COURSE INFORMATION Course Title Code Semester L+P Hour Credits ECTS GEOMETRY OF MANIFOLDS OF MAPS Math 613 1-2 3 + 0 3 10 Prerequisites Language of Instruction English Course Level Graduate Course Type Course Coordinator Instructors Prof. Hasan Gümral Assistants Goals Develope insight for structure of manifolds of maps and, computational tools for applications. Content Space of maps between finite dimensional manifolds. Realization as space of sections of a trivial bundle. Vector fields and forms, tangent and cotangent bundles. Metric and symplectic structure. Actions of Lie groups. Applications from continuum theories. Learning Outcomes Teaching Methods Assessment Methods 1) Induces basic structures on space of maps by means of those of a trivial bundle 1,2 A,B,C 2) Constructs geometric objects on manifolds of maps 1,2 A,B,C 3) Endows manifolds of maps with additional geometric structures 1,2 A,B,C 4) Computes actions of finite and infinite dimensional groups on finite dimensional manifolds 1,2 A,B,C 5) Finds actions of Lie groups on space of maps, their tangent and cotagent bundles, computes momentum maps 1,2 A,B,C 6) Applies techniques to fluid and plasma theories 1,2 A,B,C Teaching Methods: 1: Lecture, 2:Problem solving Assessment Methods: A: Written Examination, B: Homework, C: Oral examination COURSE CONTENT Week Topics Study Materials 1 Manifolds, vector bundles, sections, 1 2 Space of maps between finite dimensional manifolds 1 3 Realization as space of sections of a trivial bundle 1,5,7
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COURSE INFORMATION
Course Title Code Semester L+P
Hour Credits ECTS
GEOMETRY OF MANIFOLDS OF MAPS Math 613 1-2 3 + 0 3 10
Prerequisites
Language of
Instruction English
Course Level Graduate
Course Type
Course Coordinator
Instructors Prof. Hasan Gümral
Assistants
Goals Develope insight for structure of manifolds of maps and, computational
tools for applications.
Content
Space of maps between finite dimensional manifolds. Realization as space of sections of a trivial bundle. Vector fields and forms, tangent and
cotangent bundles. Metric and symplectic structure. Actions of Lie groups. Applications from continuum theories.
Learning Outcomes Teaching
Methods
Assessment
Methods
1) Induces basic structures on space of maps by means of
those of a trivial bundle 1,2 A,B,C
2) Constructs geometric objects on manifolds of maps 1,2 A,B,C
3) Endows manifolds of maps with additional geometric
structures 1,2 A,B,C
4) Computes actions of finite and infinite dimensional groups
on finite dimensional manifolds 1,2 A,B,C
5) Finds actions of Lie groups on space of maps, their
tangent and cotagent bundles, computes momentum maps 1,2 A,B,C
6) Applies techniques to fluid and plasma theories 1,2 A,B,C
Teaching
Methods: 1: Lecture, 2:Problem solving
Assessment
Methods: A: Written Examination, B: Homework, C: Oral examination
COURSE CONTENT
Week Topics Study
Materials
1 Manifolds, vector bundles, sections, 1
2 Space of maps between finite dimensional manifolds 1
3 Realization as space of sections of a trivial bundle 1,5,7
4 Jets and Whitney topologies 1,2
5 Vector fields and forms, tangent and cotangent bundles 1,5
6 Metric and symplectic structure 1,8
7 Actions of Lie groups 1,8
8 Diffeomorphism groups and their algebras 1,3,7
9 Diffeomorphisms on circle and KdV equation 1
10 Volume preserving diffeomorphisms and incompressible fluids 8
11 Group of canonical diffeomorphisms and plasma dynamics 5,6,7
12 Space of displacement fields and relation to canonical diffeomorphisms 9
13 Actions of diffeomorphism groups to space of displacement mappings 9
14 Further discussions
RECOMMENDED SOURCES
Textbook
1. A Kriegl and P W Michor, The Convenient Setting of Global Analysis, AMS 1997 2. P. Michor, Manifolds of smooth maps, Cahiers Top. Geo. Diff., 19 (1978), 47--78.
Additional Resources
3. T. S. Ratiu and R. Schmid, The differentiable structure of three remarkable diffeomorphism groups, Mathematische Zeitschrift, 177
(1981), 81--100. 4. T. Swift, A note on the space of lagrangian submanifolds of a symplectic 4-manifold, Journal of Geometry and Physics, 35 (2000), 183--192. 5. H. Gümral, Geometry of Plasma Dynamics I: Group of Canonical
Diffeomorphisms, J. Math. Phys. 51 (2010) 083501 (23pp). 6. O. Esen, H. Gümral, Lifts, Jets and Reduced Dynamics, Int. J. of Geom.
Meth. in Mod. Phys. 8 (2011) 331-344. 7. O. Esen, H. Gümral, Geometry of Plasma Dynamics II: Lie Algebra of Hamiltonian Vector Fields, J. Geom. Mech. 4 (2012) 239-269. 8. D. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92 (1970) 102-163 9. H.Gümral, Geometry of Plasma Dynamics IV: Space of Displacement
Mappings, work in progress
MATERIAL SHARING
Documents
Assignments
Exams
ASSESSMENT
IN-TERM STUDIES NUMBER PERCENTAGE
Mid-terms 2 100
Quizzes
Assignments
Total 100
CONTRIBUTION OF FINAL EXAMINATION TO OVERALL
GRADE 40
CONTRIBUTION OF IN-TERM STUDIES TO OVERALL
GRADE 60
Total 100
COURSE CATEGORY
COURSE'S CONTRIBUTION TO PROGRAM
No Program Learning Outcomes Contribution
1 2 3 4 5
1 Acquires a rigorous background about the fundamental fields in mathematics and the topics that are going to be specialized.
x
2 Acquires the ability to relate, interpret, analyse and synthesize on fundamental fields in mathematics and/or mathematics and other sciences.