2024 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Special lectures on current topics in Mathematics P
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Yasushi Homma / Kiyonori Gomi
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Intensive
- Class
- -
- Course Code
- MTH.E645
- Number of credits
- 200
- Course offered
- 2024
- Offered quarter
- 2Q
- Syllabus updated
- Mar 17, 2025
- Language
- Japanese
Syllabus
Course overview and goals
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.
Course description and aims
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
- 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/Objectives
| Course schedule | Objectives | |
|---|---|---|
| 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. |
Study advice (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 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）
Evaluation methods and criteria
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
None required.