Issue No. 3, Autumn 2004

Leicester The Maths Wrangler From the mathematicians at the University of Leicester http://www.math.le.ac.uk/ email: [email protected]

WELCOME TO The Wrangler This is the third issue of our maths newsletter from the University of Leicester’s Department of Mathematics. You can pick it up on-line and all the back issues at our new web address http://www.math.le.ac.uk/WRANGLER. If you would like to be added to our mailing list or have suggestions, just drop us a line at [email protected], we’ll be flattered! What do we have here in this new issue? You may wonder what wallpaper and frying pans have to do with each other. We will tell you a fascinating story how they come up in new interdisciplinary research linking geometry and physics. We portrait our chancellor Sir Michael Atiyah, who recently has received the Abel prize, the maths equivalent of the Nobel prize. Sir Michael Atiyah will also open the new building of the MMC, which will be named after him, with a public maths lecture. Everyone is very welcome! More info at our new homepage http://www.math.le.ac.uk/

Contents

Welcome to The Wrangler

Wallpaper and Frying Pans - Linking Geometry and Physics

K-Theory, Geometry and Physics - Sir Michael Atiyah gets Abel Prize

The GooglePlexor The Mathematical Modelling Centre MMC

The chancellor of the University of Leicester and world famous mathematician Sir Michael Atiyah (middle) receives the Abel Prize of the year 2004 together with colleague Isadore Singer (left) during the prize ceremony with King Harald of Norway. (Photo: Courtesy of Abel Prize Comittee) WALLPAPER AND FRYING PANS - LINKING GEOMETRY AND A more unusual kind of pattern, such as the Penrose tiling shown on the left, is quasiperiodic. Like wallpaper PHYSICS patterns, there are motifs which repeat (such as the In everyday life we see geometric patterns all around ones coloured green in the us. Decorative items, such as wallpaper or tiles include picture on the left), but the patterns in their designs. Patterns also appear in nature, patterns do not repeat in a in the arrangements of leaves on a plant or petals periodic way, since if we shift on flowers. Also, animals often have patterns in the one of the motifs on to the colourings on their fur, feathers or skin. other one, then the shapes coloured red in the picture do The patterns described above contain symmetry. not match up. Symmetries can be reflections, such as is seen in the wings of a butterfly, rotations, seen in the arrangement There are also three-dimensional structures which are of petals on a flower,translations , such as in snakeskin, or quasiperiodic. Such objects are called quasicrystals. glide reflections, combining reflection with translation, These first began to be investigated in the 1980s, which can also be seen in snakeskin. after the discovery of minerals whose atoms were not arranged in any of the known patterns for crystals. Some quasicrystals, such as the one shown on the right, are metal alloys. For example, certain alloys of aluminium, copper and iron have quasiperiodic atomic arrangements. As they are metals, they can be shaped and moulded for practical use. The fact that there is no translational symmetry in their atomic arrangements, and hence no fault lines in their Patterns involving translational symmetry are called structure, means that quasicrystals are often quite hard. periodic. Motifs in periodic patterns can be shifted Also, their unusual structure means that other substances on to other motifs so that every one matches up with cannot find ways to grip on to quasicrystals, so they have another. To see this, you could copy the pictures in this non-stick properties. This has led to their use as coatings article on to a transparency and then move the copy for frying pans! over the pictures to find when motifs coincide. Patterns with translational symmetry in one direction are called Currently, research is being carried out in Leicester friezes. By considering the different symmetries which University, looking at quasiperiodic patterns and using friezes can have, mathematical tools to study their geometric properties. it can be shown These tools could also help to describe the physical that there are properties exhibited by quasicrystals. The ultimate aim only seven distinct is to catalogue all patterns which could be used as models types. Patterns of quasicrystals, in the same way that the periodic with translational crystals have been classified. This is an exciting and symmetry in two relatively new area of work, in which progress is being directions, such made all the time. as in the picture above, are called wallpaper patterns. The symmetries That’s how physics and geometry can be used to link of these patterns have also been studied, and it is wallpaper to frying pans! well known that there are seventeen different types of wallpaper pattern. Claire Irving is a postgraduate Patterns with translational symmetry in three directions student in Pure Mathematics in are called crystals. An example is the atomic lattice for Leicester, currently writing her diamond, shown below on the right. PhD thesis under the supervision of Professor Hunton in which she Crystal types were first classified applies methods from algebraic completely in the 1890s, by German topology to study invariants for mathematician Arthur Schönflies and aperiodic tilings. You can find Russian mineralogist Eugraf Fedorov. more on her homepage: http:// They discovered that there are 230 www.math.le.ac.uk/~cirving/ different types of crystal. HEORY EOMETRY Michael Francis Atiyah and Isadore M Singer are among K-T , G the most influential mathematicians of the world. With AND PHYSICS the index theorem they changed the landscape of math- ematics. Over a period of 20 years they worked together on the index theorem and its ramifications. Atiyah and Singer came originally from different fields of SIR MICHAEL ATIYAH mathematics - Atiyah from algebraic geometry and to- pology, Singer from analysis. Their main contributions in GETS ABEL PRIZE their respective areas are also highly recognized. Atiyah’s early work on meromorphic forms on algebraic varieties, and his important 1961 paper on Thom complexes, are such examples. The University of Leicester Chancellor and world famous mathematician Sir Michael Atiyah has received the Atiyah’s pioneering work together with Friedrich Hirze- maths world equivalent of the Nobel Prize. bruch on the development of the topological analogue of Grothendieck’s K-theory, had numerous applications to The Norwegian Academy of Science and Letters classical problems of topology and turned out later to be awarded the Abel Prize for the year 2004, jointly to deeply connected with the index theorem. Sir Michael Francis Atiyah and Isadore M Singer of the Massachusetts Institute of Technology “for their Michael Francis Atiyah was discovery and proof of the index theorem, bring- born in 1929 in London. Atiyah ing together topology, geometry and analysis, got his BA and doctorate from and their outstanding role in building new bridges Trinity College, Cambridge. between mathematics and theoretical physics.” The greatest part of his aca- The award was presented by King Harald of Norway at a demic career Atiyah has spent ceremony in Oslo in May earlier this year. in Cambridge and Oxford. He has held many prominent po- The Department of Mathematics in Leicester joins in sitions; among them the high- congratulating Sir Michael Atiyah for his achievements ly prestigious Savilian Chair of and recognition of his outstanding scientific work! Geometry at Oxford and that of Master of Trinity College, Cambridge. Atiyah has also been professor of mathemat- The Atiyah-Singer index theo- ics at the Institute for Advanced Study in Princeton. rem is one of the great land- marks of 20th mathematics, Atiyah rejuvenated British mathematics during his years influencing profoundly many at Oxford and Cambridge. He was also the driving force of the most important later behind the creation of the Isaac Newton Institute for developments in topology, dif- Mathematical Sciences in Cambridge and became its first ferential geometry and math- director. Atiyah is now retired and an honorary professor ematical physics. Its authors, at the University of Edinburgh and Chancellor of the Uni- both jointly and individually, versity of Leicester. have been instrumental in repairing a rift between the Michael Francis Atiyah has received many honours during worlds of pure mathematics his career, including the Fields Medal, which he received and theoretical particle physics, initiating a cross-fertili- at the International Mathematical Congress of Mathema- zation which has been one of the most exciting develop- ticians in Moscow in 1966. He was elected a Fellow of the ments of the last decades. Royal Society in 1962 at the age of 32. He was awarded the Royal Medal of the Society in 1968 and its Copley The statement of the Norwegian Academy of Science Medal in 1988. Atiyah was also president of the Royal and Letters says: “It was the culmination and crowning Society and of the London Maths Society. Atiyah has also achievement of a more than 100-year old development been president of Pugwash Conferences on Science and of ideas, from Stokes’s theorem, which students learn in World Affairs. Among the prizes he has received are the calculus classes, to sophisticated modern theories like Feltrinelli Prize from the Accademia Nazionale dei Lincei Hodge’s theory of harmonic integrals and Hirzebruch’s (1981) and the King Faisal International Prize for Science signature theorem. The problem solved by the Atiyah- (1987). Michael Francis Atiyah was knighted in 1983 and Singer theorem is truly ubiquitous. In the 40 years since made a member of the Order of Merit in 1992. its discovery, the theorem has had innumerable applica- tions, first within mathematics and then, beginning in the late 70s, in theoretical physics: gauge theory, in- The Wrangler stantons, monopoles, string theory and in the theory of anomalies. At first, the applications in physics came as a Coordinator Ben Leimkuhler complete surprise to both the mathematics and physics Editor Frank Neumann communities. Now the index theorem has become inte- Assistant Ushma Chauhan gral part of their cultures. THE GOOGLEPLEXOR THE MATHEMATICAL John Reardon of St Francis Xavier College, London was MODELLING CENTRE MMC again first to submit correct solutions to both problems. We also obtained correct solutions to Problem 4 from Peter Macgregor of James Watt College in Greenok, Stephen Mason from Leeds, and Peter Dickinson of Salford City Council. Well done!

And here are the solutions of our last problems:

Problem 3: Star Triangle The condition that the three stars are close by implies that the problem can be considered on an infinite plane. Choose the two stars that are farthest apart and label them A and B. Let the distance AB = 2r. The third star, labelled C, must then lie within two intersecting arcs of radius 2r centred at A and B. The triangle ABC will be obtuse if C is http://www.mmc.le.ac.uk/ inside the circle with diameter AB, since the angle ACB > 90° in this Founded in 1999, the University of Leicester’s Centre case. Therefore, assuming uniform distribution of stars in the sky, the for Mathematical Modelling (MMC) unites research- probability for the triangle ABC to ers with shared interests in advanced modelling and be obtuse is equal to the ratio of simulation methods. Cutting across traditional disci- the area of the circle and the area plinary boundaries, the MMC organizes a broad-based within the arcs. That is research colloquium series, funds visiting scholars and small workshops on Centre-related themes, and hosts P(obtuse) = an annual distinguished lectureship. In Summer 2003, the MMC moved to a new building, adjacent to the University’s Space Research Centre and overlooking the magnificent Victoria Park. Problem 4: The Jogger Let x be the distance along the horizontal path and y the distance on The new MMC building will be named after University of the hill. Then the duration of the run is Leicester’s chancellor Sir Michael Atiyah and officially opens with a public lecture by the chancellor himself on October 20th (4pm, Ken Edwards LT 1). The title of the lecture emphasizing the multidisciplinary cutting-edge hours. From this equation we obtain that the total distance of the run 2(x research at the MMC is “Polyhedra in Geometry, Chem- + y) = 20 miles. The curiosity of this problem is that we have one equation istry and Physics”. Everyone is very welcome! with two unknowns, yet we can solve the problem due to the fact that The MMC unites staff from many departments of the University of Leicester, coming together as neces- sary to exchange ideas, develop new computing or modelling methodology, or for the purpose of training and because we only need to determine the sum x + y, not each of the distances separately. MSc and PhD students. The departments involved are Biology, Chemistry, Computer Science, Economics, And here come the new challenges. They are again taken Engineering, Genetics, Geosciences, Mathematics and from an excellent collection published in a popular Physics and Astronomy. Russian magazine “KBAHT”(Quantum). Solutions should be sent as usual via email to: Postgraduate students are encouraged to participate [email protected] in the activities of the MMC, either from a base in a particular department, or working as part of a cross-dis- Problem 5: Inequality (by S. Zlobin) ciplinary team. The MMC gives PhD students unique op- Given three positive numbers a, b, and c, such that portunities to broaden their exposure to modern ideas 1/a + 1/b + 1/c ≥ a + b + c. of computational science, improving both the potential Prove that: a + b + c ≥ 3abc. visibility and appeal of their research work.

By emphasizing interdisciplinary teamwork and com- Problem 6: Weights (by V. Proizvolov) munication, and through provision of enhanced parallel Given a set of 20 different weights, such that from any computing and visualization training, the MMC aims to collection of eleven weights it is possible to choose two provide motivated research students with a balanced with the combined weight equal 100g. Prove that the mix of skills that can translate into stronger career total weight of all 20 weights is equal 2000g. prospects, whether in academia or in industry.