Preface This volume contains a number of selected survey papers on the topics pre- sented and discussed at the “Lluis A. Santal´ o Summer School”, that was devoted to Mathematics of Development and Cancer. The School was held in Santander (Spain) on September 11–15, 2006, as part of the activities of the Universidad Internacional Men´endez Pelayo (UIMP), in collaboration with the Real Sociedad Matem´atica Espa˜nola (RSME). Lecturers came from different scientific fields, in- cluding Biology, Mathematics, Medicine and Physics. They were selected in an attempt to present an outline of ongoing research in selected areas of mathematics and biology, in a manner that could be widely accessible to an audience consisting mainly of advanced undergraduates and graduate students on Mathematics. These were thus given a front seat at the research currently done by a number of groups worldwide working in Biology and Mathematics. The course was centred on De- velopmental Biology and Tumour Growth. These are topics where Mathematics is increasingly being used as a new and powerful technique to gain new insights. Mathematics is in its turn receiving a significant scientific pay-off in the form of new and challenging mathematical problems to be added to their own ones. For instance, Developmental Biology is an area of basic research in Biology and Medicine that has fascinated mankind since the earliest recorded scientific thought. Developmental Biology, which has experienced a great impulse during last years, deals with the basic problem of understanding the unfolding of utterly complex living structures “from egg to embryo”. In this manner, it has been seen to raise a number of challenging quantitative problems that immediately appeal at the imagination of mathematicians. This is the case of some central problems in Biology related to pattern formation, where the issue of how space and time evolution is first coded in the genome, and then set in action during development, has generated a great interest in the use of mathematical tools. The first part of this volume contains three views on different aspects of De- velopmental Biology as seen with the eyes of biologists interested in incorporating Mathematics to their technical tools. More precisely, Alsina et al. summarise the state of the art in Developmental Biology by describing some of the basic questions in the field, and by commenting on some examples of formal approaches to the spe- cific problem of patterning in development. Arques and Torres describe the problem of how three dimensional patterns are established in the limbs of vertebrates, and show how an interesting model on mesenchymal compartments formation can be derived from a clever clonal analysis of cell lineages. Finally, Henrique and Fior discuss on one of the key signalling pathways in development, the Notch pathway, a subject on which a good deal of work is being currently done both by biologists and mathematicians. vii
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