Description

This volume examines state of the art research in finite geometries and designs. This volume examines finite geometries and designs, a key area in modern applicable mathematics. It includes state-of-the-art surveys from leading international mathematicians in their fields, and will be a useful reference for researchers in many aspects of combinatorics. This volume examines finite geometries and designs, a key area in modern applicable mathematics. It includes state-of-the-art surveys from leading international mathematicians in their fields, and will be a useful reference for researchers in many aspects of combinatorics. This volume presents up-to-date research on finite geometries and designs, a key area in modern applicable mathematics. An introductory chapter discusses topics presented in each of the main chapters, and is followed by articles from leading international figures in this field. These include a discussion of the current state of finite geometry from a group-theoretical viewpoint, and surveys of difference sets and of small embeddings of partial cycle systems into Steiner triple systems. Also presented are successful searches for spreads and packing of designs, rank three geometries with simplicial residues and generalized quadrangles satisfying Veblen's Axiom. In addition, there are articles on new 7-designs, biplanes, various aspects of triple systems, and many other topics. This book will be a useful reference for researchers working in finite geometries, design theory or combinatorics in general. Preface; Introduction; 1. On maximum size anti-Pasch sets of triples; 2. Some simple 7-designs; 3. Inscribed bundles, Veronese surfaces and caps; 4. Embedding partial geometries in Steiner designs; 5. Finite geometry after Aschbacher's theorem: PGL(n,q) from a Kleinian viewpoint; 6. The Hermitian function field arising from a cyclic arc in a Galois plane; 7. Intercalates everywhere; 8. Difference sets: an update; 9. Computational results for the known biplanes of order 9; 10. A survey of small embeddings for partial cycle systems; 11. Rosa triple systems; 12. Searching for spreads and packings; 13. A note of Buekenhout–Metz unitals; 14. Elation generalized quadrangles of order (q^2, q); 15. Uniform parallelisms of PG(3,3); 16. Double-fives and partial spreads in PG(5,2); 17. Rank three geometries with simplicial residues; 18. Generalized quadrangles and the Axiom of Veblen; Talks; Participants.