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Programme & Unit Catalogues

## MA30231: Projective geometry

Owning Department/School: Department of Mathematical Sciences
Credits: 6
Level: Honours (FHEQ level 6)
Period: Semester 2
Assessment Summary: EX 100%
Assessment Detail:
• Examination (EX 100%)
Supplementary Assessment: MA30231 Mandatory Extra Work (where allowed by programme regulations)
Requisites: Before taking this unit 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.
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
• USCM-AFB20 : BSc(Hons) Computer Science and Mathematics (Year 3)
• USCM-AAB20 : BSc(Hons) Computer Science and Mathematics with Study year abroad (Year 4)
• USCM-AKB20 : BSc(Hons) Computer Science and Mathematics with Year long work placement (Year 4)
• USCM-AFM14 : MComp(Hons) Computer Science and Mathematics (Year 3)
• USCM-AFM14 : MComp(Hons) Computer Science and Mathematics (Year 4)
• USCM-AAM14 : MComp(Hons) Computer Science and Mathematics with Study year abroad (Year 4)
• USCM-AAM14 : MComp(Hons) Computer Science and Mathematics with Study year abroad (Year 5)
• USCM-AKM14 : MComp(Hons) Computer Science and Mathematics with Year long work placement (Year 4)
• USCM-AKM14 : MComp(Hons) Computer Science and Mathematics with Year long work placement (Year 5)
• USCM-AAB14 : BSc(Hons) Computer Science with Mathematics with Study year abroad (Year 4)
• USCM-AKB14 : BSc(Hons) Computer Science with Mathematics with Year long work placement (Year 4)
• USCM-AFB01 : BSc Computing (Year 3)
• USCM-AKB01 : BSc Computing with Year long work placement (Year 4)
Department of Mathematical Sciences
• USMA-AFB15 : BSc(Hons) Mathematical Sciences (Year 3)
• USMA-AAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
• USMA-AKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
• USMA-AFB13 : BSc(Hons) Mathematics (Year 3)
• USMA-AAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
• USMA-AKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
• USMA-AFM14 : MMath(Hons) Mathematics (Year 3)
• USMA-AFM14 : MMath(Hons) Mathematics (Year 4)
• USMA-AAM15 : MMath(Hons) Mathematics with Study year abroad (Year 4)
• USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)
• USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 5)
• USMA-AFB01 : BSc(Hons) Mathematics and Statistics (Year 3)
• USMA-AAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 4)
• USMA-AKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 4)
• TSMA-AFM09 : MSc Mathematical Sciences
• TSMA-APM09 : MSc Mathematical Sciences
• USMA-AFB05 : BSc(Hons) Statistics (Year 3)
• USMA-AAB06 : BSc(Hons) Statistics with Study year abroad (Year 4)
• USMA-AKB06 : BSc(Hons) Statistics with Year long work placement (Year 4)
Department of Physics
• USXX-AFB03 : BSc(Hons) Mathematics and Physics (Year 3)
• USXX-AAB04 : BSc(Hons) Mathematics and Physics with Study year abroad (Year 4)
• USXX-AKB04 : BSc(Hons) Mathematics and Physics with Year long work placement (Year 4)
• USXX-AFM01 : MSci(Hons) Mathematics and Physics (Year 3)
• USXX-AAM01 : MSci(Hons) Mathematics and Physics with Study year abroad (Year 4)
• USXX-AKM01 : MSci(Hons) Mathematics and Physics with Year long work placement (Year 4)

Notes:
* This unit catalogue is applicable for the 2014/15 academic year only. Students continuing their studies into 2015/16 and beyond should not assume that this unit will be available in future years in the format displayed here for 2014/15.
* 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.