National Taiwan Normal University Course Outline
Fall , 2019
Please respect the intellectual property rights, and shall not copy the textbooks arbitrarily.
|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|
|Time / Location||Mon. 2-4 Gongguan 11111|
|II. General Syllabus|
|Instructor(s)||Ulrich Menne/ 孟悟理|
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].
This syllabus, including description and references, is available as PDF.
|Formal lecture||The lecture will be conducted in English. Weekly updated LaTeXed lecture notes shall be made available.|
|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.
[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.