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Course Description Course Name : Introduction to Differential Geometry Course Code : MT-521 Course Type : Optional Level of Course : Second Cycle Year of Study : 1 Course Semester : Fall (16 Weeks) ECTS : 6 Name of Lecturer(s) : Assoc.Prof.Dr. ALİ ARSLAN ÖZKURT Learning Outcomes of the Course : Knows the concept of differentiable manifold and gives some examples Does some analysis on differentiable manifolds Knows the concept of submanifold Knows the concept of a Lie group Can investigate actions of a Lie group on a differentiable manifold Knows the concept of the vector field on a differentiable manifold. Mode of Delivery : Face-to-Face Prerequisites and Co-Prerequisites : YLMT-201 Special Area Course Recommended Optional Programme Components : None Aim(s) of Course : to teach the concept of differentiable manifolds and submanifolds. To teach the concept of vector field on a diferentiable manifold and Lie groups Course Contents : Differentiability for functions of several variables, vector fields on open subsets of an Euclidean space, the rank of a mapping, Definition of differentiable manifolds and examples, Rank of a differentiable function defined on a differentiable manifold, immersions and submersions, submanifolds, Lie groups and vector fields Language of Instruction : Turkish Work Place : classroom   Course Outline /Schedule (Weekly) Planned Learning Activities Week Subject Student's Preliminary Work Learning Activities and Teaching Methods 1 Topological manifolds Read the relevant sections in the textbook and solve problems Lecture and Discussion 2 Analysis on the functions of several variables Read the relevant sections in the textbook and solve problems Lecture and Discussion 3 Differentiable manifolds Read the relevant sections in the textbook and solve problems Lecture and Discussion 4 Differentiable manifolds Read the relevant sections in the textbook and solve problems Lecture and Discussion 5 Differentiable functions Read the relevant sections in the textbook and solve problems Lecture and Discussion 6 The rank of a mapping Read the relevant sections in the textbook and solve problems Lecture and Discussion 7 Submanifolds of a differentiable manifold Read the relevant sections in the textbook and solve problems Lecture and Discussion 8 Submanifolds of a differentiable manifold Read the relevant sections in the textbook and solve problems Lecture and Discussion 9 Lie groups Read the relevant sections in the textbook and solve problems Lecture and Discussion 10 Action of a Lie group on a manifold Read the relevant sections in the textbook and solve problems Lecture and Discussion 11 Action of a Lie group on a manifold Read the relevant sections in the textbook and solve problems Lecture and Discussion 12 The tangent space at a point of a manifold and tangent bundle Read the relevant sections in the textbook and solve problems Lecture and Discussion 13 Vector fields Read the relevant sections in the textbook and solve problems Lecture and Discussion 14 One parameter groups acting on a manifold Read the relevant sections in the textbook and solve problems Lecture and Discussion 15 The existence theorem for ordinary differential equations Read the relevant sections in the textbook and solve problems Lecture and Discussion 16/17 evaluations and solutions of assignments Review the topics discussed in the lecture notes and source Lecture and Discussion   Required Course Resources Resource Type Resource Name Recommended Course Material(s)  An Introduction to Differentiable Manifolds and Riemannian Geometry, William M. Boothby Required Course Material(s)   Assessment Methods and Assessment Criteria Semester/Year Assessments Number Contribution Percentage     Mid-term Exams (Written, Oral, etc.) 0 0     Homeworks/Projects/Others 1 100 Total 100 Rate of Semester/Year Assessments to Success 40   Final Assessments 100 Rate of Final Assessments to Success 60 Total 100   Contribution of the Course to Key Learning Outcomes # Key Learning Outcome Contribution* 1 Aquires sufficient knowledge to enable one to do research over and above the undergraduate level 4 2 Learns theoretical foundations of his/her field thoroughly 4 3 Uses the knowledge in his/her field to solve mathematical problems 4 4 Proves basic theorems in different areas of Mathematics 2 5 Creates a model for the problems in his/her field and expresses the relationship between objects in a simple and comprehensive way. 3 6 Uses technical tools in his/her field 3 7 Works independently in his/her field requiring expertise 4 8 Works with other colleagues collaboratelly, takes responsibilities if necessary during the research process 4 9 Argues and analyzes knowledge in his/her field and applies them in other fields if necessary 3 10 Has sufficient knowledge to follow literature in his/her field and communicate with stakeholders 3 11 Uses the language and technology to develop knowledge in his/her field and shares these with stakeholders systematically when necessary 1 12 Knows and abides by the ethical rules in analyzing, solving problems and publishing results 5 * Contribution levels are between 0 (not) and 5 (maximum).   Student Workload - ECTS Works Number Time (Hour) Total Workload (Hour) Course Related Works     Class Time (Exam weeks are excluded) 14 4 56     Out of Class Study (Preliminary Work, Practice) 14 5 70 Assesment Related Works     Homeworks, Projects, Others 1 20 20     Mid-term Exams (Written, Oral, etc.) 0 0 0     Final Exam 0 0 0 Total Workload: 146 Total Workload / 25 (h): 5.84 ECTS Credit: 6