MAT 3015 Differential Geometry I,
MAT 3016 Differential Geometry II
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Recommended optional programme components:
None
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Language:
Turkish
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Mode of Delivery:
Face to face
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Course Coordinator:
Prof. Dr. Kadri Arslan
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Course Lecturers:
Doç. Dr. Betül BULCA
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Contactinformation of the Course Coordinator:
arslan@uludag.edu.tr (0 224) 294 17 75 Bursa Uludağ Üniversitesi, Fen-Edebiyat Fakültesi, Matematik Bölümü
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Website:
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Objective of the Course:
The aim of the course is to introduce the basic concepts of singularity theory to the student at the graduate level. Defining the concept of submanifold to the student, and to compute the singularities in surfaces and hypersurfaces. In addition, by giving the definition of contact between submanifolds, it is also to contribute to the solutions of the basic problems related to the contacts between hypersurfaces and the hyperplane and hypercapsphere. It is also to examine the applications on surfaces by giving a classification of singularities. Defining height and distance functions on submanifolds and examining their effects on surfaces and hyper surfaces.
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Contribution of the Course to Professional Development
It contributes to give geometric approaches for the classification of singularities with the help of the concept of singularity.
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Learning Outcomes:
1
He/She defines surfaces in R ^ n.;
2
He/She can establish the orthonormal frame of the surfaces in R ^ 4.;
3
He/ She can calculates the mean curvature of the surfaces in R ^ 5.;
4
He/ She can calculate the singularities of curves.;
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He/ She can define the contact between hypersurfaces and hyperspheres.;
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He can classify singularities.;
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He/She can obtain the classification of critical points. ;
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He/she will have ability to build a family of functions on hypersurfaces in R ^ 4.;
9
He/she can determine the asymptotic directions on the surfaces. ;
10
He/She can give a classification of critical points on the surfaces. ;
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Course Content:
Week
Theoretical
Practical
1
Singularity theory for curves
2
Surfaces in R^n
3
Smooth mappings
4
Quadratic forms
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Surfaces in R^4
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Surfaces in R^5
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Submanifolds in Euclidean spaces
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Contact between submanifolds
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Contact of hypersurfaces with hyperplanes
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Contact of hypersurfaces with hyperspheres
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Family of functions on hypersurfaces in R^n
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Family of height functions
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Classification of singularities
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Classification of Critical points
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Textbooks, References and/or Other Materials:
1) Shyuichi Izumiya et al. - Differential Geometry from Singularity Theory Viewpoint (2015, World Scientific) - libgen.lc 2) J. W. Bruce, P. Giblin - Curves and Singularities_ A Geometrical Introduction to Singularity Theory (1985) - libgen.lc 3) [Encyclopedia of Mathematical Sciences 6] V. I. Arnold, V. V. Goryunov, O. V. Lyashko, V. A. Vasil’ev (auth.) - Singularity Theory I (1998, Springer-Verlag Berlin He
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Assesment
TERM LEARNING ACTIVITIES
NUMBER
PERCENT
Midterm Exam
0
0
Quiz
0
0
Homeworks, Performances
2
50
Final Exam
1
50
Total
3
100
Contribution of Term (Year) Learning Activities to Success Grade
50
Contribution of Final Exam to Success Grade
50
Total
100
Measurement and Evaluation Techniques Used in the Course
The system of relative evaluation is applied.
Information
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ECTS / WORK LOAD TABLE
Activites
NUMBER
TIME [Hour]
Total WorkLoad [Hour]
Theoretical
14
3
42
Practicals/Labs
0
0
0
Self Study and Preparation
14
6
84
Homeworks, Performances
2
12
24
Projects
0
0
0
Field Studies
0
0
0
Midtermexams
0
0
0
Others
0
0
0
Final Exams
1
23
23
Total WorkLoad
173
Total workload/ 30 hr
5,77
ECTS Credit of the Course
6
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CONTRIBUTION OF LEARNING OUTCOMES TO PROGRAMME QUALIFICATIONS
