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. 24 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 kth 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 kth 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.


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. SpringerVerlag New York Inc., New York, 1969. URL: https://doi.org/10.1007/9783642620102. [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/s1045501896420. 