COURSE UNIT TITLE

: VECTOR BUNDLES AND CHARACTERISTIC CLASSES

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 6034 VECTOR BUNDLES AND CHARACTERISTIC CLASSES 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
Mathematics

Course Objective

The purpose of this course is to give an introduction to the classical Chern-Weil theory of characteristic classes with real coefficients. We define a singular cohomology class which turns out to be a differential topological invariant for a topological vector bundle.

Learning Outcomes of the Course Unit

1   will be able to know Chern-Weil theory
2   will be able to know characteristic classes
3   will be able to vector bundles
4   will be able to compute topological invariants of vector bundles and principle bundles in terms of connections and curvatures
5   will be able to explain the relation between theory of chracteristic classes and cohomology groups of BG

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Differential Forms
2 Cohomology: the de Rham complex and the de Rham cohomology group, star-shaped, Poincarés lemma, singular simplex, cohomology group, homotopy property, excision property, double complexes
3 The simplicial de Rham Complex: introduction, simplicial set, differential form, the simplicial de Rham algebra, cochain complex, Whitney theorem
4 Principal bundles: definition and examples, transition function, cocycle condition
5 Principal bundles: an extension of a bundle, a reduction of a bundle
6 Connections: definition, examples and some properties, curvature, distribution
7 Midterm
8 The Chern-Weil homomorphism: introduction, invariant polynomials, cohomology class
9 Definition of the Chern-Weil homomorphism, Complex Chern-Weil homomorphism
10 Topological bundles and Classifying space: characteristic class, the classifying space
11 Simplicial space, Fat and thin realizations
12 Simplicial manifold, Simplicial form, Simplicial bundle, Chern-Weil homomorphism for BG
13 Chern classes, Complex line bundle, Canonical line bundle
14 Pontrjagin class, the Euler class

Recomended or Required Reading

Textbook:
1. J. L. Dupont, Curvature and Characteristic Classes, Springer, 1978, ISBN-13: 978-3540086635
Supplementary Book:
2. John Milnor and James D. Stasheff, Characteristic Classes, Princeton University Press, 1974, ISBN-13: 978-0691081229
3. Shigeyuki Morita, Geometry of Characteristic Classes, AMS, 2001, ISBN-13: 978-0821821398
References:
4. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry Vol. I, Wiley-Interscience, 1996, ISBN-13: 978-0471157335
5. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry Vol. II, Wiley-Interscience, 1996, ISBN-13: 978-0471157328
6. Frank W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer, 1983, ISBN-13: 978-0387908946
7. Saunders MacLane, Homology, Springer, 1995, ISBN-13: 978-3540586623

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


*** Resit Exam is Not Administered in Institutions Where Resit is not Applicable.

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)
Lectures 13 3 39
Tutorials 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
Preparation for quiz etc. 0
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.1333533333
LO.2333533333
LO.3333533333
LO.4343534343
LO.5343534343

CONTACT INFORMATION
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