PROJECTIVE AND POLAR SPACES

By BARBARA SMITH

CONTENTS PROJECTIVE SPACES FIELDS AND VECTOR SPACES THE “FUNDAMENTAL THEOREM OF PROJECTIVE GEOMETRY” FINITE PROJECTIVE SPACES PROJECTIVE PLANES DESARGUESIAN AND PAPPIAN PLANES PROJECTIVITIES COORDINATISATION OF PROJECTIVE SPACES THE GF CASE AN APPLICATION THE GENERAL CASE LATTICES AF?NE SPACES TRANSITIVITY OF PARALLELISM VARIOUS TOPICS SPREADS AND TRANSLATION PLANES SOME SUBSETS OF PROJECTIVE SPACES SEGRE’S THEOREM OVOIDS AND INVERSIVE PLANES PROJECTIVE LINES GENERATION AND SIMPLICITY BUEKENHOUT GEOMETRIES SOME SPECIAL DIAGRAMS POLAR SPACES DUALITIES AND POLARITIES HERMITIAN AND QUADRATIC FORMS CLASSI?CATION OF FORMS CLASSICAL POLAR SPACES FINITE POLAR SPACES AXIOMS FOR POLAR SPACES GENERALISED QUADRANGLES DIAGRAMS FOR POLAR SPACES TITS AND BUEKENHOUT–SHULT RECOGNISING HYPERBOLIC QUADRICS RECOGNISING QUADRICS OVER GF THE KLEIN QUADRIC AND TRIALITY THE PFAF?AN THE KLEIN CORRESPONDENCE SOME DUALITIES DUALITIES OF SYMPLECTIC QUADRANGLES REGULI AND SPREADS TRIALITY AN EXAMPLE GENERALISED POLYGONS SOME GENERALISED HEXAGONS THE GEOMETRY OF THE MATHIEU GROUPS THE GOLAY CODE THE WITT SYSTEM SEXTETS THE LARGE MATHIEU GROUPS THE SMALL MATHIEU GROUPS 10 EXTERIOR POWERS AND CLIFFORD ALGEBRAS 10 1 TENSOR AND EXTERIOR PRODUCTS 10 2 THE GEOMETRY OF EXTERIOR POWERS 10 3 NEAR POLYGONS 10 3 1 EXERCISES 10 4 DUAL POLAR SPACES 10 5 CLIFFORD ALGEBRAS AND SPINORS 10 6 THE GEOMETRY OF SPINORS