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.

x

3 Follows contemporary scientific developments, analyses, synthesizes and evaluates novel ideas.

x

4 Uses the national and international academic sources, and computer and related IT.

x

5

Participates in workgroups and research groups, scientific meetings, contacts by oral and written communication at national and international levels.

x

6

Acquires the potential of creative and critical thinking, problem solving, research, to produce a novel and original work, self-development in areas

of interest. x

7

Acquires the consciousness of scientific ethics and responsibility. Takes responsibility about the solution of professional problems as a requirement

of the intellectual consciousness. x

ECTS ALLOCATED BASED ON STUDENT WORKLOAD BY THE COURSE DESCRIPTION

Activities Quantity Duration

(Hour)

Total

Workload

(Hour)

Course Duration (14x Total course hours) 14 3 42

Hours for off-the-classroom study (Pre-study, practice) 14 10 140

Mid-terms (Including self study) 2 20 40

Quizzes

Assignments

Final examination (Including self study) 1 30 30

Total Work Load

252

Total Work Load / 25 (h) 10,08

ECTS Credit of the Course 10