(MATH 614)
| Course Code | Course Name | Semester | Theory | Practice | Lab | Credit | ECTS |
|---|---|---|---|---|---|---|---|
| MATH 614 | 2 | 3 | 0 | 0 | 3 | 7 |
| Prerequisites | |
| Admission Requirements |
| Language of Instruction | English |
| Course Type | Elective |
| Course Level | Doctoral Degree |
| Course Instructor(s) | Serap GÜRER serapgurer@gmail.com (Email) |
| Assistant | |
| Objective |
-This course bridges the gap between abstract algebraic topology and data science. It covers the mathematical foundations of persistent homology, stability theory, and the algorith- mic implementation of topological summaries. The course concludes with modern applications in machine learning, including vectorization methods and topological deep learning. |
| Content | This doctoral course provides a rigorous introduction to Topological Data Analysis (TDA). It covers the mathematical foundations of simplicial complexes and algebraic homology, followed by the theory and computation of Persistent Homology. Key topics include the construction of filtrations (Vietoris-Rips, Cech), stability theorems (Bottleneck and Wasserstein distances), and efficient algorithms for computing topological invariants. The course concludes by integrating TDA with Machine Learning through vectorization methods (Persistence Landscapes, Images), the Mapper algorithm, and Topological Deep Learning applications |
| Course Learning Outcomes |
1.Understand the mathematical foundations of simplicial homology and persistent homology. 2. Master the algorithms used to compute topological invariants from point cloud data. 3. Analyze the stability and statistical properties of topological summaries. 4. Apply TDA tools (Mapper, Persistence Landscapes) to real-world datasets and integrate them with Machine Learning pipelines. |
| Teaching and Learning Methods | |
| References |
•Edelsbrunner, H., & Harer, J. (2010). Computational Topology: An Introduction. AMS. • Zomorodian, A. J. (2005). Topology for Computing. Cambridge University Press. (Ex- cellent for algorithms and complexity). • Hatcher, A. (2002). Algebraic Topology. Cambridge University Press. (The standard reference for rigorous Homology theory). • Oudot, S. Y. (2015). Persistence Theory: From Quiver Representations to Data Analysis. AMS. |
Theory Topics
| Week | Weekly Contents |
|---|
Practice Topics
| Week | Weekly Contents |
|---|
Contribution to Overall Grade
| Number | Contribution | |
|---|---|---|
| Contribution of in-term studies to overall grade | 3 | 50 |
| Contribution of final exam to overall grade | 1 | 50 |
| Toplam | 4 | 100 |
In-Term Studies
| Number | Contribution | |
|---|---|---|
| Assignments | 2 | 20 |
| Presentation | 0 | 0 |
| Midterm Examinations (including preparation) | 1 | 30 |
| Project | 0 | 0 |
| Laboratory | 0 | 0 |
| Other Applications | 0 | 0 |
| Quiz | 0 | 0 |
| Term Paper/ Project | 0 | 0 |
| Portfolio Study | 0 | 0 |
| Reports | 0 | 0 |
| Learning Diary | 0 | 0 |
| Thesis/ Project | 0 | 0 |
| Seminar | 0 | 0 |
| Other | 0 | 0 |
| Make-up | 0 | 0 |
| Toplam | 3 | 50 |
| No | Program Learning Outcomes | Contribution | ||||
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | ||
| Activities | Number | Period | Total Workload |
|---|---|---|---|
| Total Workload | 0 | ||
| Total Workload / 25 | 0.00 | ||
| Credits ECTS | 0 | ||