National Taiwan Normal University Course Outline
 Fall , 2019

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I.Course information
Serial No. 2676 Course Level
Course Code MAC0146 Chinese Course Name 幾何分析專題（一）
Course Name Topics on Geometric Analysis (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. 6-8 Gongguan 11111
Curriculum Goals Corresponding to the Departmental Core Goal
1.　Verify research-level mathematics Master:
　1-1 Equipped with professional mathematics competences
　4-2 Possessing a consistent and firm attitude toward pursuing truths
Doctor:
　1-1 Equipped with professional mathematics competences
　4-2 Possessing a consistent and firm attitude toward pursuing truths
2.　Conceptually understand research-level mathematics 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
3.　Concisely present research-level mathematics Master:
　2-1 Being able to communicate and express mathematically
　3-2 Possessing the abilities to think independently, criticize, and reflect
Doctor:
　2-1 Being able to communicate and express mathematically
　3-2 Possessing the abilities to think independently, criticize, and reflect

II. General Syllabus
Instructor(s) LIN, Chun-Chi/ 林俊吉 孟悟理 Ulrich Menne
Schedule

Topics in Geometric Analysis I+II

Prerequisites We assume familiarity with the concepts of abstract measure and Lebesgue integration.

Course outline Topics are presented by the participants and are assigned in consultation with the teachers taking into account the prior knowledge of the individual participants. Amongst the possible topics are the following.

  1. Computations and geometric measure theory, see for instance [BLM17] and [CT13]; this leaves plenty of choices.

  2. Functions of bounded variation, see [AFP00] or [Mag12]; this also leaves plenty of choices.

  3. Topological vector spaces, see [Bou87], with emphasis on the items summarised in [Men16, Chapter 2].

  4. An example concerning approximate differentiation, see [Koh77].

  5. Pointwise differentiability of higher order for sets, see [Men19].

  6. Rectifiability and approximate differentiability of higher order for sets, see [San19].

  7. An example of a Borel set in R2 ≃ R × R whose orthogonal projection onto R is not a Borel set, see [Fed69, 2.2.9, 2.2.11].

Items 5., 6., and 7. may be particularly suitable for students who followed the course Special Topics in Analysis in the preceding term.

This syllabus, including description and references, is available as PDF.
Instructional Approach
Methods Notes
Other: Reading seminar
Grading assessment
Methods Percentage Notes
Class discussion involvement 20 % Posing, in English, a number of good questions to presentations by other students.
Presentation 80 % Presenting, in English, a topic in one or more lectures in the course.
Required and Recommended Texts/Readings with References

References

[AFP00] Luigi Ambrosio, Nicola Fusco, and Diego Pallara. Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 2000.

[BLM17] Blanche Buet, Gian Paolo Leonardi, and Simon Masnou. A varifold approach to surface approximation. Arch. Ration. Mech. Anal., 226(2):639–694, 2017. URL: https://doi.org/10.1007/s00205-017-1141-0.

[Bou87] N. Bourbaki. Topological vector spaces. Chapters 1–5. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1987. Translated from the French by H. G. Eggleston and S. Madan. URL: https://doi.org/10.1007/978-3-642-61715-7.

[CT13] Nicolas Charon and Alain Trouvé. The varifold representation of nonoriented shapes for diffeomorphic registration. SIAM J. Imaging Sci., 6(4):2547–2580, 2013. URL: https://doi.org/10.1137/130918885.

[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.

[Koh77] Robert V. Kohn. An example concerning approximate differentiation. Indiana Univ. Math. J., 26(2):393–397, 1977. URL: https://www. iumj.indiana.edu/docs/26030/26030.asp.

[Mag12] Francesco Maggi. Sets of finite perimeter and geometric variational problems, volume 135 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2012. An introduction to geometric measure theory. URL: https://doi.org/10.1017/ CBO9781139108133.

[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.

[Men19] Ulrich Menne. Pointwise differentiability of higher order for sets. Ann. Global Anal. Geom., 55(3):591–621, 2019. URL: https://doi.org/10.1007/s10455-018-9642-0.

[San19] Mario Santilli. Rectifiability and approximate differentiability of higher order for sets. Indiana Univ. Math. J., 68(3):1013–1046, 2019. URL: https://doi.org/10.1512/iumj.2019.68.7645.

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