National Taiwan Normal University Course Outline
 Fall , 2025

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I.Course information
Serial No. 2597 Course Level Master / PhD
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
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
Day & Class Period/Location Mon. 6-8 Gongguan M210
Curriculum Goals Corresponding to the Departmental Core Goal
1.　Develop mathematical expertise to prepare for advanced analysis courses. Master:
　1-1 Equipped with professional mathematics competences
　1-2 Being able to reason and induct with mathematical logic
　2-1 Being able to communicate and express mathematically
Doctor:
　1-1 Equipped with professional mathematics competences
　1-2 Being able to reason and induct with mathematical logic
2.　Enhance the level of abstract thinking. Master:
　1-3 Being able to think mathematically and critically
　3-4 Having insights, intuitions, and senses of mathematics
3.　Interpret the connections between mathematics and other disciplines from a higher perspective. Master:
　1-6 Possessing the capacities to view elementary mathematics from an advanced viewpoint

II. General Syllabus
Instructor(s) Myles Workman/ Myles Workman
Schedule

Course Background and Purpose
The study of differentiability properties of functions (and more generally maps) is of central importance in classical and modern day analysis, with applications in the study of partial differential equations, geometry, and functional analysis. The purpose of this course is to develop the theory of pointwise differentiability of functions in Euclidean space, and as such we will develop fundamental ideas and tools used in many modern fields of research within mathematical analysis.

Course Outline and Description
The content of this one semester course is divided into four main topics, with the first two topics covering some relevant background material needed in the later two:
• Descriptive set theory: Lipschitz maps, Borel sets, Borel functions
• Multilinear algebra: tensor algebras, graded algebras, the symmetric algebra of a vector space, symmetric forms and polynomial functions
• Higher order differentiation theory: first and higher order differentiation, partitions of unity, differentiable extensions of functions (Whitney extension theorem)
• Pointwise differentiation theory: approximate differentiation

Prerequisites
Some knowledge of linear algebra, first order differentiation, metric spaces, measures and Lebesgue integration are required.

 
Instructional Approach
Methods Notes
Formal lecture Lecture in English.
Grading assessment
Methods Percentage Notes
Assignments 40 % exercise sheets in English
Midterm Exam 30 % written mid-term exam in English
Final exam 30 % written final exam in English
Required and Recommended Texts/Readings with References

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.

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