COURSE UNIT TITLE

: DIFFERENTIAL TOPOLOGY

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 6039 DIFFERENTIAL TOPOLOGY ELECTIVE 3 0 0 8

Offered By

Graduate School of Natural and Applied Sciences

Level of Course Unit

Second Cycle Programmes (Master's Degree)

Course Coordinator

Offered to

Mathematics (English)
Mathematics (English)

Course Objective

This course aims to describe the geometric methods of differential topology, and relate properties of topological manifolds with properties of differentiable manifolds.

Learning Outcomes of the Course Unit

1   will be able to define smooth manifolds as certain subspaces of euclidean spaces
2   will be able to define smooth maps between manifolds and give the implicit function theorem
3   will be able to study regular and singular values of smooth maps
4   will be able to define the degree of a smooth map and give standard topological applications
5   will be able to define and study the Euler characteristic of a compact orientable manifold, including the classification of compact oriented surfaces

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Manifolds, Submanifolds, Differential structures
2 Differentiable maps and the tangent bundle, Tangent space
3 Embeddings and Immersions, Manifolds with boundary
4 The weak and strong topologies on smooth function spaces , Approximations
5 Aproximations on manifolds wth boundary and manifold pairs, Jets and the Baire property, Analytic approximations
6 The Morse-Sard theorem, Transversality
7 Midterm
8 Vector bundles, Constructions with vector bundles, The classification of vector bundles
9 Oriented Vector bundles, Tubular neighborhoods
10 Degrees of maps, Intersection number and the Euler characteristic
11 Morse functions, Differential equations and regular level surfaces, Passing critical levels and attaching cells, CW-complexes
12 Cobordism and Transversality, The Thom homomorphism
13 Extending isotopies, Gluing manifolds together, Isotopies of disks
14 Models of surfaces, Characterization of the disk, The classification of compact surfaces

Recomended or Required Reading

Textbooks:
1. Morris W. Hirsch, Differential Topology, Springer, 1976, ISBN-13: 978-0387901480
Supplementary Books:
2. Victor Guillemin and Alan Pollack, Differential Topology, Prentice Hall, 1974, ISBN-13: 978-0132126052
3. T. Bröcker, K. Jänich, C. B. Thomas and M. J. Thomas, Introduction to Differential Topology, Cambridge University Press, 1982, ISBN-13: 978-0521284707
4. John Willard Milnor, Topology from the Differentiable Viewpoint, Princeton University Press, 1997, ISBN-13: 978-0691048338

Planned Learning Activities and Teaching Methods

Lecture notes, Presentation, Problem solving

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA
1 ASG ASSIGNMENT
2 MTE MIDTERM EXAM
3 FIN FINAL EXAM
4 FCG FINAL COURSE GRADE ASG * 0.30 + MTE * 0.30 + FIN * 0.40
5 RST RESIT
6 FCGR FINAL COURSE GRADE (RESIT) ASG * 0.30 + MTE * 0.30 + RST * 0.40


Further Notes About Assessment Methods

None

Assessment Criteria

To be announced.

Language of Instruction

English

Course Policies and Rules

Attending at least 70 percent of lectures is mandatory.

Contact Details for the Lecturer(s)

bedia.akyar@deu.edu.tr

Office Hours

To be announced.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours)
Tutorials 0
Lectures 13 3 39
Preparation for quiz etc. 0
Preparing presentations 0
Preparations before/after weekly lectures 13 4 52
Preparation for midterm exam 1 23 23
Preparation for final exam 1 30 30
Preparing assignments 5 5 25
Preparing Group Assignments 5 5 25
Quiz etc. 0
Final 1 3 3
Midterm 1 3 3
TOTAL WORKLOAD (hours) 200

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11
LO.1334444443
LO.2334444443
LO.33344433443
LO.4333443443
LO.5333443443

CONTACT INFORMATION
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Phone: +90(232) 412 12 12 - Fax: +90 (232) 464 81 35

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