An introduction to fusion categories and their applications to physics. Students will learn to work with fusion categories and their algebraic precursors with an emphasis on graphical and computational methods with the goal of preparing for research in theoretical/mathematical physics. Applications to be covered may include but are not limited to: anyon models and topological phases of matter in (2+1)D, topological quantum field theory, generalized symmetries in (1+1)D, quantum error correction, topological quantum computation, superselection theory, and (1+1)D conformal field theory.
Click here to download the public-facing syllabus. See Brightspace for the student-facing syllabus.
The pace of the course will be based on the participants and the schedule will be adjusted accordingly. Suggestions for further reading to reinforce each week's lecture content will be posted as the course progresses.
| Tuesday | Thursday | Supporting Reading | |
| Week 1 |
8/26. Course overview |
8/28. Groups, Invertible Symmetries, and Categories (Part I) |
I.1-I.2 in Zee's Group Theory in a Nutshell for Physicists Chapters 1-3 in Abstract Algebra, Dummit and Foote Chapter 1 in Category Theory in Context, Riehl |
| Week 2 |
9/2. Groups, Invertible Symmetries, and Categories (Part II)
|
9/4. The \( \mathbb{Z}_2 \) toric code (Part I) |
Section 1 and 2 of Kitaev's Fault-tolerant quantum computation by anyons Section 3.1 of Kong and Zhang's An invitation to topological orders and category theory Section 2 of Purdue Professor Shawn Cui's lecture notes on Topological Quantum Computation Chapter 27 of Simon's Topological Quantum (Chapter 24 in the Proto-Book) |
| Week 3 |
9/4. The \( \mathbb{Z}_2 \) toric code (Part II)
|
9/9. Representation theory, Noninvertible Symmetries, and Fusion Rings (Part I) |
|
| Week 4 |
9/9. Representation theory, Noninvertible Symmetries, and Fusion Rings (Part II) |
9/11. Kitaev Quantum Double Model (Part I) |
|
| Week 5 |
9/16. Fusion Categories, TQFTs, and Anyons (Part I) |
9/18. Fusion Categories, TQFTs, and Anyons (Part II) Unit 2 Exercises Due |
|
| Week 6 |
9/23. Fusion Categories, TQFTs, and Anyons (Part III) |
9/25. Levin-Wen Models |
|
| Week 7 |
9/30. Turaev-Viro-Barrett-Westbury state-sum TQFT |
10/2. Fusion Categories, TQFTs, and Anyons (Part IV) Unit 3 Exercises Due |
|
| Week 8 |
10/7. No class (Fall Break) |
10/9. Fusion Categories, TQFTs, and Anyons (Part V) |
|
| Week 9 |
10/14. Topological Quantum Computation (Part I) |
10/16. Topological Quantum Computation (Part II) Unit 4 Exercises Due |
|
| \(\vdots\) |
\(\vdots\) |
\(\vdots\) |
\(\vdots\) |
| Week 15 |
12/2. Advanced Topics Presentations |
12/4. Advanced Topics Presentations |
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| Week 16 |
12/9. Advanced Topics Presentations |
12/11. Advanced Topics Presentations |