National Taiwan Normal University Course Outline
 Fall , 2019

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I.Course information
Serial No. 2679 Course Level
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
Time / Location Mon. 2-4 Gongguan 11111
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

Geometric Measure Theory I+II

Prerequisites We assume sound familiarity with the concepts of measure, measurable function, Lebesgue integration, and product measure.  Students wishing to review this material before the course are encouraged to contact the teacher by email for relevant lecture notes on "Real Analysis", see [Men19].

Course outline In the initial part of the course, we focus on developing the relevant concepts from advanced measure theory (Riesz representation theorem, covering theorems, derivatives of measures), functional analysis (locally convex spaces, weak topology), multilinear algebra (exterior algebra, alternating forms) and basic submanifold geometry (second fundamental form, Grassmann manifold). Then, we develop Hausdorff measures and the area of Lipschitzian maps to treat rectifiable sets. Finally, the basic theory of varifolds is developed (first variation, monotonicity identity) to treat the isoperimetric inequality and compactness theorems.

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].

  1. The course is offered both at the National Taiwan Normal University and in the programme of the Taiwan Mathematics School.
  2. Video-recordings of the lectures will be made available to the participants of the course.

This syllabus, including description and references, is available as PDF.
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

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.

[Men19] Ulrich Menne. Real Analysis, 2019. 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.

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