Title Pages New Perspectives in Stochastic Geometry Wilfrid S. Kendall and Ilya Molchanov Print publication date: 2009 Print ISBN-13: 9780199232574 Published to Oxford Scholarship Online: February 2010 DOI: 10.1093/acprof:oso/9780199232574.001.0001 Title Pages (p.i) New Perspectives in Stochastic Geometry (p.ii) (p.iii) New Perspectives in Stochastic Geometry (p.iv) Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. 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Kendall and Ilya Molchanov Print publication date: 2009 Print ISBN-13: 9780199232574 Published to Oxford Scholarship Online: February 2010 DOI: 10.1093/acprof:oso/9780199232574.001.0001 Dedication (p.v) In memory of David Kendall, 15 January 1918–23 October 2007 (p.vi) Page 1 of 1 Preface New Perspectives in Stochastic Geometry Wilfrid S. Kendall and Ilya Molchanov Print publication date: 2009 Print ISBN-13: 9780199232574 Published to Oxford Scholarship Online: February 2010 DOI: 10.1093/acprof:oso/9780199232574.001.0001 (p.vii) Preface The roots of stochastic geometry derive from classical highlights of geometric probability. Examples include concepts arising from explorations of the notion of ‘natural' probability, questions raised in nineteenth‐century UK recreational mathematics, and the famous problem of the eighteenth‐century French polymath Georges‐Louis Leclerc, Comte de Buffon. The actual phrase ‘stochastic geometry' appears first to have been employed in its current sense by D.G. Kendall and K. Krickeberg in 1969, arising from their planning for an Oberwolfach workshop (Stoyan, Kendall and Mecke, 1987, foreword), though Andrew Wade has pointed out its use by Frisch and Hammersley (1963, p. 895) as one of two possible terms to describe the study of ‘random irregular structures' motivated by percolation problems. It is now plain that stochastic geometry has woven together various strands from a wide‐ranging variety of sources, of which we mention just four representative examples: • impetus arising from the creation and development of geostatistics and mathematical morphology by G. Matheron and J. Serra and other French workers; • the German school of point process and queueing theory, involving such names as J. Kerstan, K. Matthes, J. Mecke and K. Krickeberg; • British and Australian research, growing out of the study of classical geometric probabilities, and developing relationships with stochastic analysis; • the Armenian school of combinatorial geometry led by R.V. Ambartzumian. After 40 years stochastic geometry is now an established part of probability theory, expounded in several well‐established monographs, and finding vital employment in numerous application areas such as spatial statistics, image analysis, materials science, and even finance. In common with many areas of Page 1 of 3 Preface applied mathematics, the subject has been heavily influenced by the availability of cheap and powerful computing, justifying deeper study of models which are flexible but computationally demanding, and providing opportunities for simulation study of systems which would otherwise be inaccessible. A subtle and expressive mathematical vocabulary has been developed, allowing us to describe and investigate random patterns of points, lines, planes, fibres, surfaces, tessellations and sets. A wealth of hard theoretical problems can be found in the wide variety of application areas, and it is an ongoing challenge for stochastic geometry to contribute effectively to the intense and powerful scientific effort devoted to dealing with these problems. The present volume has been motivated by the sense that stochastic geometry is now poised for a further phase of development, acquiring new perspectives (p.viii) from revival of its classical roots, from new connections with recent developments in the main body of probability, and stimulated by the plethora of rapidly developing applications. The purpose of the volume is therefore to present recent developments so as to form an entry point into stochastic geometry for mathematical scientists from other areas who are curious about what it might offer, and to provide a resource for young researchers who wish to engage with the subject and seek out areas in which they might make an original contribution. The editors gathered together a group of experts, each of whom was tasked with writing a chapter to cover a particular aspect of stochastic geometry, reflecting the current state of the art but also commenting on and providing new developments. To this end, the volume is divided into a chapter on the fundamentals, and then into four parts. The initial Chapter 1 surveys what can be described as classical stochastic geometry, summarizing its historical development and clarifying the core mathematical concepts which arise throughout stochastic geometry as a whole. Classical stochastic geometry has itself recently seen a revival; Part I (‘New Developments in Classical Stochastic Geometry’) contains four chapters describing new developments in random polytopes, random measures, limit theory and tessellations (Chapters 2, 3, 4, 5). In addition, recently there has arisen a number of strong connections with other areas of mathematics such as percolation, random network theory and fractals, forming the three Chapters 6, 7 and 8 of Part II ‘Stochastic Geometry and Modern Probability'. There has always been a strong statistical theme in stochastic geometry, represented here by Part III ‘Statistics and Stochastic Geometry'. This commences with Chapter 9, discussing inferential issues arising when one seeks to estimate parameters of random pattern models, and continues with three further Chapters 10, 11 and 12, respectively covering statistical shape, estimation of sets, and notions of data depth. Finally Part IV ‘Applications’ contains five chapters surveying applications in image analysis (Chapter 13), stereology (Chapter 14), materials science (Chapter 15), telecommunications (Chapter 16) and finance (Chapter 17); the growth of stochastic geometry has always been fuelled by ever‐changing demands from Page 2 of 3 Preface application areas, and these chapters demonstrate strong possibilities for future growth. The authors and editors offer this volume to the mathematical community as a stepping stone towards the new perspectives which stochastic geometry now offers. References in the Preface Frisch, H. L. and Hammersley, J. M. (1963). Percolation processes and related topics. J. Soc. Indust. Appl. Math., 11, 894–918. Stoyan, D., Kendall, W. S., and Mecke, J. (1987). Stochastic Geometry and its Applications. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons Ltd., Chichester. With a foreword by D. G. Kendall. Page 3 of 3 Acknowledgements New Perspectives in Stochastic Geometry Wilfrid S. Kendall and Ilya Molchanov Print publication date: 2009 Print ISBN-13: 9780199232574 Published to Oxford Scholarship Online: February 2010 DOI: 10.1093/acprof:oso/9780199232574.001.0001 (p.ix) Acknowledgements The editors and authors would like to thank the Oxford University Press staff for help with the production of this volume. They are also grateful to Mathematisches Forschungsinstitut Oberwolfach for the opportunity to organize a workshop under the same name, which greatly aided the process of preparation of this volume, nearly 40 years after the Oberwolfach workshop at which the phrase ‘Stochastic Geometry' first saw the light of day. (p.x) Page 1 of 1 List of Contributors New Perspectives in Stochastic Geometry Wilfrid S. Kendall and Ilya Molchanov Print publication date: 2009 Print ISBN-13: 9780199232574 Published to Oxford Scholarship Online: February 2010 DOI: 10.1093/acprof:oso/9780199232574.001.0001 (p.xix) List of Contributors Dr. Pierre Calka (pierre.calka©math‐info.univ‐ paris5.fr) MAP5, Université Paris Descartes, 45, rue des Saints‐Pères, 75270 Paris Cedex 06, France. Dr. Ignacio Cascos (ignacio.cascos©uc3m.es) Department of Statistics, Universidad Carlos III de Madrid, Av. Universidad 30, E‐28911 Leganés (Madrid), Spain. Prof. Dr. Antonio Cuevas (antonio.cuevas©uam.es) Department of Mathematics, Facultad de Ciencias, Universidad Autonoma de Madrid, 28049 Madrid, Spain. Prof. Dr. R. Fraiman (rfraiman©cmat.edu.uy) Centro de Matematica, Universidad de la Republica, Eduardo Acevedo, 1139, Montevideo, Uruguay. Prof. Dr. Remco van der Hofstad (rhofstad©win.tue.nl) Eindhoven University of Technology Department of Mathematics and Page 1 of 4 List of Contributors Computer Science, HG 9.04 P.O. Box 513, 5600 MB Eindhoven, the Netherlands. Prof. Dr. W.S. Kendall (w.s.kendall©warwick.ac.uk) Dept of Statistics, University of Warwick Coventry CV4 7AL, United Kingdom. Prof. Dr. Günter Last (last©math.uni‐karlsruhe.de) Institut für Stochastik, Universität Karlsruhe, D‐76128 Karlsruhe, Germany. Dr. Huiling Le (huiling.le©nottingham.ac.uk) School of Mathematical Sciences, University of Nottingham Nottingham NG7 2RD, United Kingdom. Dr. M.N.M. van Lieshout (colette©cwi.nl) CWI, Science Park 123, NL‐1098, XG Amsterdam, The Netherlands. Prof. Dr. Klaus Mecke (klaus.mecke©physik.uni‐ erlangen.de) Institut für Theoretische Physik, Universität Erlangen‐Nürnberg, Staudtstrasse 7, 91058 Erlangen, Germany. Prof. Dr. Ilya Molchanov (ilya©stat.unibe.ch) IMSV, University of Bern Sidlerstrasse 5, 3012 Bern, Switzerland. (p.xx) Prof. Dr. Jesper Møller (jm math.aau.dk) Department of Mathematical Sciences, Aalborg University, F. Bajers Vej 7G, DK‐9220 Aalborg Øst, Denmark. Prof. Dr. Peter Mörters (maspm©bath.ac.uk) Page 2 of 4