Applied Differential Geometry - University of Kent

UNIVERSITY OF KENT
SCHOOL OF MATHEMATICS, STATISTICS AND ACTUARIAL
SCIENCE
MODULE SPECIFICATION FOR MA969
1. The Title of the Module: Applied differential geometry (MA969)
2. Department Responsible for Management: SMSAS.
3. The Start Date of the Module: October, 2010
4. Cohort of students (onward) to which the module will be applicable: 2010
5. The number of students expected to take the module: 15
6. Modules to be withdrawn on the introduction of this proposed module and consultation
with other relevant Departments and Faculties regarding the withdrawal: Not
applicable: this is a new program.
7. The Level of the Module : Masters [M]
8. The number of credits which the Module Represents: 15
9. In which Term(s) the Module is to be Taught: Autumn/Spring
10. Prerequisite and co-requisite modules: none
11. The Programme(s) of Study to Which the Module Contributes: This is an optional
module for the MSc in Mathematics and its Applications.
12. The Intended Subject Specific Learning Outcomes and, as Appropriate, Their Relationship
to Programme Learning Outcomes:
On successful completion of this module students will:
(i) understand basic geometric objects such as curves and surfaces and be able to determine
their intrinsic properties (A2, A5, A8)
(ii) be able to derive the geometric evolution equations for curves and surfaces and understand
the connection with nonlinear integrable systems (A1, A4, C3)
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(iii) have broadened their experience with the basic concepts in Riemannian geometry such as
metrics, connections and curvatures(A5, A6)
(iv) have developed awareness of modern applications to mathematical physics, computer vision
and image processing (A1, B2)
13. The Intended Generic Learning Outcomes and, as Appropriate, Their Relationship
to Programme Learning Outcomes:
On successful completion of the module students will
(i) have an enhanced ability to reason and deduce confidently from given definitions and constructions
(A2, B4)
(ii) have an enhanced ability to associate abstract geometrical concepts with their applications
(A5, B4)
(iii) have matured in their problem formulation and solution skills (B1, B2, D4)
(iv) have consolidated their grasp of a wide variety of mathematical skills and methods (A1,
A4, A8, A9, C3)
14. A synopsis of the Curriculum The main aim of this module is to give an introduction to the
basics of differential geometry, keeping in mind the recent applications in the analysis of pattern
recognition, image processing and computer graphics.
(a) Theory of curves: Plane and space curves. Euclidean and affine invariants of plane curves.
(b) Geometry of surfaces: Metrics on regular surface, Curvature of a curve on a surface, Gaussian
curvature and mean curvature, Covariant derivative and geodesics, The Euler-Lagrange
equations, Minimal surfaces.
(c) Evolution of curves and surfaces as integrable systems: Invariant curve evolution. The mean
curvature flows. The connection with integrable systems. The modified Korteweg de-Vries
equation. Soliton solutions and the corresponding curves.
(d) Curves in Riemannian manifolds: Riemannian metrics, connections, curvatures and geodesics.
Curves evolution in Riemannian manifold with constant curvature.
(e) Modern applications include: Moving frames, invariant signatures in pattern recognition,
Poisson manifolds and Hamiltonian systems, and multiple view geometry in computer vision
(the emphasis and the topics may vary from year to year).
15. Indicative Reading List:
Books:
Lectures on differential geometry, I.A. Taimanov, EMS series of lectures in Mathematics, 2008.
Bäcklund and Darboux transformations: Geometry and modern applications in soliton theory,
C. Rogers and W.K. Schief. Cambridge university press, 2002.
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Numerical geometry of images, theory, algorithms and applications, R. Kimmel, Springer Verlag,
2003.
Multiple View geometry in computer vision, R. Hartley and A. Zisserman, Cambridge university
press, 2nd ed. 2003.
Articles:
Lectures on moving frames, P.J. Olver, preprint, University of Minnesota, 2008.
On integrable systems in 3-dimensional Riemannian geometry, G. Marí Beffa, J.A. Sanders, J.P.
Wang. Journal of Nonlinear Science, 12:143–167, 2002.
16. Learning and teaching methods
Number of contact hours: 24-36
Total study hours: 150
Contact hours comprise a mix of lecture and example class.
Student study hours would be distributed between consolidation of lecture material, the working
of exercises on exercise sheets, assessed exercises, using Maple to obtain results and check their
work, independent reading and exam preparation.
17. Assessment methods and how these relate to testing achievement of the intended
learning outcome:
Assessment: The module is assessed on the basis of examination (70%), coursework (30%)
Examination: A 2 hour written examination in Term 3 that consists of multi-part questions
requiring a mix of long and short answers to test at varying levels of proficiency the learning
outcomes (12(i-iv), 13(i-iv)).
Coursework: This consists of open-book assignments of unseen problems, a mid-term class test,
and exercises based on material the students read independently (12(i-iv), 13(i-iv)).
18. Implications for Learning Resources
Staff: Convener and moderator
Library: books as requested
IT and Space: suitable rooms for lectures and classes (max of 3 per week)
19. A statement confirming that, as far as can be reasonably anticipated, the curriculum,
learning and teaching methods and forms of assessment do not present any nonjustifiable
disadvantage to students with disabilities
The expectations of SENDA have been incorporated in the curriculum design and students
having documented individual learning needs are advised by the Dyslexia & Disability Support
Service. Therefore, as far as can be reasonably anticipated, the curriculum, learning and teaching
methods and forms of assessment do not present any non-justifiable disadvantage to students
with disabilities.
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Statement by the Director of Graduate Studies
“I confirm I have been consulted on the above module proposal and have given advice on the
correct procedures and required content of module proposals”
................................................................
Director of Graduate Studies
Print Name
..............................................
Date
Statement by the Head of School
“I confirm that the Department has approved the introduction of the module and, where the module
is proposed by Departmental staff, will be responsible for its resourcing”
.................................................................
Head of School
Print Name
..............................................
Date
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