National Taiwan Normal University Course Outline
 Spring , 2020

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I.Course information
Serial No. 2547 Course Level
Course Code MAC0147 Chinese Course Name 幾何分析專題（二）
Course Name Topics on Geometric Analysis (II)
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 Fri. 2-4 Gongguan MA1-04
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:
　1-3 Being able to think mathematically and critically
　2-1 Being able to communicate and express mathematically
　3-2 Possessing the abilities to think independently, criticize, and reflect
Doctor:
　1-3 Being able to think mathematically and critically
　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

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. Mass and comass, see [Fed69, Section 1.8].

  2. The Besicovitch property for parabolic balls, see [AF02].

  3. An example concerning Vitali relations: If F is the family of all compact nondegenerate axis-parallel rectangles in R^2, then {(x, S) : x ∈ S ∈ F } is not a Vitali relation with respect to two-dimensional Lebesgue measure, see [Car68, pp. 689–692].

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

  5. Pointwise differentiability of higher order for distributions, see [Men19a].

  6. An example of a Borel set in R^2 ≃ R × R whose orthogonal projection onto R is not a Borel set, see [Fed69, 2.2.9, 2.2.11].

  7. A bi-Lipschitzian embedding the space Q_Q(R^n), consisting of all unordered Q tuples in R^n, into Euclidean space, see [Alm00, 1.1(1)–(4), 1.2].

  8. Estimating distances in Q_Q(R^n) by means of Hall’s theorem on perfect matching, see [Men10, 2.14] and [LP86, Theorem 1.1.3].
Instructional Approach
Methods Notes
Other: Reading course
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

[AF02] Hugo Aimar and Liliana Forzani. On the Besicovitch property for parabolic balls. Real Anal. Exchange, 27(1):261–267, 2001/02.

[Alm00] Frederick J. Almgren, Jr. Almgren’s big regularity paper, volume 1 of World Scientific Monograph Series in Mathematics. World Scientific Publishing Co. Inc., River Edge, NJ, 2000. Q-valued functions minimizing Dirichlet’s integral and the regularity of area-minimizing rectifiable currents up to codimension 2, With a preface by Jean E. Taylor and Vladimir Scheffer. URL: https://doi.org/10.1142/9789812813299.

[Car68] Constantin Carathéodory. Vorlesungen über reelle Funktionen. Third (corrected) edition. Chelsea Publishing Co., New York, 1968.

[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.

[LP86] László Lovász and Michael D. Plummer. Matching theory, volume 121 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1986. Annals of Discrete Mathematics, 29.

[Men10] Ulrich Menne. A Sobolev Poincaré type inequality for integral varifolds. Calc. Var. Partial Differential Equations, 38(3-4):369–408, 2010. URL: https://doi.org/10.1007/s00526-009-0291-9.

[Men19a] Ulrich Menne. Pointwise differentiability of higher order for distributions, 2019. arXiv:1803.10855v2.

[Men19b] 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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