MATH543 LOW DIMENSIONAL TOPOLOGY
| Course Code: | 2360543 |
| METU Credit (Theoretical-Laboratory hours/week): | 3 (3.00 - 0.00) |
| ECTS Credit: | 8.0 |
| Department: | Mathematics |
| Language of Instruction: | English |
| Level of Study: | Graduate |
| Course Coordinator: | Prof.Dr. MOHAN LAL BHUPAL |
| Offered Semester: | Fall or Spring Semesters. |
Course Objectives
The objective of this course is to prepare the student for research in low dimensional topology. In particular, students will learn:
- how the intersection form plays a key role in the classification of topological and smooth 4-manifolds;
- how to use the theorems of Freedman, Donaldson and Rokhlin in the study of 4-manifolds;
- constructions of elliptic surfaces and operations on them such as the logarithmic transform;
- how to describe smooth 4-manfolds using Kirby diagrams;
- how to manipulate Kirby diagrams via Kirby calculus;
- descriptions of plumbings using Kirby diagrams;
- some advanced topics such as branched coverings and resolution of singularities.
Course Content
4-manifolds, surfaces in 4-manifolds, complex surfaces, complex curves and their desingularizations. Elliptic surfaces; classification of complex surfaces and logarithmic transform. Handle decomposition, Heegard splitting and Kirby diagrams. Linking numbers and framings. Kirby calculus, handle moves and Dehn surgery. Spin structures, plumbings and related constructions. Embedded surfaces and branched covers.
Course Learning Outcomes
After successfully completing the course, students will know:
- how to use the intersection form in the study of topological and smooth 4-manifolds;
- how to appy the Theorems of Freedman, Rokhlin and Donaldson;
- some constructions of elliptic surfaces and the logarithmic transform;
- how to draw and manipulate Kirby diagrams for some 4-manifolds.
Program Outcomes Matrix
| Level of Contribution | |||||
| # | Program Outcomes | 0 | 1 | 2 | 3 |
| 1 | Acquires mathematical thinking skills (problem solving, generating ways of thinking, forming correspondence, generalizing etc.) and can use them in related fields. | ✔ | |||
| 2 | Gains academic maturity through self-study. | ✔ | |||
| 3 | Can design mathematics related problems, devise solution methods and apply them when appropriate. | ✔ | |||
| 4 | Carries out parts of a mathematical research program independently. | ✔ | |||
| 5 | Has a command of Turkish and English languages so that he/she can actively communicate (read, write, listen and speak). | ✔ | |||
| 6 | Contributes to solving global, environmental and social problems either individually or as being part of a social group. | ✔ | |||
| 7 | Respects ethical values and rules; applies them in professional and social issues. | ✔ | |||
| 8 | Can work cooperatively in a team and also individually. | ✔ | |||
| 9 | Gets exposed to academic culture through interaction with others. | ✔ | |||
| 10 | Comprehends necessity of knowledge, can define it and acquires it; uses knowledge effectively and shares it with others. | ✔ | |||
0: No Contribution 1: Little Contribution 2: Partial Contribution 3: Full Contribution