MA30231: Projective geometry
[Page last updated: 23 October 2023]
Academic Year:  2023/24 
Owning Department/School:  Department of Mathematical Sciences 
Credits:  6 [equivalent to 12 CATS credits] 
Notional Study Hours:  120 
Level:  Honours (FHEQ level 6) 
Period: 
 Semester 2

Assessment Summary:  EX 100% 
Assessment Detail:  
Supplementary Assessment: 
 Likeforlike reassessment (where allowed by programme regulations)

Requisites: 
Before taking this module you must take MA20216

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.

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 3dimensional space and a 4dimensional quadric called 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 3space, 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.

Course availability: 
MA30231 is Optional on the following courses:
Department of Computer Science
 USCMAFB20 : BSc(Hons) Computer Science and Mathematics (Year 3)
 USCMAAB20 : BSc(Hons) Computer Science and Mathematics with Study year abroad (Year 4)
 USCMAKB20 : BSc(Hons) Computer Science and Mathematics with Year long work placement (Year 4)
 USCMAFM14 : MComp(Hons) Computer Science and Mathematics (Year 3)
 USCMAFM14 : MComp(Hons) Computer Science and Mathematics (Year 4)
 USCMAAM14 : MComp(Hons) Computer Science and Mathematics with Study year abroad (Year 3)
 USCMAAM14 : MComp(Hons) Computer Science and Mathematics with Study year abroad (Year 5)
 USCMAKM14 : MComp(Hons) Computer Science and Mathematics with Year long work placement (Year 3)
 USCMAKM14 : MComp(Hons) Computer Science and Mathematics with Year long work placement (Year 5)
Department of Economics
 UHESAFB04 : BSc(Hons) Economics and Mathematics (Year 3)
 UHESAAB04 : BSc(Hons) Economics and Mathematics with Study year abroad (Year 4)
 UHESAKB04 : BSc(Hons) Economics and Mathematics with Year long work placement (Year 4)
 UHESACB04 : BSc(Hons) Economics and Mathematics with Combined Placement and Study Abroad (Year 4)
Department of Mathematical Sciences
 USMAAFB15 : BSc(Hons) Mathematical Sciences (Year 3)
 USMAAAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
 USMAAKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
 USMAAFB13 : BSc(Hons) Mathematics (Year 3)
 USMAAAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
 USMAAKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
 USMAAFB01 : BSc(Hons) Mathematics and Statistics (Year 3)
 USMAAAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 4)
 USMAAKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 4)
 USMAAFB20 : BSc(Hons) Mathematics, Statistics, and Data Science (Year 3)
 USMAAAB06 : BSc(Hons) Statistics with Study year abroad (Year 4)
 USMAAKB06 : BSc(Hons) Statistics with Year long work placement (Year 4)
 USMAAFM14 : MMath(Hons) Mathematics (Year 3)
 USMAAFM14 : MMath(Hons) Mathematics (Year 4)
 USMAAAM15 : MMath(Hons) Mathematics with Study year abroad (Year 4)
 USMAAKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)
 USMAAKM15 : MMath(Hons) Mathematics with Year long work placement (Year 5)
Department of Physics
 USXXAFB03 : BSc(Hons) Mathematics and Physics (Year 3)
 USXXAAB04 : BSc(Hons) Mathematics and Physics with Study year abroad (Year 4)
 USXXAKB04 : BSc(Hons) Mathematics and Physics with Year long work placement (Year 4)
 USXXAFM01 : MSci(Hons) Mathematics and Physics (Year 3)
 USXXAAM01 : MSci(Hons) Mathematics and Physics with Study year abroad (Year 4)
 USXXAKM01 : MSci(Hons) Mathematics and Physics with Year long work placement (Year 4)

Notes:  This unit catalogue is applicable for the 2023/24 academic year only. Students continuing their studies into 2024/25 and beyond should not assume that this unit will be available in future years in the format displayed here for 2023/24.
 Courses and units are subject to change 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 prerequisite rules.
 Find out more about these and other important University terms and conditions here.
