The slot with the title Mathematical Physics / 数理物理学 in the Graudate School of Physics was dormant for about twenty years. I decided to revive it this year to have an introductory course on algebraic topology meant for theoretical physicists. The rationale is as follows.
In the last ten years, theoretical physicsits realized that algebraic topology is useful and necessary to have a systematic understanding of symmetry-protected topological phases of matter and of anomalies in particle physics. Before that, only a very rudimentary part of algebraic topology was used in theoretical physics. As such, the techniques of algebraic topology required to follow recent advances are not adequately covered in the standard courses of mathematics for physicisists.
This set of lectures is an effort to fill this gap. I plan to use various physics topics to illustrate how concepts from algebraic topology are used. Lectures consisted of
You'll need to submit the term-end report through the link above.
Suggestions for the topics for the term-end paper can be found here. To get credits, please pick at least one, and submit it to the UTokyo LMS page linked below. The deadline was January 26 (Sun), 2025, 23:59.
The histogram of the submission times relative to the deadline can be found here: Record of submission times
2nd slot of Thursdays, from 10:25 to 12:10.
| Date | Summary | Comments | |
|---|---|---|---|
| 1. | Oct. 3 | General introduction to the course | |
| 2. | Oct. 10 | Manifolds 1 | Covered Sec.2.1, 2.2, 2.3 and 2.7 |
| 3. | Oct. 17 | Manifolds 2 | Covered the rest of Sec.2 |
| 4. | Oct. 24 | Fiber bundles 1 | Talked about principal bundles. |
| 5. | Oct. 31 | Fiber bundles 2 | Talked about vector bundles. |
| 6. | Nov. 7 | Basic homotopy theory 1 | Handout |
| 7. | Nov. 14 | Basic homotopy theory 2 | Mostly done except for solitons. |
| Nov. 21 | no class due to Komaba Festival | ||
| 8. | Nov. 28 | Differential forms and de Rham cohomology 1 | Up to Sec.5.1 |
| 9. | Dec. 5 | Differential forms and de Rham cohomology 2 | Up to Sec.5.4 |
| 10. | Dec. 12 | de Rham and other (co)homology theories | Up to Sec.6.3 |
| 11. | Dec. 19 | Other (co)homology theories and characteristic classes I | |
| 12. | Dec. 26 | Characteristic classes II | Up to 7.4 |
| Jan. 2 | no class due to New Year Holidays | ||
| 13. | Jan. 9 | Abelian Chern-Simons terms and the integer quantum Hall effect I | to 8.1 |
| 14. | Jan. 16 | Abelian Chern-Simons terms and the integer quantum Hall effect II | to 8.4 |
2nd slot of Fridays, from 10:25 to 12:10.
| Date | Summary | Comments | |
|---|---|---|---|
| 1. | May 9 | General introduction to the course | |
| May 16 | canceled due to family reasons | ||
| 2. | May 23 | Classifying space for finite groups and group cohomology | |
| 3. | May 30 | Fermions and spinors | |
| June 6 | canceled due to family reasons | ||
| 4. | June 13 | Fermions and spinors | |
| 5. | June 20 | Fermions and spinors | |
| 6. | June 27 | Periodic tables of topological insulators and superconductors | |
| 7. | July 4 | Periodic tables of topological insulators and superconductors | |
| 8. | July 11 | Interacting fermionic invertible phases and bordism groups | |
| 9. | July 18 | Interacting fermionic invertible phases and bordism groups | |
| 10. | July 25 | Interacting fermionic invertible phases and bordism groups |
email: yuji.tachikawa_at_ipmu.jp