Research Problems in Discrete Geometry

Although discrete geometry has a rich history extending more than 150 years, it abounds in open problems that even a high-school student can understand and appreciate. Some of these problems are notoriously difficult and are intimately related to deep questions in other fields of mathematics. But many problems, even old ones, can be solved by a clever undergraduate or a high-school student equipped with an ingenious idea and the kinds of skills used in a mathematical olympiad.

Research Problems in Discrete Geometry is the result of a 25-year-old project initiated by the late Leo Moser. It is a collection of more than 500 attractive open problems in the field. The largely self-contained chapters provide a broad overview of discrete geometry, along with historical details and the most important partial results related to these problems. This book is intended as a source book for both professional mathematicians and graduate students who love beautiful mathematical questions, are willing to spend sleepless nights thinking about them, and who would like to get involved in mathematical research.

Important features include:

- More than 500 open problems, some old, others new and never before published

- Each chapter divided into self-contained sections, each section ending with an extensive bibliography

- A great selection of research problems for graduate students looking for a dissertation topic

- A comprehensive survey of discrete geometry, highlighting the frontiers and future of research

- More than 120 figures

- A preface to an earlier version written by the late Paul Erdos

Peter Brass is Associate Professor of Computer Science at the City College of New York. William O. J. Moser is Professor Emeritus at McGill University. Janos Pach is Distinguished Professor at The City College of New York, Research Professor at the Courant Institute, NYU, and Senior Research Fellow at the Rényi Institute, Budapest.

• Lattice
• combinatorial geometry
• discrete geometry

From the reviews:

"Research Problems in Discrete Geometry has recently been released by Springer. This book collects hundreds of questions and open conjectures in the field, and discusses partial results … . The book very efficiently and clearly describes the relevant definitions … . The book is very well annotated, with a comprehensive bibliography after every set of problems … . Brass, Moser, and Pach have done a very nice job in putting this collection together … an excellent addition to any library or math department lounge." (Daren Glass, MathDL – Online, October, 2006)

"Many problems in discrete geometry have an intuitive appeal … . This book is a large collection of such problems … . the collection is quite impressive and is likely to be appreciated by a large audience of mathematicians … . an up-to-date bibliography is given, making this book a valuable source of reference. The whole text is written in a pleasant, easy to read style. The great number of beautiful illustrations help with the explanations and make the book even more attractive." (W. Kuperberg, Mathematical Reviews, 2006 i)

"The book is written in the style of Paul Erdös’s problem articles. … Each section is a self-contained essay that provides the history of a problem, known partial results, motivation and bibliography – everything needed for the reader, especially a young mathematician seeking a problem to conquer … . Professionals will find the book to be a valuable encyclopedic source for their research. Indeed, this is an exceptional book, a must for all active mathematicians." (Alexander Soifer, Zentralblatt MATH, Vol. 1086, 2006)

"Research Problems in Discrete Geometry is the result of a 25-year-old project initiated by the late Leo Moser. It is a collection of more than 500 attractive open problems in the field. The largely self-contained chapters provide a broad overview of discrete geometry, along with historical details and the most important partial results related to these problems. This book is intended as a source book for both professional mathematicians and graduate students who love beautiful mathematical questions." (L'Enseignement Mathématique, Vol. 52 (2), 2006)

"The origin of this book is an informal list of problems, first distributed by Leo Moser (1921-1970) in 1963 (in Boulder, Colorado) and then in 1977 (in Oberwolfach). It soon became famous in the discrete geometry community, was widely distributed and updated several times. … this is a ‘long-awaited’ book which turns out to be not only a valuable source for researchers, but is also a very readable panorama of discrete geometry for all who are interested in this subject." (P. Schmitt, Monatshefte für Mathematik, Vol. 151 (4), 2007)

"This is a book that gathers together an enormous number of open problems in Discrete Geometry. … The intended audience is both researchers and graduate students … . This is a very nice collection of problems. If you want to learn some open problems in Discrete Geometry this is clearly the book to go to. … An interested undergraduate, or someone in a different field, can get a start in this field through this book." (William Gasarch, SIGACT News, Vol. 38 (4), 2007)

• Department of Computer Science, City College, City University of New York, New York

  Peter Brass

• Department of Mathematics and Statistics, McGill University, Montreal

  William O. J. Moser

• Courant Institute, New York University, New York

  János Pach

Pach is a distinguished researcher at NYU, his book "Combinatorial Geometry," 1995, Wiley, is considered "the bible" in the area of discrete geometry. He has also published several books with Springer-Verlag.

William O.J. Moser is a Springer author as well. He has been awarded the CMS 2003 Distinguished Service Award for his sustained and significant contributions to the Canadian mathematical community.

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