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.
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. I haven't decided the deadline, but it'll be around the last week of January.
The latest stable version of the lecture notes is here. (Last update: Dec 18) The unstable version is kept at this git repository.
Please also see the course page provided by the university. The Zoom info, the link to the Slack channel, and the YouTube channel with recordings can be found there.
You'll need to submit the term-end report through the link above.
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 | |
| Jan. 2 | no class due to New Year Holidays | ||
| 13. | Jan. 9 | Abelian Chern-Simons terms and the integer quantum Hall effect I | |
| 14. | Jan. 16 | Abelian Chern-Simons terms and the integer quantum Hall effect II |
email: yuji.tachikawa_at_ipmu.jp