Our work suggests a promising way to minimize adhesion between two surfaces by tuning asperity height to feature-size in MEMS devices,” Liu told physicsworld.com. “We didn’t quantify by how much stiction could be reduced, but our model can provide a useful predictor of the behaviour of adhesive contacts down to the nano scale.”

X-ray holography breaks the femtosecond barrier

In their technique, Chapman and co-workers start by firing a coherent pulse of light from the X-ray free-electron laser at the DESY lab in Hamburg through a small hole in a “detector” mirror. This pulse then encounters a thin, translucent membrane that has been covered with a sample material – in this case 140-nm-diameter polystyrene balls – that lies just in front of a second, “backing” mirror (see Quick as a flash).

If the pulse hits one of the balls, it strips the polystyrene chains of their electrons, causing the material to explode under the repulsion of the remaining positive charge. The X-rays then scatter off the ball before travelling to the backing mirror, which reflects it.

Because the pulse has a finite width, however, not all of the X-rays interact with the ball – some of them bypass it and continue in their original direction before bouncing off the backing mirror. These X-rays can then scatter off the ball, albeit a fraction of a second after the initial scattering event, by which time the ball has got bigger as a result of the explosion.

Both parts of the now-scattered pulse – known as the “reference” and “object” beams – then travel back to the detector mirror, which reflects them onto a digital camera whereupon they interfere with each other to produce a “holographic” pattern. The researchers then analyze this pattern to reveal the structure of the sample’s material and how it evolved during the explosion.

Although other X-ray holography techniques have been used as far back as the 1970s, Chapman and co-workers’ technique is much faster, having a temporal resolution of just one femtosecond. This is the timescale of atomic motion, meaning that different samples could be imaged to see the stages in, for example, chemical reactions. “This is certainly the fastest hologram ever recorded,” Chapman told physicsworld.com.

Since a relatively long wavelength of 32.5 nm was used in this experiment, the spatial resolution of the technique currently stands at 50 nm, but Chapman explains that with future, shorter-wavelength free-electron lasers it should be possible to resolve features as small as 1 nm.

The researchers say they were inspired to make holographic patterns in this way by Isaac Newton, who noticed that sunlight produced “strange and surprising” light and dark bands on a screen after he had bounced it off a mirror speckled with dust particles.

Dirac medal honours charm-quark physicists

The charm quark was predicted in 1970 by Iliopoulos and Maiani when, with future Nobel laureate Sheldon Glashow, they formulated the now-famous “GIM mechanism” in an attempt to understand the weak interaction. This quark – the fourth predicted to exist – is now known to have a positive charge of two-thirds of that of an electron. “The GIM mechanism was a seminal contribution to the developing theory of the electroweak interaction,” David Gross, a member of the Dirac medal selection committee, told physicsworld.com.

Their theory was confirmed in November 1974 with the discovery of the J/Ψ particle – a bound state of a charm quark and a charm antiquark – at both the Brookhaven National Laboratory and the Stanford Linear Accelerator Centre in the US. The discovery persuaded many physicists for the first time to realize that quarks exist.

Maiani says he is extremely honoured to win the medal. “Dirac has been my hero since the beginning of [my] physics studies,” he said. “I will never forget the impression made upon me by the hole theory of the positron, and reading his book – together with [Richard] Feynman’s – is the way I learned quantum mechanics.”

The Dirac Medal is awarded to scientists previously unrecognized by the Nobel prize, Fields medal or Wolf Foundation prize who “have made significant contributions to physics.”

Phoenix blasts off to Mars

The $420 million Phoenix mission, the result of an international collaboration led by the University of Arizona, US, is the first project in NASA’s Mars Scout mission. It started life in 2003 as an attempt to revive the 2001 Mars Surveyor Lander, which was cancelled after the Mars Climate Orbiter and Mars Polar Lander failed in 1999. “We have worked for four years to get to this point, so we are all very excited,” said project manager Barry Goldstein at NASA’s Jet Propulsion Laboratory.

Phoenix will land using descent engines on a site in the northern hemisphere at 68.35° north latitude and 233.0° east longitude. Although these engines have not been used successfully since 1976 – NASA has recently favoured airbag systems – they enable the craft to carry the weight of seven different instruments, some of which were those mothballed from the Mars Surveyor Lander.

First, Phoenix will use a 3.35-metre-long robotic arm to dig into the surface and reach the icy layer residing a few centimetres beneath. Mounted on the end of the arm is a visible-light camera, which will provide high-resolution colour images of the soil and ice.

Samples delivered to the lander by the arm will be heated by a “thermal and evolved gas analyzer” to see how much water vapour, carbon dioxide and volatile organic compounds are contained in the soil. The samples will also be distributed to optical and atomic-force microscopes, which will examine mineral grains, while electrochemistry cells will be able to measure properties such as acidity or alkalinity, and a conductivity probe can check thermal and electrical properties.

Other instruments will look at the wider environment. During descent, a camera will record the geography around the landing site, and after Phoenix has landed a stereo camera will observe the local terrain in 3D. Finally, meteorological equipment will monitor changes in water abundance, dust, temperature and other variables.

“[Saturday’s] launch is the first step in the long journey to the surface of Mars,” said principal investigator Peter Smith of the University of Arizona. “We certainly are excited about launching, but we are still concerned about our actual landing, the most difficult step of this mission.”

NASA is still reviewing proposals for the second Mars Scout mission, due to fly in 2011.

Laser flips magnetic bit without any help

Most computers store data on magnetic hard disk drives, in which the direction – “up” or “down” – of the magnetic moments in a small region of the disk corresponds to a binary bit. Data are read by a magneto-resistance element and written by heating the bit with a laser and then flipping the moments with a magnetic field pulse from a tiny coil.

The cost and complexity of hard drives could be reduced significantly if data could instead be read and written using light alone. While some commercial hard drives now use light to read data from magnetic bits, a technique for writing data using only light had remained elusive.

Now, Theo Rasing and colleagues at Radboud University Nijmegen in the Netherlands along with researchers at Nihon University in Japan have shown that a single laser pulse can flip the magnetization of a 5 µm spot on a thin magnetic film from up to down and vice versa – without the need for an external magnetic field.

The pulse was only 40 fs (10-15 s) long – much shorter than the magnetic field pulses used in hard drives, which cannot be made much shorter than about 2 ns. Indeed, the 40 fs switching time had been thought to be impossible because in 2004, a 2 ps lower limit on controlled magnetic switching had been established by another team of physicists.

The laser pulse was circularly polarized, which means that it creates an intense but highly localized magnetic field within the material. The pulse was switched between two polarization states, which flips the direction of the field.

The researchers did their experiments on an alloy of gadolinium, iron and cobalt, which is used widely in magneto-optic data storage devices. The team is now checking to see if the switching occurs in materials with higher coercivity, which could allow an all-optical memory to achieve the same storage density as a conventional hard drive.

Rasing has patented the write process and he is confident that it will be commercialized. However, he admits that anyone wanting to build a hard drive using the technology would have to overcome the significant challenge of how to build a tiny laser that can also produce an intense pulse of circularly-polarized light that can be focussed down to a spot 50 nm in diameter, which is much smaller than the wavelength of the laser light. “But these are solvable problems,” he says.

Islamic science

Most physicists, particularly those in western nations, probably do not give religion a great deal of thought. A minority of physicists do, however, have firmly held religious beliefs and think long and hard about reconciling those beliefs with their scientific knowledge, as our report on a recent meeting on “God and physics” in Cambridge makes clear (p10, print edition only). There is much to admire in their deep thinking, which has been recognized by physicists being awarded the Templeton prize for progress in religion six times in the last eight years.

But in Muslim nations, religion plays a far bigger role in everyday life than it does in the West. Indeed, today Islam is actually holding back scientific progress by placing too great an emphasis on studying and interpreting the pages of the Koran, as the leading Iranian physicist Reza Mansouri points out (see “A way forward for Islamic science”). Those students in his country who do study science at university tend to learn a very narrow curriculum by rote, rather than being encouraged to think for themselves. Low investment in science – even in oil-rich Gulf states – and restrictions on freedom of expression compound the problem.

It was not always thus. Muslim scholars made huge contributions in areas like astronomy, optics and mathematics between the 8th and the 13th centuries, with Islam encouraging rigorous intellectual enquiry. Why science in the Islamic world fell from grace is a topic of considerable debate among historians, with the advances made in Renaissance Europe certainly playing a part in halting progress.

But whatever the reasons, the key for Muslim nations now is to rebuild their scientific strengths through increased public funding – no mean feat when their governments fail to see the merits of such investment – and by encouraging links between scientists in those countries and in the West. Placing a greater focus on a few, world-class labs rather than spreading money thinly around will help too. These solutions are essentially no different to what is needed in other parts of the developing world. But given the great untapped potential in the 1.3 billion or so people who live in the Islamic world, that rebuilding – long though it may take – is a worthwhile task.

Once a physicist: Sergi Jordà


How did you first become interested in physics?

As a child I was always inventing weird artefacts or constructing houses out of balsa wood with the idea of becoming an architect. Then as a teenager I had a kind of ideological conflict, thinking that the nice architectural projects that were worth working on were mostly for the rich. Somehow this brought me into physics, although at that time, the subject didn’t mean much more to me than bricks slipping on inclined planes.

Where did you study physics and how much did you enjoy it?

I studied at the University of Barcelona in the early 1980s. The sad truth is that by concentrating on passing the exams, I didn’t have much time to enjoy the deeper concepts. Near the end of my studies I was tutored by a disciple of the Belgian chemistry Nobel laureate Ilya Prigogine and I started to really appreciate nonlinear thermodynamics, and complex and chaotic systems.

How and when did you become interested in computer music?

While studying physics, I also played the saxophone – a sort of free jazz – and in my third year at university I discovered that I loved computer programming. Then, in my fourth year, I came across a snapshot of an audio spectrogram on the back cover of an album by Laurie Anderson. I had studied the Fourier transform in an abstract way (no-one ever talked about sound during my five years of physics), so I could intuitively understand what the image was about: sound, and therefore music, could be “understood” by computers. I soon imagined that computers could be used for making music – even free jazz. And believing that computers were far better suited than me to repetitive and unexciting tasks, I gave up practising scales on the saxophone and started programming.

How did your career develop after you graduated?

By then I was already sure that I wanted to become a computer musician, although I didn’t know how to proceed. I first survived as a computer programmer, then started teaching programming in private schools. Meanwhile, I studied anything I could find on computer music and made my own music programs that I started using in performances. In the 1990s I worked on multimedia projects and computer art, before returning to academia to teach in the computer-science faculty and do research into real-time musical interaction between humans and computers.

What are you working on at the moment?

For the last four years I have been working on the reactable, together with Günter Geiger, Martin Kaltenbrunner and Marcos Alonso from the Music Technology Group at my university. The reactable is an electronic musical instrument conceived for collaborative computer-music performance and improvisation. It is based on a circular table around which several musicians share control of the instrument by rotating, moving and caressing physical artefacts on its luminous surface.

What are some of your career highlights?

In the 1990s I had a successful collaboration with Catalan theatre group la Fura dels Baus, which gave birth to FMOL, a software program for online musical collaboration that can be considered as the precursor of the reactable. But the reactable itself is by far my most successful creation and also the most accomplished. It is the fruit of 20 years of work in the field; and the fact that an artist such as Björk used it extensively for her last world tour is enormously gratifying.

How has your background in physics helped you in your career?

Somehow it made me feel confident about the potential of human knowledge and understanding. It gave me the illusion that anything, with the possible exception of humans themselves, can be understood – no matter how complex it seems or how long it may take.

Physicists at Aldermaston

The Atomic Weapons Establishment (AWE) is the home of the UK’s nuclear deterrent and is responsible for the entire lifecycle of the country’s warheads from research and design through assembly to in-service support and, finally, decommissioning and disposal. AWE also plays a vital role in national security and international monitoring of the Comprehensive Test Ban Treaty. Its core mission is to build and maintain the warheads for the submarine-launched Trident ballistic-missile system that forms the UK’s sole nuclear deterrent. It is also required to maintain the capability to design a warhead to replace Trident, should it ever be required.

In order to perform all these tasks, AWE carries out world-class science in some of the most challenging fields, including explosive detonation, hydrodynamics, high-strain-rate deformation behaviour, radiation physics and computer modelling. The AWE site at Aldermaston in Berkshire includes all the facilities needed to carry out this science – extremely fast supercomputers, areas for explosive trials and experimental facilities for high-energy-density physics. It employs researchers in not just nuclear physics, but all branches of the subject: from atomic and condensed-matter physics to astrophysics and quantum physics.

AWE is currently investing in its building and facilities in order to support Trident safely and reliably for the next 20 years. But behind all these facilities are the people. AWE currently employs 4300 staff and 1500 contractors across its sites in Aldermaston and nearby Burghfield, and it prides itself on recruiting only the best people in science, engineering and technology. Maintaining the UK’s nuclear deterrent is not textbook science – everything AWE does is innovative. It carries out experiments on materials under extreme temperatures, strain rates and pressures that are over in the blink of an eye. AWE needs technical experts in a wide variety of physics fields to be able to understand and model the phenomena of interest.

Explosive research

I applied to work at AWE after completing my undergraduate degree in physics at Lancaster University in 1993. I was looking for a career in physics research, and AWE seemed to offer everything I wanted. I was recruited into the hydrodynamics department as a research scientist and spent the first few years carrying out experiments in the explosive facilities researching the detonation of condensed high explosives. During this time AWE sponsored me to carry out a part-time MSc in numerical methods at the Royal Military College at Shrivenham. I also had the opportunity to publish my work and travel to many conferences around the world.

In 2000 my team and I moved to the theoretical- physics area on site, where I started leading a team of scientists looking into models of shear strength in condensed matter. At this time I also embarked on a company-sponsored part-time PhD in non-equilibrium thermodynamics. Within two years I was asked to lead the theoretical material-modelling group, consisting of some 25 staff – quite a leap for a mere scientist! In January this year I moved back to the hydrodynamics division to head up its science group of 50 full-time employees. I have lots of opportunities to develop in both technical and business matters, and I travel regularly to work with AWE’s international counterparts. No two days are ever the same.

Most of the physicists at AWE work within the Directorate for Research and Applied Science. The directorate has about 1100 staff, of whom some 600 are scientists – a mixture of physicists, chemists, materials scientists, computational scientists and mathematicians. There is a roughly even split between theoretical and experimental staff, although we all work closely together to deliver integrated programmes.

Although the science and technology we need is self-contained at AWE, we do have active external collaborations, consultancies and contracts. We work with UK universities to employ summer students and foster graduate and postgraduate research, thus helping to develop the scientists of the future. We also contract work out to industry, thereby helping to invigorate technological advances in the UK as a whole. We have regular audits of our technical work, which allow top academics and experienced workers in industry access to our work, and we take on board their suggestions for future research directions. There is also an ongoing peer-review process with our colleagues in the US weapons laboratories that allows the exchange of data, ideas and staff under the auspices of the 1958 US/UK Mutual Defence Agreement.

Continuing development

Many scientists at AWE say that working in a technical area at Aldermaston is like being back at university but with higher pay and greater job security. AWE encourages its researchers to publish externally, to attend and speak at prestigious international conferences, to write textbooks, to assist the research councils with paper reviews and funding approvals, and to advise and direct UK technical policy. Substantial funding is also allocated to blue-sky work, where it is relevant to the core business.

AWE has a recognized graduate scheme for all recently graduated scientists and engineers that lasts about two years. The focus is on learning more about the wider company and workplace skills. An attractive remuneration package goes along with this. AWE continues to take professional development very seriously beyond the first few years, with on-the-job mentoring by experienced staff, funding for higher education and postdoctoral work, and placements at international facilities. The company vision is to be “internationally recognized for science, engineering and technology”, and as such AWE constantly strives for excellence – mediocrity simply will not do.

Poincaré, Perelman and proof

The public’s view of the mathematician as a reclusive genius toiling away for years at an arcane problem is one that will not go away. The media have enthusiastically seized on this image since the depictions of Andrew Wiles in Simon Singh’s best-seller Fermat’s Last Theorem and John Nash in the film A Beautiful Mind. Last summer in Madrid it surfaced again in the form of Grigori Perelman and the Poincaré conjecture. Not that Perelman was in Madrid – that was exactly the point.

The occasion was the award of the Fields medals, the mathematicians’ equivalent of the Nobel prizes. King Juan Carlos was centre stage, flanked by political and mathematical dignitaries, as an audience of thousands strained to identify the dark-suited young men at the front waiting to receive the four medals. The second name to be announced was that of Perelman, “for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow”.

“I deeply regret”, the announcer continued, “that Dr Perelman has declined to accept the Fields medal”. Where was Perelman? Presumably he was back in his apartment in St Petersburg, working hard on the next challenging problem.

The events leading up to this point form the subject of this book by Donal O’Shea, a mathematician at Mount Holyoke College, Massachusetts. It would be impossible to describe the step-by-step evolution of Perelman’s result. What might his diary say? “Morning: proved A implies B; afternoon: B implies C; evening: found a counterexample to A.” So instead the author gives a broad history of this area of mathematics, tailored to culminate with the famous conjecture that states that “a simply connected three-dimensional manifold is homeomorphic to the sphere”.

Therein lies the challenge for the author: to explain both the concepts and the history for a general readership in the manner of Singh’s successful book on Wiles and Fermat. While Singh’s task was to explain number theory to the layperson, O’Shea’s is to deal with geometry. This should be easier and less dependent on formulae, but quite quickly it becomes clear that pictures and diagrams have their limitations when two dimensions give way to three. Fortunately, the author’s lively style carries the reader quite successfully through this short book.

Geometry for many readers means the ancient Greeks, and that is where O’Shea starts. The Poincaré conjecture concerns a “three-sphere”, the analogue of a sphere in 4D space. We cannot sit in four dimensions and look down on such an object, but nor could the Greeks see the two-sphere that is the Earth. Nevertheless, they not only knew our planet’s shape but measured its diameter. O’Shea focuses too on Euclid, in particular his clumsy “parallel postulate”. The discovery in the 19th century of non-Euclidean geometry where the postulate fails is a theme of the book: it led mathematicians to rethink what geometry consisted of. By 1850 the existence of different types of geometry was recognized, but they were all homogeneous – all points looked the same.

Then in 1854 came Bernhard Riemann, clearly the author’s hero. His controversial lecture of that year, “On the foundations that underlie geometry”, portrayed a different world: non-linear, non-homogeneous and existing in any number of dimensions. This was the concept of a manifold: a mathematical space that is constructed by “gluing together” local patches that are Euclidean into a more complex global structure. My local space, yours and everybody else’s join together to give a 3D manifold: it might be a three-sphere, it might be more complicated. O’Shea’s subtitle, “In search of the shape of the universe”, suggests that Perelman has solved that problem too, but that is hardly likely to be the case.

Henri Poincaré made his name in celestial mechanics and differential equations, but he was also one of the founders of algebraic topology. Topology concerns the properties of objects that are unchanged by continuous deformation. The wording “simply connected” in the Poincaré conjecture is such a property: it means that a closed curve can be continuously shrunk to a point. One can easily visualize doing this on the surface of a two-sphere like the Earth, but not on a doughnut, where a curve looping round the hole cannot be shrunk. In two dimensions the sphere is the only surface that is simply connected, and Poincaré’s famous conjecture of 1904 asked whether the same was true in three dimensions.

As the field of topology developed in the 20th century, reputations were made and then broken by attempts to prove this conjecture. Finally it was proved in all higher dimensions, and only Poincaré’s original problem was left; it was clear that there was something special about three dimensions. The prevailing view of 3D topology changed radically in the 1980s when US mathematician Bill Thurston showed that many three-manifolds could be systematically broken up into pieces, each of which had a type of homogeneous geometry. He conjectured that this procedure should apply much more widely and it is this that Perelman actually succeeded in proving. It implies the Poincaré conjecture as a special case (just as Wiles’ proof of Fermat’s theorem was a special case of the Shimura–Taniyama– Weil conjecture).

What technique did Perelman use that evaded the other topologists? Not to use topology! He used instead the “Ricci flow equation” – a geometrical version of the heat equation pioneered by mathematician Richard Hamilton. Given an irregular distribution of temperature in a body, the heat equation describes the temperature at subsequent times. The flow of heat tends to smooth out the initial irregularities. Perelman’s idea was to start with an arbitrary manifold and “follow the flow”, thereby hoping to get something regular and homogeneous like a sphere or a non-Euclidean geometry. But it does not work like that, as Hamilton knew, because the solution blows up in finite time. Perelman showed that when this happens, you can modify the manifold in a controlled way and start the flow again, then repeat. Eventually you have either cut the space up into Thurston’s pieces or arrived at a homogeneous geometry – simple connectivity then gives you a sphere.

O’Shea’s book describes well the progress and personalities involved in this long process, and sensibly puts anything vaguely technical into 45 pages of notes at the end. Disappointingly for a subject so geometrical, the illustrations are of poor quality. Maps are shrunk so much as to be illegible and many illustrations are idiosyncratic line drawings. On the other hand, O’Shea includes an intriguing engraving from The Divine Comedy – I had no idea that Dante had described the three-sphere.

The book goes on to describe other events in Madrid last summer, as articles in the press suggested that the proof might be incomplete and that other researchers were claiming credit. (Note the absence of the words “Poincaré conjecture” in the citation – even the Fields medal committee could not decide.) Such disputes in the abstract world of pure mathematics rarely make the news, but journalists were on hand and the leading character was intriguingly absent.

Perelman in fact never published his proof in a peer-reviewed journal, but instead placed his articles on the preprint server arXiv.org. He followed this up by answering questions in detail on a lecture tour of the US, and confirmation of the validity of his proof is now emerging from various research groups around the world.

Proof is a delicate issue and always has been: the venerated logic of Euclid’s Elements had to be adjusted by David Hilbert; Poincaré’s own work was littered with errors; and there was a famous (though soon corrected) gap in Wiles’ proof. But time, like the flow equation, smooths these out. The same is true of accreditation. Who was Euclid anyway? For all we know he may have been a committee. But who cares now? We possess the propositions, lemmas and theorems to appreciate and use. So when the barbarians have come and gone again, we will still have a theorem, not a conjecture, which tells us that a simply connected three-manifold is a sphere.

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