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MA30231: Projective geometry

Follow this link for further information on academic years Academic Year: 2019/0
Further information on owning departmentsOwning Department/School: Department of Mathematical Sciences
Further information on credits Credits: 6      [equivalent to 12 CATS credits]
Further information on notional study hours Notional Study Hours: 120
Further information on unit levels Level: Honours (FHEQ level 6)
Further information on teaching periods Period:
Semester 2
Further information on unit assessment Assessment Summary: EX 100%
Further information on unit assessment Assessment Detail:
  • Examination (EX 100%)
Further information on supplementary assessment Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Further information on requisites Requisites: Before taking this module you must take MA20216
Further information on descriptions 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.
Further information on programme availabilityProgramme 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 3)
  • 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 3)
  • USCM-AKM14 : MComp(Hons) Computer Science and Mathematics with Year long work placement (Year 5)
Department of Economics
  • UHES-AFB04 : BSc(Hons) Economics and Mathematics (Year 3)
  • UHES-AAB04 : BSc(Hons) Economics and Mathematics with Study year abroad (Year 4)
  • UHES-AKB04 : BSc(Hons) Economics and Mathematics with Year long work placement (Year 4)
  • UHES-ACB04 : BSc(Hons) Economics and Mathematics with Combined Placement and Study Abroad (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-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)
  • 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)
  • 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)
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)

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