| EUROPEAN UNION RELATIONS | |||||
| Bachelor | TR-NQF-HE: Level 6 | QF-EHEA: First Cycle | EQF-LLL: Level 6 | ||
| Course Code | Course Name | Semester | Theoretical | Practical | Credit | ECTS |
| MAT4053 | Differentiable Manifolds | Spring | 3 | 0 | 3 | 6 |
| This catalog is for information purposes. Course status is determined by the relevant department at the beginning of semester. |
| Language of instruction: | English |
| Type of course: | Non-Departmental Elective |
| Course Level: | Bachelor’s Degree (First Cycle) |
| Mode of Delivery: | Face to face |
| Course Coordinator : | |
| Recommended Optional Program Components: | None |
| Course Objectives: | The differentiable manifolds course aims to give the fundamental knowledge for the studies of graduate students who intends to study at geometry. |
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The students who have succeeded in this course; upon succeeding this course 1)be able to test a differentiable structure given on a set 2)be able to give examples of Differentiable structures on a set 3) be able to check differentiablity of a function 4) be able to solve problems involving the derived map of a transformation between two manifolds 5) be able to use the properties of induced topology on a manifold, 6) be able to coordinatize Grassmann manifolds and can evaluate their dimensions, 7) be able to understand the existence problems by using the unity of partition 8)be able to explain the derived function of a function by using the Leibniz rule, 9) be able to explain submanifolds as images under Immersions 10) be able to coordinatize quotient manifolds and calculate their dimensions, 11) be able to construct Klein bottle and Mobius strip as an example of a quotient manifold |
| Differentiable (diff.able) functions, Atlas, diff.able structures on a set, Examples of diff.able structures, diff.able manifolds, diff.able functions, The induced topology on a manifold, diff.able varieties, Grassmann manifolds, Manifold structure on a topological space, properties of the induced topology, Topological restrictions on a manifold, Partitions of unity, Partial differentiation, tangent vectors, The invers function Theorem, Leibniz's rule. İmmersions, submanifolds, regular submanifolds, some topological properties of submanifolds. Submersions, The fibres of submersions, Quotient manifolds, Transformation groups, Examples of quotient manifolds. |
| Week | Subject | Related Preparation |
| 1) | Preliminaires | |
| 2) | Some classical theory of differentiable functions | |
| 3) | Atlas, differentiable structures on a set | |
| 4) | Examples of differentiable structures on a set | |
| 5) | Differentiable manifolds | |
| 6) | Differentiable functions | |
| 7) | The induced topology on a manifold | |
| 8) | Differentiable varieties, Grassmann manifolds | |
| 9) | Topological restrictions on a manifold, Partitions of unity | |
| 10) | Manifold structure on a topological space, properties of the induced topology | |
| 11) | Partial differentiation, tangent vectors, derived linear functions, The invers function Theorem, Leibniz's rule. | |
| 12) | İmmersions, submanifolds, regular submanifolds, some topological properties of submanifolds. | |
| 13) | Submersions, The fibres of submersions, Quotient manifolds | |
| 14) | Transformation groups, Examples of quotient manifolds. |
| Course Notes / Textbooks: | Differentiable Manifolds an Introduction ,F Brickell, R. S. Clark. |
| References: | . |
| Semester Requirements | Number of Activities | Level of Contribution |
| Midterms | 2 | % 45 |
| Final | 1 | % 55 |
| Total | % 100 | |
| PERCENTAGE OF SEMESTER WORK | % 45 | |
| PERCENTAGE OF FINAL WORK | % 55 | |
| Total | % 100 | |
| Activities | Number of Activities | Duration (Hours) | Workload |
| Course Hours | 14 | 3 | 42 |
| Study Hours Out of Class | 7 | 2 | 14 |
| Midterms | 2 | 20 | 40 |
| Final | 1 | 30 | 30 |
| Total Workload | 126 | ||
| No Effect | 1 Lowest | 2 Low | 3 Average | 4 High | 5 Highest |
| Program Outcomes | Level of Contribution | |
| 1) | To be able to examine, interpret data and assess ideas with the scientific methods in the area of EU studies. | 2 |
| 2) | To be able to inform authorities and institutions in the area of EU studies, to be able to transfer ideas and proposals supported by quantitative and qualitative data about the problems. | 2 |
| 3) | To be introduced to and to get involved in other disciplines that EU studies are strongly related with (political science, international relations, law, economics, sociology, etc.) and to be able to conduct multi-disciplinary research and analysis on European politics. | 3 |
| 4) | To be able to evaluate current news on European Union and Turkey-EU relations and identify, analyze current issues relating to the EU’s politics and policies. | 2 |
| 5) | To be able to use English in written and oral communication in general and in the field of EU studies in particular. | 1 |
| 6) | To have ethical, social and scientific values throughout the processes of collecting, interpreting, disseminating and implementing data related to EU studies. | 1 |
| 7) | To be able to assess the historical development, functioning of the institutions and decision-making system and common policies of the European Union throughout its economic and political integration in a supranational framework. | 2 |
| 8) | To be able to evaluate the current legal, financial and institutional changes that the EU is going through. | 2 |
| 9) | To explain the dynamics of enlargement processes of the EU by identifying the main actors and institutions involved and compare previous enlargement processes and accession process of Turkey. | 2 |
| 10) | To be able to analyze the influence of the EU on political, social and economic system of Turkey. | 2 |
| 11) | To acquire insight in EU project culture and to build up project preparation skills in line with EU format and develop the ability to work in groups and cooperate with peers. | 2 |
| 12) | To be able to recognize theories and concepts used by the discipline of international relations and relate them to the historical development of the EU as a unique post-War political project. | 3 |