▶Japanese
Post to SNS
- Home
- > School of Science
- > Graduate major in Mathematics
- > Special lectures on current topics in Mathematics P
- School of Science
-
School of Engineering
- Undergraduate major in Mechanical Engineering
- Undergraduate major in Systems and Control Engineering
- Undergraduate major in Electrical and Electronic Engineering
- Undergraduate major in Information and Communications Engineering
- Undergraduate major in Industrial Engineering and Economics
- Common courses
- School of Materials and Chemical Technology
- School of Computing
- School of Life Science and Technology
- School of Environment and Society
- 工学院，物質理工学院，環境・社会理工学院共通科目
- Liberal arts and basic science courses
- First-Year Courses
- School of Science
- School of Engineering
- School of Materials and Chemical Technology
- School of Computing
- School of Life Science and Technology
-
School of Environment and Society
- Department of Architecture and Building Engineering
- Department of Civil and Environmental Engineering
- Department of Transdisciplinary Science and Engineering
- Department of Social and Human Sciences
- Department of Innovation Science
- Graduate major in Technology and Innovation Management
- Common courses
- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Homma Yasushi Gomi Kiyonori
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- Day/Period(Room No.)
- Intensive ()
- Group
- -
- Course number
- MTH.E645
- Credits
- 2
- Academic year
- 2024
- Offered quarter
- 2Q
- Syllabus updated
- 2024/3/14
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
-
Course description and aims
Spin geometry means one of fields in geometry and global analysis, where the Dirac operator acting on the spinor fields and special spinors such as Killing spinors play important roles. In this course, we learn fundamental tools in spin geometry such as Clifford algebra, spinor fields, the Dirac operator and the twistor operator. We also study Friedrich's eigenvalue estimate, and know the limiting case of the estimate gives Killing spinors. Then we study the classification of spin manifolds with parallel spinors and Killing spinors. If we have time, then we would take a look at the spin 3/2 geometry, a recent research of the instructor.
Through this course, you find that many kinds of geometric structures (Einstein manifolds, Ricci flat manifolds etc.) are related to spin geometry so that spin geometry is necessary in recent geometry.
Student learning outcomes
Be familiar with Clifford algebra, spin group and their representations.
Be familiar with spin structures, the Dirac and Penrose operators.
Understand Killing spinors and Einstein manifolds.
Understand relations between Killing spinors and a variety of geometric structures.
Keywords
Clifford algebra, spinor fields, the Dirac operator, Killing spinor fields, Einstein manifolds.
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
This is a standard lecture course. There will be some assignments.
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | 1. Clifford algebra, spin group and their representations 2. spin structures 3. Levi-Civita connection, spin connection and their holonomy group 4. the Dirac operator and index theorem 5. Eigenvalue estimate and Killing spinor fields 6. Einstein manifolds and geometric structures 7. Classification theorem 8. Spin 3/2 geometry | Details will be provided during each class session. |
Textbook(s)
None required
Reference books, course materials, etc.
``スピン幾何学 -スピノール場の数学-'' 本間泰史 著 森北出版
``Real Killing spinors and holonomy’’, C Bär, Comm. Math. Phys. 154 (1993), 509–521.
``spin geometry'' by J. Figueroa-O'Farrill (@ https://empg.maths.ed.ac.uk/Activities/Spin/SpinNotes.pdf)
``spin geometry'' by C. Bär (＠ https://www.math.uni-potsdam.de/en/professuren/geometry/teaching/lecture-notes）
Assessment criteria and methods
Assignments (100%).
Related courses
- MTH.B341 ： Topology
- MTH.B301 ： Geometry I
- MTH.B302 ： Geometry II
- MTH.B331 ： Geometry III
- MTH.C305 ： Real Analysis I
- MTH.C306 ： Real Analysis II
Prerequisites (i.e., required knowledge, skills, courses, etc.)
None required.
