On Dubrovin Topological Field Theories

I show that the new topological field theories recently associated by Dubrovin with each Coxeter group may be all obtained in a simple way by a ``restriction'' of the standard ADE solutions. I then study the Chebichev specializations of these topological algebras, examine how the Coxeter graphs and matrices reappear in the dual algebra and mention the intriguing connection with the operator product algebra of conformal field theories. A direct understanding of the occurrence of Coxeter groups in that context is highly desirable.Comment: 15 page

Generalised twisted partition functions

We consider the set of partition functions that result from the insertion of twist operators compatible with conformal invariance in a given 2D Conformal Field Theory (CFT). A consistency equation, which gives a classification of twists, is written and solved in particular cases. This generalises old results on twisted torus boundary conditions, gives a physical interpretation of Ocneanu's algebraic construction, and might offer a new route to the study of properties of CFT.Comment: 12 pages, harvmac, 1 Table, 1 Figure . Minor typos corrected, the figure which had vanished reappears

Conformal Boundary Conditions and what they teach us

The question of boundary conditions in conformal field theories is discussed, in the light of recent progress. Two kinds of boundary conditions are examined, along open boundaries of the system, or along closed curves or ``seams''. Solving consistency conditions known as Cardy equation is shown to amount to the algebraic problem of finding integer valued representations of (one or two copies of) the fusion algebra. Graphs encode these boundary conditions in a natural way, but are also relevant in several aspects of physics ``in the bulk''. Quantum algebras attached to these graphs contain information on structure constants of the operator algebra, on the Boltzmann weights of the corresponding integrable lattice models etc. Thus the study of boundary conditions in Conformal Field Theory offers a new perspective on several old physical problems and offers an explicit realisation of recent mathematical concepts.Comment: Expanded version of lectures given at the Summer School and Conference Nonperturbative Quantum Field Theoretic Methods and their Applications, August 2000, Budapest, Hungary. 35 page

Knot theory and matrix integrals

The large size limit of matrix integrals with quartic potential may be used to count alternating links and tangles. The removal of redundancies amounts to renormalizations of the potential. This extends into two directions: higher genus and the counting of "virtual" links and tangles; and the counting of "coloured" alternating links and tangles. We discuss the asymptotic behavior of the number of tangles as the number of crossings goes to infinity.Comment: chapter of the book Random Matrix Theory, Eds Akemann, Baik and Di Francesc

A Mars orbital laser altimeter for rover trafficability: Instrument concept and science potential

Limited information on the types of geologic hazards (boulders, troughs, craters etc.) that will affect rover trafficability on Mars are available for the two Viking Lander sites, and there are no prospects for increasing this knowledge base in the near future. None of the instrument payloads on the upcoming Mars Observer or Soviet PHOBOS missions can directly measure surface obstacles on the scales of concern for rover safety (a few meters). Candidate instruments for the Soviet Mars 92 orbiter/balloon/rover mission such as balloon-borne stereo imaging, rover panoramic imaging, and orbital synthetic aperature imaging (SAR) are under discussion, but data from this mission may not be available for target areas of interest for the U.S. Mars Rover Sample Return (MRSR) mission. In an effort to determine how to directly measure the topography of surface obstacles that could affect rover trafficability on Mars, we are studying how to design a laser altimeter with extremely high spatial and vertical resolution that would be suitable for a future Mars Orbiter spacecraft (MRSR precursor or MRSR orbiter). This report discusses some of the design issues associated with such an instrument, gives examples of laser altimeter data collected for Mars analog terrains on Earth, and outlines the scientific potential of data that could be obtained with the system