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2024　Topics in Geometry

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Academic unit or major
Graduate major in Mathematical and Computing Science
Instructor(s)
Umehara Masaaki  Nishibata Shinya  Arai Zin  Murofushi Toshiaki  Suzuki Sakie 
Class Format
Lecture    (Face-to-face)
Media-enhanced courses
Day/Period(Room No.)
Mon5-6(S3-207(S322))  Thr5-6(S3-207(S322))  
Group
-
Course number
MCS.M426
Credits
2
Academic year
2024
Offered quarter
2Q
Syllabus updated
2024/3/14
Lecture notes updated
-
Language used
English
Access Index

Course description and aims

Singularities often appear when considering the time evolution of planar curves and surfaces as wavefronts. In this lecture, we will introduce some important differential geometry techniques for curves and surfaces, including singularities, and explain how to determine and describe the properties of important singularities. It is desirable for students to have some knowledge of the theory of curves and surfaces, but the lecture will start from the basics so that students without basic knowledge can understand the lecture. The lecture will be explained from the basics so that the students can understand the lecture without basic knowledge of curves and surfaces.

Student learning outcomes

The objective of this lecture is to enable students to understand the fundamentals of geometrical methods essential for dealing with curves and surfaces, and to be able to apply them to various concrete problems.

Keywords

Curves, Surfaces, Singularities, Gaussian curvatrue, Wave fronts

Competencies that will be developed

✔ Specialist skills ✔ Intercultural skills Communication skills Critical thinking skills ✔ Practical and/or problem-solving skills

Class flow

This lecture will explain the basic matters of curves in the plane and surfaces in space, such as curvature, and introduce useful methods for treating these objects as mathematics. We will introduce typical concrete examples of singularities appearing on curves and surfaces. Useful criteria of the representative singularities of curves and surfaces are introduced. In order to deepen the understanding of the contents, computer graphics will be used to illustrate concrete examples in each lecture.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Fundamentals of plane curves (singular points, regular points, curvature functions) Understand the content of the lecture.
Class 2 Plane curves with singular points (envelope, evolutes, cusp singular points) Understand the content of the lecture.
Class 3 Relation between parallel curves and evolutes and involutes (application to pendulum clocks) Understand the content of the lecture.
Class 4 Curves as wave fronts Understand the content of the lecture.
Class 5 Properties of wavefronts that giving closed curves Understand the content of the lecture.
Class 6 Fundamentals of Surface Theory 1 (Introduction of important Singularities on ｓurfaces, first fundamental Form) Understand the content of the lecture.
Class 7 Fundamentals of Surface Theory 2 (Gaussian curvature, mean curvature, principal curvature) Understand the content of the lecture.
Class 8 Parallel Surfaces and Wave front (A relationship between constant Gaussian ccurvature surfaces and constant mean ccurvature surfaces) Understand the content of the lecture.
Class 9 Surfaces as wave fronts (Definition of abstract wave fronts, etc) Understand the content of the lecture.
Class 10 Properties of smooth function and a criterion of cross caps Understand the content of the lecture.
Class 11 Proof of the criterion of cusps and the riterion of cross caps Understand the content of the lecture.
Class 12 Geometry of cuspidal edges Understand the content of the lecture.
Class 13 Gauss-Bonnet type Theorem for suraces with singulaities Understand the content of the lecture.
Class 14 developmental content Understand the content of the lecture.

Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

Textbook(s)

None in particular.

Reference books, course materials, etc.

Differential Geometry of Curves and Surfaces with Singularities (Umehara, Saji, Yamada, translated by W. Rossman) Maruzen Publishing, Curves and Surfaces (co-authored by M. Umehara and K. Yamada) Shokabo, 2002.
Differential Geometry of Curves and Surfaces with Singularities (Umehara, Saji, Yamada, translated by W. Rossman) World Scientific.

Assessment criteria and methods

The course will be judged comprehensively based on the results of the quizzes and reports at the lectures.

Related courses

  • MCS.T331 ： Discrete Mathematics

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Some familiarity with general topology, vector analysis, etc.

Other

none in particular

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