National Taiwan Normal University Course Outline Fall , 2020 |
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I.Course information |
Serial No. | 2685 | Course Level | |
Course Code | MAC8026 | Chinese Course Name | 幾何測度論專題(一) |
Course Name | Topics in Geometric Measure Theory (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 | Fri. 7-9 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 |
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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 |
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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) | Ulrich Menne/ 孟悟理 | |||||||||||||||||||||||||||||
Schedule | ||||||||||||||||||||||||||||||
Course outline of Topics in Geometric Measure Theory I+II Topics are presented by the participants and are assigned in consultation with the teacher taking into account the prior knowledge of the individual participants. Remote participants may employ a virtual whiteboard for their presentation. Each participant usually presents two sessions and all topics should take two sessions unless indicated otherwise. The following seven topics are of preparatory nature. (1) Locally convex spaces induced by a nonempty family of real valued semi- norms, see [Men16c, § 2]; one session. (2) Locally convex spaces of continuous functions with and without compact support, see [Fed69, 2.5.19] and [Men16b, 2.10–2.12, 2.23]. (3) Decomposition of Daniell integrals and weak convergence of linear functionals on spaces of continuous functions with compact support, see [Fed69, 2.5.20] and [AFP00, 1.61–1.63]; one session. (4) Distributions, regularisation, and distributions representable by integration, see [Fed69, 4.1.1–4.1.5], [Men16b, 2.13–2.21, 2.24, 3.1], and [Men16a]. (5) The Grassmann manifolds, Jacobians, and curvatures of submanifolds, see [Fed69, 3.2.18 (1) (2) (4)], [All72, 2.3–2.5], and [Men18, § 6]. (6) Some structure theory, see [Fed69, 3.3.1, 3.3.5, 3.3.6, 3.3.17] and [Men18, § 13]. (7) Regularity of solutions of certain partial differential equations: L_2 and Hölder conditions, strongly elliptic systems, Sobolev’s inequality, and strongly elliptic systems, see [Fed69, 5.2.1–5.2.6]; four sessions. The following six main topics are devoted to foundational results on varifolds. (8) Basic properties of varifolds, see [All72, 3.1–3.5], [MS18, 4.1], and [Men18, § 15]. (9) The first variation (with respect to the area integrand) of a varifold, see [All72, 4.1–4.6, 4.8 (2) (3), 4.10 (1), 4.11–4.12] and [Men18, § 16]. (10) Radial deformations and the rectifiability theorem, see [All72, 2.6 (3), § 5], [Kol15, § 3], [Men16b, § 4], and [Men18, § 17]. (11) The compactness theorem for integral varifolds, see [All72, § 6] and [HM86]. (12) The isoperimetric inequality, see [MS18, § 3], [Men09, 2.5], [All72, 8.6], and [Men16b, 5.1, § 7]; with regard to [All72, 8.6], see also [Sim83, 17.8]. (13) The regularity theorem, see [All72, § 8]; four sessions. Finally, the following two supplementary topics are independent of the main line of development and serve to complete the picture. (14) Curves of finite length, see [Fed69, 2.9.21–2.9.23, 2.10.10–20.10.14, 3.2.6]. (15) Almgren’s bi-Lipschitzian embedding of the space Q_Q(R^n), consisting of all unordered Q tuples in R^n, into a suitable Euclidean space, see [Alm00, 1.1 (1)–(4), 1.2]. Topics are assigned on a first-come-first-served basis. Please contact the instructor by email. Participants are recommended to use the time before the start of the lecture period to prepare the topic chosen to present. |
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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 |
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