Donato

“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 driving force for much of the development described in this book. The dichotomy between discrete and continuous structures is driven by opposing goals: machine computation and human understanding. The tension between the abstract and the concrete as well as between the intrinsic and the extrinsic has everything to do with the human approach to knowledge. An example close to home is the step from geometry to topology in which we remove the burdens of size to focus on the phenomenon of connectivity. The more abstract the context the more general the insight. Now, generality is good, but it is not a substitute for the concrete steps that have to be taken to build bridges to applications. Zooming in and out of generality leads to unifying viewpoints and suggests meaningful integrations where they exist.”
― Herbert Edelsbrunner, John Harer, Computational Topology: An Introduction