National Taiwan Normal University Course Outline
 Spring , 2021

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I.Course information
Serial No. 2648 Course Level
Course Code MAC8027 Chinese Course Name 幾何測度論專題（二）
Course Name Topics in 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 11111
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

Course outline Topics are presented by the participants and are assigned in consultation with the teacher taking into account the prior knowledge of the individual participants. Remote participants may employ a virtual whiteboard for their presentation. Each participant usually presents two sessions and all topics should take two sessions unless indicated otherwise.

The following two topics are of preparatory nature.

(1)  Distributions, regularisation, and distributions representable by integration, see [Fed69, 4.1.1–4.1.5], [Men16b, 2.13–2.21, 2.24, 3.1], and [Men16a].

(2)  Regularity of solutions of certain partial differential equations: Sobolev’s inequality, and strongly elliptic systems, see [Fed69, 5.2.4–5.2.6].

The following five main topics are devoted to foundational results on varifolds.

(3)  The first variation (with respect to the area integrand) of a varifold, see [All72, 4.1–4.6, 4.8 (2) (3), 4.10 (1), 4.11–4.12] and [Men18, § 16].

(4)  Radial deformations and the rectifiability theorem, see [All72, 2.6 (3), § 5], [Kol15, § 3], [Men16b, § 4], and [Men18, § 17].

(5)  The compactness theorem for integral varifolds, see [All72, § 6] and [HM86].

(6)  The isoperimetric inequality, see [MS18, § 3], [Men09, 2.5], [All72, 8.6], and [Men16b, 5.1, § 7]; with regard to [All72, 8.6], see also [Sim83, 17.8].

(7)  The regularity theorem, see [All72, § 8]; four sessions.

Finally, the following supplementary topic is independent of the main line of development and serves to complete the picture.

(8) Curves of finite length, see [Fed69, 2.9.21–2.9.23, 2.10.10–20.10.14, 3.2.6].

Topics are assigned on a first-come-first-served basis. Please contact the instructor by email (see below). Participants are recommended to use the time before the start of the lecture period to prepare the topic chosen to present.
Instructional Approach
Methods Notes
Other: Reading course
Grading assessment
Methods Percentage Notes
Class discussion involvement 20 % Posing, in English, a number of good questions to presentations by other students.
Presentation 80 % Presenting, in English, a topic in one or more lectures in the course.
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.

[HM86]

John E. Hutchinson and Michael Meier. A remark on the nonuniqueness of tangent cones. Proc. Amer. Math. Soc., 97(1):184–185, 1986. URL: https://doi.org/10.2307/2046103.

[Kol15]

Jan Kolář. Non-unique conical and non-conical tangents to rectifiable stationary varifolds in R^4. Calc. Var. Partial Differential Equations, 54(2):1875–1909, 2015. URL: https://doi.org/10.1007/ s00526-015-0847-9.

[Men09]

Ulrich Menne. Some applications of the isoperimetric inequality for integral varifolds. Adv. Calc. Var., 2(3):247–269, 2009. URL: https://doi.org/10.1515/ACV.2009.010.

[Men16a]

Ulrich Menne. Topologies on D(U,Y), 2016. Max Planck Institute for Gravitational Physics and University of Potsdam, 3 pages, unpublished.

[Men16b]

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.

[Men18]

Ulrich Menne. Geometric variational problems, 2018. Lecture notes, University of Leipzig.

[Men20]

Ulrich Menne. Geometric measure theory, 2020. Lecture notes, National Taiwan Normal University.

[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.

[Sim83]

Leon Simon. Lectures on geometric measure theory, volume 3 of Proceedings of the Centre for Mathematical Analysis, Australian National University. Australian National University Centre for Mathematical Analysis, Canberra, 1983. URL: https://maths-proceedings.anu. edu.au/CMAProcVol3/CMAProcVol3-Complete.pdf.

 

 

 

 

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