BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Talks.cam//talks.cam.ac.uk// X-WR-CALNAME:Talks.cam BEGIN:VEVENT SUMMARY:ONE DAY MEETING - Geometry in Science - Professor Sir Michael Berr y FRS\; Professor Chris Calladine FRS \; Professor Gabor Domokos\; Profess or Jan Koenderink\; Professor Gabriel Paternain\; Professor Denis Weaire F RS DTSTART:20120113T090000Z DTEND:20120113T173000Z UID:TALK31919@talks.cam.ac.uk CONTACT:Beverley Larner DESCRIPTION:Professor Chris Calladine FRS \nDepartment of Engineering\, Un iversity of Cambridge\n\n“Some spiral structures in biology”\nBiologic al structures\, unlike those designed and constructed by human structural engineers\, are built by a process of self-assembly. The internal machiner y of biological cells produces molecular building-blocks in accordance wit h information encoded in the DNA\, and these then find their appointed pla ce in the assembly under construction. The simplest possible self-assemb led structure is a uniform helix\, or spiral: repeated addition of identic al building-blocks in a regular pattern make a uniform helix\, which is th e simplest space-curve.\nIn this talk I shall discuss three particular spi ral structures built from molecular components.\n1. The α-helix\, an im portant motif in protein structures\; and in particular the way in which t wo α-helices can be programmed to assemble into a “coiled coil” arran gement.\n2. The DNA double-helix. How it can switch between the classic al “A” and “B” geometries\; and how this kind of change enables u s to understand sequence-dependent curvature and flexibility of the molecu le – which is important in the recognition of DNA sequences by contactin g proteins.\n3. Bacterial flagellar filaments – the corkscrew-like org anelles which\, when rotated by their motors\, enable bacteria such as E.c oli to swim in their watery environment and navigate towards nutrients. H ere the building-block\, much larger than those of examples 1 and 2\, is a protein molecule which contains a “switch” feature\; and this enables the filament to change from a left-handed to a right-handed corkscrew whe n it is driven in reverse.\nThese examples illustrate the power of evoluti on to develop highly sophisticated variants of the simplest kind of self-a ssembled biological structures. Geometry provides an indispensable tool i n elucidating the subtle structural phenomena seen in these helices. \n\n \nProfessor Denis Weaire FRS \nTrinity College Dublin\n\n“The geometry o f foam packings.”\nThe structure of a foam was first described in detail by Plateau in the nineteenth century\, and his geometrical/topological ru les have guided us ever since. The foam may be monodisperse or polydispers e\, ordered or disordered\, wet or dry (referring to the liquid fraction)\ , two or three dimensional\, and may be confined (eg in a cylinder) or ess entially infinite. Kelvin posed the question of the minimum energy structu re for a monodisperse 3D dry foam\, and this has been debated ever since. In the opposite limit\, that of a wet foam\, the bubbles are spherical and hard-sphere structures are formed. We review a variety of problems having to do with structure and stability\, including the computer simulations a nd experiments that have been brought to bear on them.\n\nProfessor Sir Mi chael Berry FRS \nDepartment of Physics\, University of Bristol \n\n“The singularities of light: intensity\, phase\, polarization”\n\nGeometry d ominates modern optics\, in which we understand light through its singular ities. These are different at different levels of description. \nAt the co arsest level\, where light is described in terms of the rays of geometrica l optics\, the singularities are caustics: focal lines and surfaces – th e envelopes of ray families. These singularities of bright light are class ified by the mathematics of catastrophe theory. Wave optics smooths these singularities and decorates them with rich interference patterns\, widely applicable\, for example to rainbows\, ship wakes and quantum scattering.\ nWave optics introduces a new quantity\, namely phase\, which has its own singularities. These are optical vortices\, a.k.a nodes or wavefront dislo cations. Geometrically these singularities of dark light are lines in spac e\, or points in the plane. They occur in all types of quantum or classica l waves. Currently\, optical phase singularities are being used to rotate small particles (optical spanners) and as a possible way to detect extra-s olar planets.\nOn a finer scale\, where the vector nature of light cannot be ignored\, the new phenomenon is polarization. This possesses its own si ngularities\, also geometrical\, describing lines where the polarization i s purely circular.\nAs well as representing interesting physics at each le vel\, these optical and wave geometries illustrate the idea of asymptotica lly emergent phenomena. \n\nProfessor Gabor Domokos \nDepartment of Mechan ics\, Materials and Structures\, Budapest University of Technology and Eco nomics\n\n“Natural numbers\, natural shapes”\n\nThe first step towards understanding natural shapes might be their systematic description. Inste ad of creating a hierarchical list of names in the spirit of Linné\, we t ry to classify shapes based on naturally assigned integers\, carrying info rmation on the number\, type and interrelation of static equilibrium point s. In mechanical language\, these are points where the body is at rest on a horizontal surface\, in mathematical language these are the singulariti es of the gradient flow associated with the surface. \nWhile at first sigh t this appears to be a rather meager source of information compared to th e abundance of three-dimensional shapes\, we found that often meaningful information is condensed here.\n\nOne advantage of this classification is that we count (instead of measure) and thus do not add observer-related n oise to the obtained data. Counting equilibria results in several\, distin ct integers describing different geometrical aspects of the investigated s hape. One can distinguish between stable and unstable equilibria\, also\, the graph (called the Morse-Smale graph) carrying the topological informat ion about their arrangement can be uniquely identified by an integer. Beyo nd physically existing equilibria we can also count imaginary ones\, corre sponding to arbitrarily fine\, equidistant polyhedral approximations\, pro viding information about curvatures.\n\nWhen looking at various shapes in Nature\, ranging from coastal pebbles to asteroids\, from extant to long-e xtinct turtles\, the integers extracted by the described means appear to c arry information relevant to natural history. One could also imagine the l ong evolution of these shapes (whether biological or mechanical) as a codi ng sequence. Whether or not equilibria are the 'true code'\, we do not kno w\, however\, these simple numbers certainly help to better understand evo lutionary history. We are also confronted by some puzzles: shapes correspo nding to some special integer combinations appear to be missing from Natur e.\n\n-----------------------------------------------------\nSome links to recent work on the subject:\nhttp://arxiv.org/abs/0904.4423\nhttp://arxiv .org/abs/1106.0626\nhttp://arxiv.org/abs/1104.4813\nhttp://www.springerlin k.com/content/6880172060n40l18/\nhttp://www.akademiai.com/content/e4646nr7 083g47w6/\nhttp://www.gomboc.eu/100.pdf\n---------------------------\n\nPr ofessor Gabriel Paternain\nDepartment of Pure Mathematics and Mathematical Statistics\, University of Cambridge\n\n“Contact geometry in dynamics: the 3-body problem”\n\nWe have known for a long time how to write down t he equations of motion of a satellite that moves under the influence of th e gravitational fields of the Earth and the Moon\, but surprisingly\, we s till do not fully understand the long term behaviour of the satellite sinc e we cannot explicitly solve the equations. At the end of the 19th century \, Poincar\\'e noticed the presence of chaos in the system and kick-starte d the modern theory of dynamical systems. There have been some remarkable applications of these chaotic motions to low fuel transfer orbits\, specia lly since the celebrated rescue of the Japanese satellite Hiten in 1991 (B elbruno). Recently a new type of geometry called contact geometry (the odd -dimensional relative of symplectic geometry) has been proposed as a tool for understanding this old problem in celestial mechanics. In the talk I w ill try to explain what contact geometry is and why it is relevant for the 3-body problem.\n\nProfessor Jan Koenderink \nDelft University of Technol ogy and Katholieke Universiteit Leuven\n\n"Pictorial space: A geometry of visual awareness"\n\n\nThe visual field is two-fold\, the visual world thr ee-fold extended. Pictorial space assumes an intermediate position\, “de pth” being a quale\, much like color. \n\nFrom an evolutionary perspecti ve vision is an optical user interface. This has important implications\, e.g.\, vision is not necessarily veridical\, since an effective interface shields the user from irrelevant complexity. This is indeed obvious in imm ediate awareness\, where one frequently “sees” objects or processes th at reflective thought reveals as illusory. This even extends to the space- time structure of the visual field. On artificial spatiotemporal scramblin g of the retinal image\, visual awareness is often coherent. \n\nDepth is a quality of “separateness from the self”\, even though the eye is not in pictorial space. Since there is no natural depth origin\, nor a unit o f length\, a convenient formal description is the full affine line. Then “pictorial space” would be a fiber bundle\, the base space being repre sented by the Euclidean plane\, the fibers the depth dimension. An analys is of the structure of “depth cues” lets one derive a group of movemen ts and scalings that transforms configurations such that the cues are resp ected. One expects observers to mutually agree modulo such a transformatio n.\n\nWe have developed methods to operationalize (“measure” is not ap t because pictorial depth configurations are only operationally defined) p ictorial reliefs (surface shapes) as well as arbitrary point configuration s. We find apparently very significant differences between different obser vers. However\, such differences are explained in considerable quantitativ e detail through the group of movements mentioned above. Apparently “pic torial space” has a tight (non-Euclidean) structure.\n\n\n\n LOCATION: Cambridge University Engineering Department\, LR0 END:VEVENT END:VCALENDAR