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National Taiwan Normal University Course Outline Spring , 2020 |
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| I.Course information |
| Serial No. | 2550 | Course Level | |
| Course Code | MAC0198 | Chinese Course Name | 幾何測度論（二） |
| Course Name | Geometric Measure Theory(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 | |||
| Time / Location | Fri. 7-9 Gongguan MA-210 | ||
| Curriculum Goals | Corresponding to the Departmental Core Goal | ||
| 1. Develop professional skills in mathematics |
Master: 1-1 Equipped with professional mathematics competences Doctor: 1-1 Equipped with professional mathematics competences |
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| 2. Improve the ability of logical reasoning and induction |
Master: 1-2 Being able to reason and induct with mathematical logic 1-3 Being able to think mathematically and critically Doctor: 1-2 Being able to reason and induct with mathematical logic 1-3 Being able to think mathematically and critically |
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| 3. Improve mathematical and critical thinking skills |
Master: 1-3 Being able to think mathematically and critically 1-6 Possessing the capacities to view elementary mathematics from an advanced viewpoint Doctor: 1-3 Being able to think mathematically and critically 1-6 Possessing the capacities to view elementary mathematics from an advanced viewpoint |
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| II. General Syllabus |
| Instructor(s) | Ulrich Menne/ 孟悟理 | ||
| Schedule | |||
Change of schedule No lecture on Friday, 21 Feb 2020, due to 2019-nCoV prevention measures – make up lecture on Monday, 4 May 2020 (same time and room); thus, the first lecture is Friday, 6 March 2020. Prerequisites We continue the course Geometric Measure Theory I. Course outline We firstly cover Carathéodory’s construction (Hausdorff measure and spherical measure, densities, Cantor sets, Steiner symmetrisation, equality of measures on Euclidean spaces, extensions of Lipschitzian functions). Next, we complete differentiation theory (curves of finite length). Then, differentials and tangents (differentiation Lebesgue almost everywhere, factorisation near generic points, submanifolds of Euclidean space, tangent vectors, relative differentials, second fundamental form) are treated; this is followed by the area of Lipschitzian maps (Jacobi, area of mappings in Euclidean space, rectifiable sets, approximate tangent vectors and differentials, area of maps of rectifiable sets, agreement of measures on rectifiable sets, Grassmann manifold). To prepare for rectifiable varifolds, selected elements of structure theory are presented. If time permits, the basic theory of varifolds (first variation, monotonicity identity) followed by the isoperimetric inequality and compactness theorems is developed; otherwise, this material could be covered in the planned subsequent reading course Topics in Geometric Measure Theory I. Sources for the course and other information on the course Our main sources are [Fed69] and, concerning varifolds, [All72] supplemented by more detailed LATEX-ed lecture notes; some simplifications from [Men16] and [MS18] will also be employed. A survey of the concept of varifold is available at [Men17].
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| Instructional Approach | |||
| Methods | Notes | ||
| Formal lecture | The lecture will be conducted in English. Weekly updated LaTeXed lecture notes shall be made available. | ||
| Grading assessment | |||
| Methods | Percentage | Notes | |
| Final exam | 100 % | Individual oral examination conducted in English. | |
| Required and Recommended Texts/Readings with References | [All72] William K. Allard. On the first variation of a varifold. Ann. of Math. (2), 95:417–491, 1972. URL: https://doi.org/10.2307/1970868. [Fed69] Herbert Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969. URL: https://doi.org/10.1007/ 978-3-642-62010-2. [Men16] Ulrich Menne. Weakly differentiable functions on varifolds. Indiana Univ. Math. J., 65(3):977–1088, 2016. URL: https://doi.org/10.1512/iumj.2016.65.5829. [Men17] Ulrich Menne. The concept of varifold. Notices Amer. Math. Soc., 64(10):1148–1152, 2017. URL: https://doi.org/10.1090/noti1589. [MS18] Ulrich Menne and Christian Scharrer. An isoperimetric inequality for diffused surfaces. Kodai Math. J., 41(1):70–85, 2018. URL: https://doi.org/10.2996/kmj/1521424824. |
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