MA30231: Projective geometry
 Academic Year: | 2019/0 |
| Owning Department/School: | Department of Mathematical Sciences |
| Credits: | 6 [equivalent to 12 CATS credits] |
| Notional Study Hours: | 120 |
| Level: | Honours (FHEQ level 6) |
| Period: |
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| Assessment Summary:
| EX-TH 100%* |
| Assessment Detail: |
*Assessment updated due to Covid-19 disruptions |
| Supplementary Assessment: |
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| Requisites: | Before taking this module you must take MA20216 |
| Description:
| Aims: 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. Learning Outcomes: 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. Skills: 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. Content: 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. |
| Programme availability: |
MA30231 is Optional on the following programmes:Department of Computer Science
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Notes:
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