Oriented matroids appear throughout discrete geometry, with applications in algebra, topology, physics, and data analysis. This introduction to oriented matroids is intended for graduate students, scientists wanting to apply oriented matroids, and researchers in pure mathematics. The presentation is geometrically motivated and largely self-contained, and no knowledge of matroid theory is assumed. Beginning with geometric motivation grounded in linear algebra, the first chapters prove the major cryptomorphisms and the Topological Representation Theorem. From there the book uses basic topology to go directly from geometric intuition to rigorous discussion, avoiding the need for wider background knowledge. Topics include strong and weak maps, localizations and extensions, the Euclidean property and non-Euclidean properties, the Universality Theorem, convex polytopes, and triangulations. Themes that run throughout include the interplay between combinatorics, geometry, and topology, and the idea of oriented matroids as analogs to vector spaces over the real numbers and how this analogy plays out topologically.

‘This is a book we have been waiting for for a long time: Laura Anderson’s Introduction to Oriented Matroids - accessible, lively, a lot to discover!’

Günter M. Ziegler - Freie Universität Berlin

‘The subject of oriented matroids is a cornerstone of modern combinatorial geometry. The author's conversational writing style brings this technically challenging topic to an easily comprehensible level, making it an excellent candidate for a textbook or for personal study. I am looking forward to utilizing it myself when I next teach a course on oriented matroids!’

Jim Lawrence - George Mason University

‘Laura Anderson expertly guides the reader through the multifaceted theory of oriented matroids, with a strong geometric motivation and a careful combinatorial exposition. Her book is a welcome and opportune addition to the literature, which will be valuable for newcomers and specialists alike.’

Federico Ardila-Mantilla - San Francisco State University

‘This book is a friendly invitation to the subject of oriented matroids, providing a geometrically inclined introduction and bridging the gap to the more demanding and encyclopaedic Red Book (Björner et al., 1999). It is not always easy to reconcile geometry with combinatorics and this textbook is certainly on the right path. For illustration, ‘combinatorial Farkas Property’ (‘Farkas’ is mentioned in the book 72 times!) offers a unified point of view throughout the book and helps the reader understand the rationale behind more technically involved parts of the subject. Both students and teachers will find a selection of relatively new developments in the book (which haven’t been covered in a textbook before!), including a glimpse into the general theory of matroids over hyperfield.’

Rade Živaljević - Mathematical Institute SASA, Belgrade