COURSE UNIT TITLE

: CURVATURE AND HOMOLOGY

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 6050 CURVATURE AND HOMOLOGY 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

This course aims to provide an introduction to theory of curvatures and homologies, analyses the appraisal of particular geometric theories; investigates the concepts, theories and basic principles of reasoning which are part and parcel of the discipline of mathematics. The course discusses the relationship between geometry and homology.

Learning Outcomes of the Course Unit

1   will be able to know how curvatures and homologies bear on each other
2   will be able to know Laplace-Beltrami operator and its consequences
3   will be able to write Laplace-Beltrami operator in local coordinates
4   will be able to know classical properties of compact Riemann surfaces
5   will be able to know Ricci curvature

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 A review of differential geometry: Riemannian surfaces
2 The topology of Riemannian manifolds and their metric geometry
3 Elementary differential geometry
4 Coordinate-free global differential geometry of Riemmannian manifolds
5 A review of singular homology and Rham cohomology: Rham theorem. Harmonic forms
6 Laplace-Beltrami operator
7 An explicit expression for the laplace-beltrami operator in local coordinates
8 Midterm
9 The particular quadratic form involving the curvature tensor
10 Betti numbers. Ricci curvature
11 Compact Lie groups, The harmonic forms on compact lie groups, Differential forms that are invariant under both left and right translations of the lie group
12 Complex structures on separable Hausdorff spaces, Riemannian metric on complex structures, Kaehler metric and Kaehler manifold, Hermitian geometry, Holomorphic p-form
13 Studies in detail how curvature and homology are related for the case of kaehler manifolds, Kodaira vanishing theorems, Classical results concerning compact Riemannian surfaces
14 Conformal transformation on Riemannian manifolds, The solution of a system of differential equations which involve Ricci curvature, The groups of holomorphic transformations, Almost Kaehler manifolds

Recomended or Required Reading

Textbook:
1. Moussa Balde, On one parameter Families of Dido Riemannian Problems, Dover Publications, 1998
Supplementary Book:
2. Samuel I Goldberg, Curvature and Homology, Dover publications, 2011, ISBN-13: 978-0486402079
References:
Materials:

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)

email: 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
Preparations before/after weekly lectures 13 4 52
Preparation for midterm exam 1 23 23
Preparation for final exam 1 30 30
Preparing assignments 10 5 50
Midterm 1 3 3
Final 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.14445434343
LO.23334444343
LO.33334444343
LO.444344333
LO.533354433

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

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