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Applied Topology and Geometry: preface
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The idea of this book is to present the basics of
- algebraic topology, specifically homology and cohomology (with a minimum of point-set topology), and
- differential geometry, specifically differential forms, both continuous and discrete,
with as many examples of applications as possible by which I mean primarily the ones outside of pure mathematics. More advanced applications are provided at the end.
Besides advanced undergraduate and beginning graduate students in mathematics the intended audience is that of students of science and engineering. Even though this is a mathematics book, some parts can be seen as physics and others as computer science.
Differential forms provide a modern view of calculus. They also give you a start with algebraic topology in the sense that one can extract topological information about a manifold from its space of differential forms. It is called cohomology.
The main prerequisite is linear algebra. By that I mean a mathematical course that is based on theory of vector spaces, not just matrix manipulation. There is no need for linear algebra in the chapters about image analysis and point-set topology but some later chapters on algebraic topology require groups theory and even advanced algebra. For some chapters on differential forms the reader can benefit from a prior exposure to vector calculus, beyond calc 3.
The title isn't computational topology and geometry as it would suggest much more attention payed to the algorithms.
Most of the content comes from actual lectures written in class on the screen of a tablet PC. As a result, the style is much looser than the usual definition-theorem-proof format. The presentation is filled with examples, discussions, and links to other topics.
There are hundreds of illustrations. Most of them are still the sketches I drew during the lectures. Meanwhile, the text is more sparse then you normally see in books at this level.
Related content frequently appears in different sections and chapters. This is meant to be a textbook not a reference book.
A number of courses can be constructed from this material, with or without an emphasis on applications. For example:
- Algebraic topology: course
- Calculus of differential forms: course
- Differential forms and cohomology: course
- Homology of cell complexes: course
- Mathematics of computer vision: course
- Point-set topology: course
- Topology 1: course
- Topology 2: course
Best of luck!
Peter Saveliev, 2013
