Preface
The last ten years have witnessed the fact that geometry, topology, and algorithms
form a potent mix of disciplines with many applications inside and outside academia.
We aim at bringing these developments to a larger audience. This book has been
written to be taught, and it is based on notes developed during courses delivered
at Duke University and at the Berlin Mathematical School, primarily to students
of computer science and mathematics. The organization into chapters, sections,
and exercises reflects the teaching style we practice. Each chapter develops a major
topic and provides material for about two weeks. The chapters are divided into
sections, each a lecture of one and a quarter hours. An interesting challenge is the
mixed background of the audience. How do we teach topology to students with a
limited background in mathematics, and how do we convey algorithms to students
with a limited background in computer science? Assuming no prior knowledge and
appealing to the intelligence of the listener are good first steps. Motivating the
material by relating it to situations in different walks of life is helpful in building
up intuition that can cut through otherwise necessary formalism. Exposing central
ideas with simple means helps, and so does minimizing the necessary amount of
technical detail.
The material in this book is a combination of topics in geometry, topology, and
algorithms. Far from getting diluted, we find that the fields benefit from each other.
Geometry gives a concrete face to topological structures, and algorithms offer a
means to construct them at a level of complexity that passes the threshold necessary
for practical applications. As always, algorithms have to be fast because time is the
one fundamental resource humankind has not yet learned to manipulate for its selfish
purposes. Beyond these obvious relationships, there is a symbiotic affinity between
algorithms and the algebra used to capture topological information. It is telling that
both fields trace their names back to the writing of the same Persian mathematician,
al-Khwarizmi, working in Baghdad during the ninth century after Christ. Besides
living in the triangle spanned by geometry, topology, and algorithms, we find it
useful to contemplate the place of the material in the tension between extremes
such as local vs. global, discrete vs. continuous, abstract vs. concrete, and intrinsic
vs. extrinsic. Global insights are often obtained by a meaningful integration of local
information. This is how we proceed in many fields, taking on bigger challenges after
mastering the small ones. But small things are big from up close, and big things
are small from afar. Indeed, the question of scale lurking behind this thought is the
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