|Owning Department/School:||Department of Mathematical Sciences|
|Level:||Honours (FHEQ level 6)|
|Supplementary Assessment:||MA30231 Mandatory Extra Work (where allowed by programme regulations)|
|Requisites:||Before taking this unit you must take MA20216 or (take MA20008 and take MA20012)|
This course introduces basic notions in projective geometry using linear algebra. It aims to strengthen understanding of linear algebra by demonstrating its geometrical significance, while also pointing towards more advanced algebraic geometry. Particular attention will be paid to quadrics (the geometric representation of quadratic forms) and the Klein correspondence between lines in 3-dimensional space and a 4-dimensional quadric called the Klein quadric.
An understanding of the following topics:
* Basic properties of projective spaces over arbitrary fields.
* Projective transformations and their uses.
* Geometry of the dual space.
* Geometry of quadrics, especially conics.
* Definition and properties of the Klein quadric.
Ability to tackle the following:
* Compute dimensions of intersections and joins.
* Find the singular conics in a pencil.
* Simultaneously diagonalize conics.
* Recognize decomposable forms in exterior algebra.
Projective spaces over arbitrary fields: projective subspaces, homogeneous and inhomogeneous coordinates, joins and intersections with dimension formula and applications.
Projective maps and transformations, perspective drawing, points in general position, Desargues' theorem and applications.
Projective lines and cross ratios.
Dual projective space, annihilators and duality, relation with joins and intersections.
Quadrics: bilinear forms and quadratic forms, singular and nonsingular quadrics, quadrics on a line, classification of conics with application to Pythagorean triples, quadric surfaces and rulings, polarity.
Pencils of quadrics, simultaneous diagonalizability and singular quadrics, simultaneous diagonalization for conics.
Exterior algebra and Klein correspondence: alternating forms and wedge product, decomposables and their characterization, the Klein quadric and its correspondence with lines in projective 3-space, alpha and beta planes and their propoerties, relevance to tomography.
Additional topics may be chosen from the following (or similar):
* Minkowski space and the celestial sphere.
* Klein geometries.
* Hyperbolic space and the parallel postulate.
MA30231 is Optional on the following programmes:Department of Computer Science