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Programme & Unit Catalogues

## MA50231: Projective geometry

Owning Department/School: Department of Mathematical Sciences
Credits: 6
Level: Masters UG & PG (FHEQ level 7)
Period: Semester 2
Assessment Summary: CW 25%, EX 75%
Assessment Detail:
• Coursework (CW 25%)
• Examination (EX 75%)
Supplementary Assessment: Like-for-like reassessment (where allowed by programme regulations)
Requisites:
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:
After taking this unit, students should be able to:
* state definitions and theorems in projective geometry and present proofs of the main theorems
* construct their own proofs of unseen results by geometric and analytic methods
* apply definitions and theorems to solve problems in linear algebra and related areas
* compute dimensions of intersections and joins
* find the singular conics in a pencil
* compute when and how quadratic forms can be simultaneously diagonalised
* recognize decomposable forms and calculate efficiently in exterior algebra. Through their coursework, students will be able to demonstrate a deep understanding of an area of projective geometry.

Skills:
Analytic skills T/F A; Problem solving T/F A; Written communication F A.

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:

#### MA50231 is Optional on the following programmes:

Department of Mathematical Sciences

Notes:
* This unit catalogue is applicable for the 2015/16 academic year only. Students continuing their studies into 2016/17 and beyond should not assume that this unit will be available in future years in the format displayed here for 2015/16.
* Programmes and units are subject to change at any time, in accordance with normal University procedures.
* Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.