National Taiwan Normal University Course Outline
 Fall , 2026

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I.Course information
Serial No. 2666 Course Level Master / PhD
Course Code MAC0146 Chinese Course Name 幾何分析專題（一）
Course Name Topics on Geometric Analysis (I)
Department Department of Mathematics
Two/one semester 1 Req. / Sel. Sel.
Credits 3.0 Lecturing hours Lecture hours: 3
Teach in English Y Teach in National Languages
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. 7-9 Gongguan M310
Curriculum Goals Corresponding to the Departmental Core Goal
1.　Verify research-level mathematics Master:
　1-1 Equipped with professional mathematics competences
　4-2 Possessing a consistent and firm attitude toward pursuing truths
Doctor:
　1-1 Equipped with professional mathematics competences
　4-2 Possessing a consistent and firm attitude toward pursuing truths
2.　Conceptually understand research-level mathematics 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
3.　Concisely present research-level mathematics Master:
　2-1 Being able to communicate and express mathematically
　3-2 Possessing the abilities to think independently, criticize, and reflect
Doctor:
　2-1 Being able to communicate and express mathematically
　3-2 Possessing the abilities to think independently, criticize, and reflect

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

Prerequisites We assume familiarity with the concepts of abstract measure and Lebesgue integration.

Course outline Topics are presented by the participants and are assigned in consultation with the teachers taking into account the prior knowledge of the individual participants. Possible topics are listed below. About half of the eleven topics are expected to be covered in the course. Topics are assigned on a first-come-first-served basis; please contact the instructor by email. It is recommended to use the summer vacation for preparation.

General measure theory Topics 1–3 supplement the material of the instructor’s courses Real Analysis (I)+(II), whereas Topic 4 relates to the study of derivatives in his course Geometric Measure Theory (I).

  1. An example of a Borel set in the Cartesian product R2 ≃ R × R whose projection onto the first factor is not a Borel set, see [Fed69, 2.2.9, 2.2.11].
  2. A representation theorem for linear functionals with applications to the decomposition of Daniell integrals, see [Fed69, 2.5.12, 2.5.20].
  3. An example concerning Vitali relations, see [Car68, pp. 689–692] and [Fed69, 2.8.20].
  4. Results on derivation using centred balls, see [Fed69, 2.9.14–2.9.18].

Differentiation The topics all supplement the course Special Topics in Analysis lectured by M. Workman (and earlier by the instructor).

  1. Whitney’s extension theorem for functions of class ∞, see [Whi34, §§ 1–12].
  2. An example concerning approximate differentiation, see [Koh77].
  3. Analytic functions, see [Fed69, 3.1.24] with the exception of the last three paragraphs therein.

Distributions The theory of distributions provides the foundation for concepts such as weak differentiability of functions and functions of bounded variation.

  1. Locally convex spaces defined by non-empty families of real valued seminorms, see [Men16b, 2.1–2.6], and distributions, see [Fed69, 4.1.1].
  2. Regularisation, see [Fed69, 4.1.2–4.1.4] and [Men16a, 3.1].
  3. Distributions representable by integration, see [Fed69, 4.1.5] and [Men16a, 2.18–2.21].
  4. Locally convex spaces arising as strict inductive limit of Fréchet spaces, see [Men16a, 2.3–2.17] and references to [Bou87] contained therein.

Topics assigned Topic 2, Topic 7, and Topic 8.
Instructional Approach
Methods Notes
Other: Reading seminar
Grading assessment
Methods Percentage Notes
Presentation 100 % Presenting, in English, one topic in one or two lectures in the course.
Adjustment methods for students
Required and Recommended Texts/Readings with References

[Bou87] N. Bourbaki. Topological vector spaces. Chapters 1–5. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1987. Translated from the French by H. G. Eggleston and S. Madan. URL: https://doi.org/10.1007/978-3-642-61715-7.

[Car68] Constantin Carathéodory. Vorlesungen über reelle Funktionen. Third (corrected) edition. Chelsea Publishing Co., New York, 1968.

[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.

[Koh77] Robert V. Kohn. An example concerning approximate differentiation. Indiana Univ. Math. J., 26(2):393–397, 1977. URL: https://doi.org/10.1512/iumj.1977.26.26030.

[Men16a] 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.

[Men16b] Ulrich Menne. Sobolev functions on varifolds. Proc. Lond. Math. Soc. (3), 113(6):725–774, 2016. URL: https://doi.org/10.1112/plms/pdw023.

[Whi34] Hassler Whitney. Analytic extensions of differentiable functions defined in closed sets. Trans. Amer. Math. Soc., 36(1):63–89, 1934. URL: https://doi.org/10.2307/1989708.

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