National Taiwan Normal University Course Outline
Spring , 2019

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I.Course information
Serial No. 2611
Course Code MAC8009 Chinese Course Name 分析專題研究
Course Name Special Topics in Analysis
Department Department of Mathematics
Two/one semester 1 Req. / Sel. Sel.
Credits 3.0 Lecturing hours Lecture hours: 3
Restrict Course ◎課程開放上修
Comment 授課教師Ulrich Menne
Course Description
Time / Location Tue. 2-4 Gongguan 11111

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

The goal of this course is to develop the theory of pointwise differentiation in the simplest case of real valued functions defined on Euclidean space. A function is k times pointwise differentiable at a point if and only if it may be approximated by a polynomial function of degree at most k to k-th order at that point; in particular, derivatives of order k − 1 need not exist in a neighbourhood of the point. Such derivatives for instance occur in the study of Sobolev functions, convex functions (k = 2), and viscosity solutions to fully nonlinear elliptic equations of second order (k = 2). Our motivation however is mainly from geometry: the theory presented acts both as a model case and a toolbox for recent characterisations of rectifiability of order k which essentially is the weakest possible sense in which a set (or, a function) may be considered to have k-th order differentials almost everywhere.

Some knowledge of linear algebra, first order differentials, measure, and Lebesgue integration is prerequisite for this course, but the treatment is selfcontained with regard to all those topics in descriptive set theory, multilinear algebra, and higher order differentiation, that occur. A particular feature of our approach is the development of the concept of symmetric algebra of a vectorspace to effectively treat polynomial functions. The topics covered are as follows.

  1. Descriptive set theory: Lipschitzian maps, Suslin sets, Borel functions (see [Fed69, 2.2.7–2.2.14]).

  2. Multilinear algebra: tensor algebras, graded algebras, the symmetric algebra of a vectorspace, symmetric forms, and polynomial functions (see [Fed69, Sections 1.2, 1.3, 1.9, 1.10]).

  3. Higher order differentiation: higher differentials, partitions of unity, differentiable extensions of functions (see [Fed69, 3.1.11–3.1.15]).

  4. Approximate differentiation and rectifiability of higher order (see [Isa87]).

  5. Pointwise differentiation of higher order (see [Men19, Sections 2, 4]).

PDF file of this course description including references.

Lecturing Methodologies
Methods Notes
Formal lecture Lecture in English with weekly updated lecture notes (LaTeX) in English.
Grading assessment
Methods Percentage Notes
Final exam 100 % Individual oral exam conducted in English determining pass or fail as well as the grade for the course.
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.

[Isa87] N. M. Isakov. On a global property of approximately differentiable functions. Mathematical Notes, 41(4):280–285, 1987. URL: https://doi.org/10.1007/BF01137673.

[Men19] Ulrich Menne. Pointwise differentiability of higher order for sets. Ann. Global Anal. Geom., 2019. URL: https://doi.org/10.1007/s10455-018-9642-0.