National Taiwan Normal University Course Outline
 Fall , 2025

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I.Course information
Serial No. 2596 Course Level Master / PhD
Course Code MAC0197 Chinese Course Name 幾何測度論（一）
Course Name Geometric Measure Theory(I)
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 Fri. 2-4 Gongguan M210
Curriculum Goals Corresponding to the Departmental Core Goal
1.　Develop professional skills in mathematics Master:
　1-1 Equipped with professional mathematics competences
　4-1 Being knowledgeable and being able to self-develop in the profession
Doctor:
　1-1 Equipped with professional mathematics competences
　4-1 Being knowledgeable and being able to self-develop in the profession
2.　Improve the ability of logical reasoning and induction Master:
　1-2 Being able to reason and induct with mathematical logic
Doctor:
　1-2 Being able to reason and induct with mathematical logic
3.　Improve mathematical and critical thinking skills Master:
　1-3 Being able to think mathematically and critically
　3-2 Possessing the abilities to think independently, criticize, and reflect
Doctor:
　1-3 Being able to think mathematically and critically
　3-2 Possessing the abilities to think independently, criticize, and reflect

II. General Syllabus
Instructor(s) Ulrich Menne/ 孟悟理
Schedule

Course outline  The content of the present course concerns four topics; namely,

  • covering theorems (adequate families, covering with enlargement, centred ball coverings, and Vitali relations),
  • derivatives (existence of derivatives, indefinite integrals, density and approximate continuity, and curves of finite length),
  • Carathéodory’s construction (the general construction, Hausdorff and spherical measure, relation to Riemann-Stieltjes integration, densities, Cantor sets, Steiner symmetrisation, equality of measures over Euclidean space, and Lipschitzian extensions of functions), and
  • differentiation (inverse function theorem, factorisation of maps near generic points).

The last topic serves as preparation for the topics submanifolds of Euclidean space and area for Lipschitzian maps treated in the course Geometric Measure Theory (II).

Details of the course  The main reference text will be the instructor’s weekly updated lecture notes written in LATEX, see [Men23]. They are based on and expand the relevant parts of Federer’s treatise [Fed69] and, regarding pointwise differentiation of higher order, the presentation of [Men19]. Grading is solely determined by individual oral examinations conducted in English.

Prerequisites  We employ measures, measurable sets, Borel regular measures, measurable functions, Lebesgue integration, linear functionals, and product measures. Familiarity with the majority of these concepts is thus expedient.

Related courses. The course Special Topics in Analysis (lectured by Myles Workman in the same term) is independent of the present course.  In spring term, the course Geometric Measure Theory (II) shall be offered (jointly by Myles Workman and Ulrich Menne) which depends on the results both of these courses.
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
[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.
[Men19]Ulrich Menne. Pointwise differentiability of higher order for sets. Ann. Global Anal. Geom., 55(3):591–621, 2019. URL: https://doi.org/10. 1007/s10455-018-9642-0.
[Men23]Ulrich Menne. Geometric measure theory, 2023. Lecture notes, National Taiwan Normal University.

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