Pattern Formation in Morphogenesis : Problems and Mathematical Issues Preface Why Would a Mathematician Care About Embryology M.Gromov. PART I.BIOLOGICAL BACKGROUND Preface to the Biological Part Pattern Formation in Regenerating Tissues. A.Hoffmann and P.A. Tsonis Gradients and Regulatory Networks of Wnt Signalling in Hydra Pattern Formation. T.W. Holstein Modeling of Planar Cell Polarity Signaling. J.D. Axelrod Integrated Molecular Circuits for Stem Cell Activity in Arabidopsis Roots. B. Scheres The Mechanics of Tissue Morphogenesis. T. Lecuit Small Regulatory RNAs and Skeletal Muscle Cell Differentiation. A. Harel Bellan A.Polesskaya I.Naguibneva M.Ameyar Zazoua C.Degerny J.Kropp N.Nonne M.Souidi G.Kratassiouk G.Pinna L.L.Pritchard Pattern Formation in Sea Urchin Endomesoderm as Instructed by Gene Regulatory Network Topologies. I.S. Peter and E.H. Davidson PART II. MATHEMATICAL MODELS Modelling Oscillator Synchronisation During Vertebrate Axis Segmentation. P.J. Murray P.K. Maini and R.E. Baker Pattern Formation in Hybrid Models of Cell Populations. N. Bessonov P. Kurbatova V. Volpert. An Integrative Approach to the Analysis of Pattern Formation in Butterfly Wings – Experiments and Models. T. Sekimura Modeling Morphogenesis in Multicellular Structures with Cell Complexes and L systems. P. Prusinkiewicz and B. Lane Multistability and Hysteresis based Mechanism of Pattern Formation in Biology. A. Marciniak Czochra and A. Köthe How to knock out feedback Circuits in Gene Networks H. Gruber A. Richard C. Soulé Formation of Evolutionary Patterns in Cancer Dynamics. M. Delitala and T. Lorenzi How Cell Decides Between Life and Death: Mathematical Modeling of Epigenetic Landscapes of Cellular Fates. A. Zinovyev L. Calzone S. Fourquet and E. Barillot PART III. IDEAS HYPOTHESIS SUGGESTIONS From Hydra to Vertebrates: Models for the Transition from Radial to Bilateral symmetric Body Plans H. Meinhardt Cell Division and Hyperbolic Geometry. M. Gromov Formalistic Representation of the Cellular Architecture in the Course of Plant Tissue Development. I.V. Rudskiy The Geometry of Morphogenesis and the Morphogenetic Field Concept. N.Morozova and M.Shubin Randomness and Geometric Structures in Biology. V. Capasso. Pattern Formation in Morphogenesis is a rich source of interesting and challenging mathematical problems. The volume offers an interdisciplinary interaction space between biologists working in this field and mathematicians who may propose solutions to the problems put forward by biologists. The main goal is to facilitate the process of cultivating a mutual recognition of the complementary skills between biologists and mathematicians to the point where the resulting synergy generates new and novel discoveries in the field of Developmental Biology. Lastly the volume shows how a combination of new discoveries in developmental biology and associated mathematical modeling and computational techniques has stimulated or may stimulate relevant advances in the field. .