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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