YEAR IN REVIEW 2001: MATHEMATICS-AND-PHYS-SCIENCES

Astronomy Mathematics In August 2000 the American Mathematical Society convoked a weeklong meeting in Los Angeles devoted to "Mathematical Challenges of the 21st Century." The gathering featured 30 plenary speakers, including eight winners of the quadrennial Fields Medal, a distinction comparable to a Nobel Prize. In assembling at the start of the new century, the participants jointly undertook a task analogous to one accomplished by a single person 100 years earlier. At the Second International Congress of Mathematicians in Paris in August 1900, the leading mathematician of the day, David Hilbert of the University of Gttingen, Ger., had set out a list of 23 "future problems of mathematics." The list included not only specific problems but also whole programs of research. Some of Hilbert's problems were completely solved in the 20th century, but others led to prolonged, intense effort and to the development of entire fields of mathematics. The talks in Los Angeles included topics of applied mathematics that could not have been imagined in Hilbert's day-for example, the physics of computation, the complexity of biology, computational molecular biology, models of perception and inference, quantum computing and quantum information theory, and the mathematical aspects of quantum fields and strings. Other topics, such as geometry and its relation to physics, partial differential equations, and fluid mechanics, were ones that Hilbert would have found familiar. Just as Hilbert could not have anticipated all the themes of mathematical progress for 100 years into the future, mathematicians at the 2000 conference expected that the emphases within their subject would be reshaped by society and the ways that it applied mathematics. The reputation and cachet of Hilbert, together with the compactness of his list, were enough to spur mathematical effort for most of the 20th century. On the other hand, major monetary rewards for the solution of specific problems in mathematics were few. The Wolfskehl Prize, offered in 1908 for the resolution of Fermat's last theorem, amounted to $50,000 when it was awarded in 1995 to Andrew Wiles of Princeton University. The Beal Prize of $50,000 was offered in 1998 for the proof of the Beal conjecture-that is, apart from the case of squares, no two powers of integers sum to another power, unless at least two of the integers have a common factor. Unlike Nobel Prizes, which include a monetary award of about $1 million each, the Fields Medal in mathematics carried only a small award-Can$15,000, or about U.S. $9,900. A major development in 2000 was the offer of $1 million each for the solution of some famous problems. In March, as a promotion for a fictional work about a mathematician, publishers Faber and Faber Ltd. and Bloomsbury Publishing offered $1 million for a proof of Goldbach's conjecture-that every even integer greater than 2 is the sum of two prime numbers. The limited time (the offer was to expire in March 2002) would likely be too short to stimulate the needed effort. More perduring prizes were offered in May by the Clay Mathematics Institute (CMI), Cambridge, Mass., which designated a $7 million prize fund for the solution of seven mathematical "Millennium Prize Problems" ($1 million each), with no time limit. The aim was to "increase the visibility of mathematics among the general public." Three of the problems were widely known among mathematicians: P versus NP (are there more efficient algorithms for time-consuming computations?), the Poincar conjecture (if every loop on a compact three-dimensional manifold can be shrunk to a point, is the manifold topologically equivalent to a sphere?), and the Riemann hypothesis (all zeros of the Riemann zeta function lie on a specific line). The other four were in narrower fields and involved specialized knowledge and terminology: the existence of solutions for the Navier-Stokes equations (descriptions of the motions of fluids), the Hodge conjecture (algebraic geometry), the existence of Yang-Mills fields (quantum field theory and particle physics), and the Birch and Swinnerton-Dyer conjecture (elliptic curves). Hilbert tried to steer mathematics in directions that he regarded as important. The new prizes concentrated on specific isolated problems in already-developed areas of mathematics. Nevertheless, as was noted at the May prize announcement by Wiles, a member of CMI's Scientific Advisory Board, "The mathematical future is by no means limited to these problems. There is a whole new world of mathematics out there, waiting to be discovered." Paul J. Campbell Chemistry Organic Chemistry. After more than a decade of effort, University of Chicago organic chemists in 2000 reported the synthesis of a compound that could prove to be the world's most powerful nonnuclear explosive. Octanitrocubane (C88) has a molecular structure once regarded as impossible to synthesize-eight carbon atoms tightly arranged in the shape of a cube, with a nitro group (NO2) projecting outward from each carbon. Philip Eaton and colleagues created octanitrocubane's nitro-less parent, cubane (C8H8), in 1964. Later, he and others began the daunting task of replacing each hydrogen atom with a nitro group. Octanitrocubane's highly strained 90 bonds, which store large amounts of energy, and its eight oxygen-rich nitro groups accounted for the expectations of its explosive power. Eaton's team had yet to synthesize enough octanitrocubane for an actual test, but its density (a measure of explosive power)-about 2 g/cc-suggested that it could be extraordinarily potent. Trinitrotoluene (TNT), in contrast, has a density of 1.53 g/cc; HMX, a powerful military explosive, has a density of 1.89 g/cc. Eaton pointed out that the research yielded many new insights into the processes underlying chemical bonding. His group also had indications that cubane derivatives interact with enzymes involved in Parkinson disease and so could have therapeutic applications. Oligosaccharides are carbohydrates made of a relatively small number of units of simple sugars, or monosaccharides. These large molecules play important roles in many health-related biological processes, including viral and bacterial infections, cancer, autoimmune diseases, and rejection of transplanted organs. Researchers wanted to use oligosaccharides in the diagnosis, treatment, and prevention of diseases, but, because of the great difficulty involved in synthesizing specific oligosaccharides in the laboratory, the potential for these compounds in medicine remained unfulfilled. Conventional synthesis techniques were labour-intensive, requiring specialized knowledge and great chemical skill. Peter H. Seeberger and associates at the Massachusetts Institute of Technology reported the development of an automated oligosaccharide synthesizer that could ease those difficulties. Their device was a modified version of the automated synthesizer that revolutionized the synthesis of peptides. Peptides are chains of amino acids-the building blocks of antibiotics, many hormones, and other medically important substances. The oligosaccharide synthesizer linked together monosaccharides. It fed monosaccharide units into a reaction chamber, added programmed amounts of solvents and reagents, and maintained the necessary chemical conditions for the synthesis. Seeberger described one experiment in which it took just 19 hours to synthesize a certain heptasaccharide (a seven-unit oligosaccharide), with an overall yield of 42%. Manual synthesis of the same heptasaccharide took 14 days and had an overall yield of just 9%. Seeberger emphasized, however, that additional developmental work would be needed to transform the machine into a commercial instrument widely available to chemists. Nuclear Chemistry. The periodic table of elements lays out the building blocks of matter into families based on the arrangement of electrons in each element's reactive outer electron shell. Although the table has been highly accurate in predicting the properties of new or as-yet-undiscovered elements from the properties of known family members, theorists believed that it might not work as well for extremely heavy elements that lie beyond uranium on the table. The heavier an element, the faster the movement of its electrons around the nucleus. According to Einstein's theory of relativity, the electrons in a very massive element may move fast enough to show effects that would give the element weird properties. Elements 105 and 106-dubnium and seaborgium, respectively-showed hints of such unusual behaviour, and many nuclear chemists suspected that element 107, bohrium, would exhibit a more pronounced strangeness. Andreas Trler of the Paul Scherrer Institute, Villigen, Switz., and co-workers reported that relativistic effects do not alter bohrium's predicted properties. Trler and associates synthesized a bohrium isotope, bohrium-267, that has a half-life of 17 seconds. It was long enough for ultrafast chemical analysis to show that bohrium's reactivity and other properties are identical to those predicted by the periodic table. How heavy, then, must an element be for relativistic effects to appear? Trler cited the major difficulty in searching for answers-the short half-lives of many superheavy elements, which often are in the range of fractions of a second, do not allow enough time for chemical analysis. Physics Particle Physics. The standard model, the mathematical theory that describes all of the known elementary particles and their interactions, predicts the existence of 12 kinds of matter particles, or fermions. Until 2000 all but one had been observed, the exception being the tau neutrino. Neutrinos are the most enigmatic of the fermions, interacting so weakly with other matter that they are incredibly difficult to observe. Three kinds of neutrinos were believed to exist-the electron neutrino, the muon neutrino, and the tau neutrino-each named after the particle with which it interacts. Although indirect evidence for the existence of the tau neutrino had been found, only during the year did an international team of physicists working at the DONUT (Direct Observation of the Nu Tau) experiment at the Fermi National Accelerator Laboratory (Fermilab) near Chicago report the first direct evidence. The physicists' strategy was based on observations of the way the other two neutrinos interact with matter. Electron neutrinos striking a matter target were known to produce electrons, whereas muon neutrinos under the same conditions produced muons. In the DONUT experiment, a beam of highly accelerated protons bombarded a tungsten target, creating the anticipated tau neutrinos among the spray of particle debris from the collisions. The neutrinos were sent through thick iron plates, where on very rare occasions a tau neutrino interacted with an iron nucleus, producing a tau particle. The tau was detected, along with its decay products, in layers of photographic emulsion sandwiched between the plates. In all, four taus were found, enough for the DONUT team to be confident of the results. Six of the fermions in the standard model are particles known as quarks. Two of them, the up quark and the down quark, make up the protons and neutrons, or nucleons, that constitute the nuclei of familiar matter. Under the low-energy conditions prevalent in the universe today, quarks are confined within the nucleons, bound together by the exchange of particles called gluons. It was postulated that, in the first few microseconds after the big bang, however, quarks and gluons existed free as a hot jumble of particles called a quark-gluon plasma. As the plasma cooled, it condensed into the ordinary nucleons and other quark-containing particles presently observed. In February physicists at the European Laboratory for Particle Physics (CERN) near Geneva reported what they claimed was compelling evidence for the creation of a new state of matter having many of the expected features of a quark-gluon plasma. The observations were made in collisions between lead ions that had been accelerated to extremely high energies and lead atoms in a stationary target. It was expected that a pair of interacting lead nuclei, each containing more than 200 protons and neutrons, would become so hot and dense that the nucleons would melt fleetingly into a soup of their building blocks. The CERN results were the most recent in a long quest by laboratories in both Europe and the U.S. to achieve the conditions needed to create a true quark-gluon plasma. Some physicists contended that unambiguous confirmation of its production would have to await results from the Relativistic Heavy Ion Collider (RHIC), which went into operation in midyear at Brookhaven National Laboratory, Upton, N.Y. RHIC would collide two counterrotating beams of gold ions to achieve a total collision energy several times higher-and thus significantly higher temperatures and densities-than achieved at CERN. Solid-State Physics. New frontiers in solid-state physics were being opened by the development of semiconductor quantum dots. These are isolated groups of atoms, numbering approximately 1,000 to 1,000,000, in the crystalline lattice of a semiconductor, with the dimensions of a single dot measured in nanometres (billionths of a metre). The atoms are coupled quantum mechanically so that electrons in the dot can exist only in a limited number of energy states, much as they do in association with single atoms. The dot can be thought of as a giant artificial atom having light-absorption and emission properties that can be tailored to various uses. Consequently, quantum dots were being investigated in applications ranging from the conversion of sunlight into electricity to new kinds of lasers. Researchers at Toshiba Research Europe Ltd., Cambridge, Eng., and the University of Cambridge, for example, announced the development of photodetectors based on quantum-dot construction that were capable of detecting single photons. Unlike present single-photon detectors, these did not rely on high voltages or electron avalanche effects and could be made small and robust. Applications could include astronomical spectrosopy, optical communication, and quantum computing. Space Exploration