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MA30231: Projective geometry

Follow this link for further information on academic years Academic Year: 2013/4
Further information on owning departmentsOwning Department/School: Department of Mathematical Sciences
Further information on credits Credits: 6
Further information on unit levels Level: Honours (FHEQ level 6)
Further information on teaching periods Period: Semester 2
Further information on unit assessment Assessment: EX 100%
Further information on supplementary assessment Supplementary Assessment: MA30231 Mandatory Extra Work (where allowed by programme regulations)
Further information on requisites Requisites: Before taking this unit 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-AFB01 : BSc Computing (Full-time) - Year 3
  • USCM-AKB01 : BSc Computing (Full-time with Thick Sandwich Placement) - Year 4
  • USCM-AFB20 : BSc (hons) Computer Science and Mathematics (Full-time) - Year 3
  • USCM-AKB20 : BSc (hons) Computer Science and Mathematics (Full-time with Thick Sandwich Placement) - Year 4
  • USCM-AAB20 : BSc (hons) Computer Science and Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
  • USCM-AFB13 : BSc (hons) Computer Science with Mathematics (Full-time) - Year 3
  • USCM-AKB14 : BSc (hons) Computer Science with Mathematics (Full-time with Thick Sandwich Placement) - Year 4
  • USCM-AAB14 : BSc (hons) Computer Science with Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
  • USCM-AFM14 : MComp (hons) Computer Science and Mathematics (Full-time) - Year 3
  • USCM-AFM14 : MComp (hons) Computer Science and Mathematics (Full-time) - Year 4
  • USCM-AKM14 : MComp (hons) Computer Science and Mathematics with Industrial Placement (Full-time with Thick Sandwich Placement) - Year 4
  • USCM-AKM14 : MComp (hons) Computer Science and Mathematics with Industrial Placement (Full-time with Thick Sandwich Placement) - Year 5
  • USCM-AAM14 : MComp (hons) Computer Science and Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
  • USCM-AAM14 : MComp (hons) Computer Science and Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 5
Department of Mathematical Sciences
  • USMA-AFB15 : BSc (hons) Mathematical Sciences (Full-time) - Year 3
  • USMA-AKB16 : BSc (hons) Mathematical Sciences (Full-time with Thick Sandwich Placement) - Year 4
  • USMA-AAB16 : BSc (hons) Mathematical Sciences with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
  • USMA-AFB13 : BSc (hons) Mathematics (Full-time) - Year 3
  • USMA-AKB14 : BSc (hons) Mathematics (Full-time with Thick Sandwich Placement) - Year 4
  • USMA-AFB01 : BSc (hons) Mathematics and Statistics (Full-time) - Year 3
  • USMA-AKB02 : BSc (hons) Mathematics and Statistics (Full-time with Thick Sandwich Placement) - Year 4
  • USMA-AAB02 : BSc (hons) Mathematics and Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
  • USMA-AAB14 : BSc (hons) Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
  • USMA-AFB05 : BSc (hons) Statistics (Full-time) - Year 3
  • USMA-AKB06 : BSc (hons) Statistics (Full-time with Thick Sandwich Placement) - Year 4
  • USMA-AAB06 : BSc (hons) Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
  • USMA-AFM14 : MMath Mathematics (Full-time) - Year 3
  • USMA-AFM14 : MMath Mathematics (Full-time) - Year 4
  • USMA-AAM15 : MMath Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
  • TSMA-AFM09 : MSc Mathematical Sciences (Full-time)
  • TSMA-APM09 : MSc Mathematical Sciences (Part-time)
Department of Physics
  • USXX-AFB03 : BSc (hons) Mathematics and Physics (Full-time) - Year 3
  • USXX-AKB04 : BSc (hons) Mathematics and Physics with Placement (Full-time with Thick Sandwich Placement) - Year 4
  • USXX-AAB04 : BSc (hons) Mathematics and Physics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
  • USXX-AFM01 : MSci (hons) Mathematics and Physics (Full-time) - Year 3

Notes:
* This unit catalogue is applicable for the 2013/4 academic year only. Students continuing their studies into 2014/15 and beyond should not assume that this unit will be available in future years in the format displayed here for 2013/14.
* 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.