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National Taiwan Normal University Course Outline Fall , 2025 |
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| I.Course information |
| Serial No. | 2598 | Course Level | Master / PhD |
| Course Code | MAC8020 | Chinese Course Name | 幾何與拓樸學（二） |
| Course Name | Geometry and Topology (II) | ||
| Department | Department of Mathematics | ||
| Two/one semester | 1 | Req. / Sel. | Sel. |
| Credits | 3.0 | Lecturing hours | Lecture hours: 3 |
| Prerequisite Course | ◎1. This is a cross-level course and is available for junior and senior undergraduate students, master's students and PhD students. 2. If the listed course is a doctroal level course, it is only available for master's students and PhD students. | ||
| Comment | |||
| Course Description | |||
| Day & Class Period/Location | Mon. 2-4 Gongguan M212 | ||
| Curriculum Goals | Corresponding to the Departmental Core Goal | ||
| 1. Train students to master the knowledge of differential topology |
Master: 1-1 Equipped with professional mathematics competences 1-2 Being able to reason and induct with mathematical logic 1-3 Being able to think mathematically and critically 1-4 Possessing the abilities to propose and solve questions in advanced mathematics 1-5 Being able to use mathematics as tools to learn other subjects 1-6 Possessing the capacities to view elementary mathematics from an advanced viewpoint 3-4 Having insights, intuitions, and senses of mathematics 3-5 Having good taste for mathematics Doctor: 1-1 Equipped with professional mathematics competences 1-2 Being able to reason and induct with mathematical logic 1-3 Being able to think mathematically and critically 1-4 Possessing the abilities to propose and solve questions in advanced mathematics 1-5 Being able to use mathematics as tools to learn other subjects 1-6 Possessing the capacities to view elementary mathematics from an advanced viewpoint 3-4 Having insights, intuitions, and senses of mathematics 3-5 Having good taste for mathematics |
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| 2. Train students to communicate, express and write in the language of differential geometry. |
Master: 2-1 Being able to communicate and express mathematically 2-4 Possessing the competences of lifelong learning Doctor: 2-1 Being able to communicate and express mathematically 2-4 Possessing the competences of lifelong learning |
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| 3. To develop students' intuition, appreciation and ability to apply differential topology to other disciplines. |
Master: 3-1 Being able to seek out answers with the attitudes of patience, diligence, concentration, and curiosity 3-2 Possessing the abilities to think independently, criticize, and reflect 3-4 Having insights, intuitions, and senses of mathematics 3-5 Having good taste for mathematics 4-1 Being knowledgeable and being able to self-develop in the profession 4-2 Possessing a consistent and firm attitude toward pursuing truths 4-3 Possessing a variety of beliefs regarding mathematics values and mathematics learning 4-4 Possessing global views from both scientific and humanistic perspectives, and being able to appreciate the values of other knowledge fields Doctor: 3-1 Being able to seek out answers with the attitudes of patience, diligence, concentration, and curiosity 3-2 Possessing the abilities to think independently, criticize, and reflect 3-4 Having insights, intuitions, and senses of mathematics 3-5 Having good taste for mathematics 4-1 Being knowledgeable and being able to self-develop in the profession 4-2 Possessing a consistent and firm attitude toward pursuing truths 4-3 Possessing a variety of beliefs regarding mathematics values and mathematics learning 4-4 Possessing global views from both scientific and humanistic perspectives, and being able to appreciate the values of other knowledge fields |
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| II. General Syllabus |
| Instructor(s) | LIN, Chun-Chi/ 林俊吉 郭庭榕 KUO, Ting-Jung | ||
| Schedule | |||
This reading seminar course focuses on two primary themes: (i) Riemann surfaces and (ii) topics in geometric analysis, emphasizing contemporary research directions and the mathematical tools essential for their study. Course materials will be curated from diverse sources, including textbooks, research papers, and ArXiv preprints. The covered topics will span a range of complexity levels, from fundamental concepts to sophisticated subjects, with foundational topics chosen specifically to provide the background knowledge necessary for deeper investigation. Topics planned to be covered include: Part I. Topics in geometric analysis (Geometric Analysis and Elliptic/Parabolic PDEs)： Part II. Riemann Surfaces 1. Korteweg-de Vries (KdV) equation 2. Finite-gap integration theory 3. cone spherical metrics and blow-up analysis 4. Isomonodromic theory in integrable systems
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| Instructional Approach | |||
| Methods | Notes | ||
| Formal lecture | participants give presentation in the scheduled weeks | ||
| Group discussion | participants raise up questions during presentation | ||
| Grading assessment | |||
| Methods | Percentage | Notes | |
| Class discussion involvement | 30 % | ||
| Presentation | 70 % | ||
| Required and Recommended Texts/Readings with References | 1. Q. Han and F. H. Lin, Elliptic Partial Differential Equations, 2nd Ed. American Mathematical Society, 2011. 2. V. A. Solonnikov, Boundary Value Problems of Mathematical Physics. III., Proceedings of the Steklov institute of Mathematics (1965), Amer. Math. Soc., Providence, R. I., No. 83, 1967. 3. G. Bellettini, Lecture Notes on Mean Curvature Flow, Barriers and Singular Perturbations, Scuola Normale Superiore Pisa, 2013. 4. L. Simon, Introduction to Geometric Measure Theory, 2018. 5. H. M. Fakas and I. Kra, Riemann Surfeces, Springer, 1992. |
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