Henry O. Pollak Chairman of the International Program Committee Bell Laboratories Murray Hill, New Jersey, USA The Fourth International Congress on Mathematics Education was held in Berkeley, California, USA, August 10-16, 1980. Previous Congresses were held in Lyons in 1969, Exeter in 1972, and Karlsruhe in 1976. Attendance at Berkeley was about 1800 full and 500 associate members from about 90 countries; at least half of these come from outside of North America. About 450 persons participated in the program either as speakers or as presiders; approximately 40 percent of these came from the U.S. or Canada. There were four plenary addresses; they were delivered by Hans Freudenthal on major problems of mathematics education, Hermina Sinclair on the relationship between the learning of language and of mathematics, Seymour Papert on the computer as carrier of mathematical culture, and Hua Loo-Keng on popularising and applying mathematical methods. Gearge Polya was the honorary president of the Congress; illness prevented his planned attendence but he sent a brief presentation entitled, "Mathematics Improves the Mind". There was a full program of speakers, panelists, debates, miniconferences, and meetings of working and study groups. In addition, 18 major projects from around the world were invited to make presentations, and various groups representing special areas of concern had the opportunity to meet and to plan their future activities.

This edited volume presents examples of social science research projects that employ new methods of quantitative analysis and mathematical modeling of social processes. This book presents the fascinating areas of empirical and theoretical investigations that use formal mathematics in a way that is accessible for individuals lacking extensive expertise but still desiring to expand their scope of research methodology and add to their data analysis toolbox. Mathematical Modeling of Social Relationships professes how mathematical modeling can help us understand the fundamental, compelling, and yet sometimes complicated concepts that arise in the social sciences. This volume will appeal to upper-level students and researchers in a broad area of fields within the social sciences, as well as the disciplines of social psychology, complex systems, and applied mathematics.

Birkhauser Boston, Inc., will publish a series of carefully selected mono graphs in the area of mathematical modeling to present serious applications of mathematics for both the undergraduate and the professional audience. Some of the monographs to be selected and published will appeal more to the professional mathematician and user of mathematics, serving to familiarize the user with new models and new methods. Some, like the present monograph, will stress the educational aspect and will appeal more to a student audience, either as a textbook or as additional reading. We feel that this first volume in the series may in itself serve as a model for our program. Samuel Goldberg attaches a high priority to teaching stu dents the art of modeling, that is, to use his words, the art of constructing useful mathematical models of real-world phenomena. We concur. It is our strong conviction as editors that the connection between the actual problems and their mathematical models must be factually plausible, if not actually real. As this first volume in the new series goes to press, we invite its readers to share with us both their criticisms and their constructive suggestions.

This IMA Volume in Mathematics and its Applications Applications of Combinatorics and Graph Theory to the Biological and Social Sciences is based on the proceedings of a workshop which was an integral part of the 1987-88 IMA program on APPLIED COMBINATORICS. We are grateful to the Scientific Committee: Victor Klee (Chairman), Daniel Kleitman, Dijen Ray-Chaudhuri and Dennis Stanton for planning and implementing an exciting and stimulating year long program. We especially thank the Workshop Organizers, Joel Cohen and Fred Roberts, for organizing a workshop which brought together many of the major figures in a variety of research fields connected with the application of combinatorial ideas to the social and biological sciences. A vner Friedman Willard Miller APPLICATIONS OF COMBINATORICS AND GRAPH THEORY TO THE BIOLOGICAL AND SOCIAL SCIENCES: SEVEN FUNDAMENTAL IDEAS FRED S. RoBERTS* Abstract. To set the stage for the other papers in this volume, seven fundamental concepts which arise in the applications of combinatorics and graph theory in the biological and social sciences are described. These ideas are: RNA chains as "words" in a 4 letter alphabet; interval graphs; competition graphs or niche overlap graphs; qualitative stability; balanced signed graphs; social welfare functions; and semiorders. For each idea, some basic results are presented, some recent results are given, and some open problems are mentioned.

This is a rich and exciting collection of examples and applications in mathematical modelling. There is broad variety, balance and highly motivating material and most of this assumes minimal mathematical training.