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Applied Differential Geometry - University of Kent

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UNIVERSITY OF KENT<br />

SCHOOL OF MATHEMATICS, STATISTICS AND ACTUARIAL<br />

SCIENCE<br />

MODULE SPECIFICATION FOR MA969<br />

1. The Title <strong>of</strong> the Module: <strong>Applied</strong> differential geometry (MA969)<br />

2. Department Responsible for Management: SMSAS.<br />

3. The Start Date <strong>of</strong> the Module: October, 2010<br />

4. Cohort <strong>of</strong> students (onward) to which the module will be applicable: 2010<br />

5. The number <strong>of</strong> students expected to take the module: 15<br />

6. Modules to be withdrawn on the introduction <strong>of</strong> this proposed module and consultation<br />

with other relevant Departments and Faculties regarding the withdrawal: Not<br />

applicable: this is a new program.<br />

7. The Level <strong>of</strong> the Module : Masters [M]<br />

8. The number <strong>of</strong> credits which the Module Represents: 15<br />

9. In which Term(s) the Module is to be Taught: Autumn/Spring<br />

10. Prerequisite and co-requisite modules: none<br />

11. The Programme(s) <strong>of</strong> Study to Which the Module Contributes: This is an optional<br />

module for the MSc in Mathematics and its Applications.<br />

12. The Intended Subject Specific Learning Outcomes and, as Appropriate, Their Relationship<br />

to Programme Learning Outcomes:<br />

On successful completion <strong>of</strong> this module students will:<br />

(i) understand basic geometric objects such as curves and surfaces and be able to determine<br />

their intrinsic properties (A2, A5, A8)<br />

(ii) be able to derive the geometric evolution equations for curves and surfaces and understand<br />

the connection with nonlinear integrable systems (A1, A4, C3)<br />

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(iii) have broadened their experience with the basic concepts in Riemannian geometry such as<br />

metrics, connections and curvatures(A5, A6)<br />

(iv) have developed awareness <strong>of</strong> modern applications to mathematical physics, computer vision<br />

and image processing (A1, B2)<br />

13. The Intended Generic Learning Outcomes and, as Appropriate, Their Relationship<br />

to Programme Learning Outcomes:<br />

On successful completion <strong>of</strong> the module students will<br />

(i) have an enhanced ability to reason and deduce confidently from given definitions and constructions<br />

(A2, B4)<br />

(ii) have an enhanced ability to associate abstract geometrical concepts with their applications<br />

(A5, B4)<br />

(iii) have matured in their problem formulation and solution skills (B1, B2, D4)<br />

(iv) have consolidated their grasp <strong>of</strong> a wide variety <strong>of</strong> mathematical skills and methods (A1,<br />

A4, A8, A9, C3)<br />

14. A synopsis <strong>of</strong> the Curriculum The main aim <strong>of</strong> this module is to give an introduction to the<br />

basics <strong>of</strong> differential geometry, keeping in mind the recent applications in the analysis <strong>of</strong> pattern<br />

recognition, image processing and computer graphics.<br />

(a) Theory <strong>of</strong> curves: Plane and space curves. Euclidean and affine invariants <strong>of</strong> plane curves.<br />

(b) <strong>Geometry</strong> <strong>of</strong> surfaces: Metrics on regular surface, Curvature <strong>of</strong> a curve on a surface, Gaussian<br />

curvature and mean curvature, Covariant derivative and geodesics, The Euler-Lagrange<br />

equations, Minimal surfaces.<br />

(c) Evolution <strong>of</strong> curves and surfaces as integrable systems: Invariant curve evolution. The mean<br />

curvature flows. The connection with integrable systems. The modified Korteweg de-Vries<br />

equation. Soliton solutions and the corresponding curves.<br />

(d) Curves in Riemannian manifolds: Riemannian metrics, connections, curvatures and geodesics.<br />

Curves evolution in Riemannian manifold with constant curvature.<br />

(e) Modern applications include: Moving frames, invariant signatures in pattern recognition,<br />

Poisson manifolds and Hamiltonian systems, and multiple view geometry in computer vision<br />

(the emphasis and the topics may vary from year to year).<br />

15. Indicative Reading List:<br />

Books:<br />

Lectures on differential geometry, I.A. Taimanov, EMS series <strong>of</strong> lectures in Mathematics, 2008.<br />

Bäcklund and Darboux transformations: <strong>Geometry</strong> and modern applications in soliton theory,<br />

C. Rogers and W.K. Schief. Cambridge university press, 2002.<br />

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Numerical geometry <strong>of</strong> images, theory, algorithms and applications, R. Kimmel, Springer Verlag,<br />

2003.<br />

Multiple View geometry in computer vision, R. Hartley and A. Zisserman, Cambridge university<br />

press, 2nd ed. 2003.<br />

Articles:<br />

Lectures on moving frames, P.J. Olver, preprint, <strong>University</strong> <strong>of</strong> Minnesota, 2008.<br />

On integrable systems in 3-dimensional Riemannian geometry, G. Marí Beffa, J.A. Sanders, J.P.<br />

Wang. Journal <strong>of</strong> Nonlinear Science, 12:143–167, 2002.<br />

16. Learning and teaching methods<br />

Number <strong>of</strong> contact hours: 24-36<br />

Total study hours: 150<br />

Contact hours comprise a mix <strong>of</strong> lecture and example class.<br />

Student study hours would be distributed between consolidation <strong>of</strong> lecture material, the working<br />

<strong>of</strong> exercises on exercise sheets, assessed exercises, using Maple to obtain results and check their<br />

work, independent reading and exam preparation.<br />

17. Assessment methods and how these relate to testing achievement <strong>of</strong> the intended<br />

learning outcome:<br />

Assessment: The module is assessed on the basis <strong>of</strong> examination (70%), coursework (30%)<br />

Examination: A 2 hour written examination in Term 3 that consists <strong>of</strong> multi-part questions<br />

requiring a mix <strong>of</strong> long and short answers to test at varying levels <strong>of</strong> pr<strong>of</strong>iciency the learning<br />

outcomes (12(i-iv), 13(i-iv)).<br />

Coursework: This consists <strong>of</strong> open-book assignments <strong>of</strong> unseen problems, a mid-term class test,<br />

and exercises based on material the students read independently (12(i-iv), 13(i-iv)).<br />

18. Implications for Learning Resources<br />

Staff: Convener and moderator<br />

Library: books as requested<br />

IT and Space: suitable rooms for lectures and classes (max <strong>of</strong> 3 per week)<br />

19. A statement confirming that, as far as can be reasonably anticipated, the curriculum,<br />

learning and teaching methods and forms <strong>of</strong> assessment do not present any nonjustifiable<br />

disadvantage to students with disabilities<br />

The expectations <strong>of</strong> SENDA have been incorporated in the curriculum design and students<br />

having documented individual learning needs are advised by the Dyslexia & Disability Support<br />

Service. Therefore, as far as can be reasonably anticipated, the curriculum, learning and teaching<br />

methods and forms <strong>of</strong> assessment do not present any non-justifiable disadvantage to students<br />

with disabilities.<br />

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Statement by the Director <strong>of</strong> Graduate Studies<br />

“I confirm I have been consulted on the above module proposal and have given advice on the<br />

correct procedures and required content <strong>of</strong> module proposals”<br />

................................................................<br />

Director <strong>of</strong> Graduate Studies<br />

Print Name<br />

..............................................<br />

Date<br />

Statement by the Head <strong>of</strong> School<br />

“I confirm that the Department has approved the introduction <strong>of</strong> the module and, where the module<br />

is proposed by Departmental staff, will be responsible for its resourcing”<br />

.................................................................<br />

Head <strong>of</strong> School<br />

Print Name<br />

..............................................<br />

Date<br />

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