See below for a course description.
| Points at least (%) | Guaranteed grade |
|---|---|
| 97 | A+ |
| 93 | A |
| 90 | A- |
| 87 | B+ |
| 83 | B |
| 80 | B- |
| 77 | C+ |
| 73 | C |
| 70 | C- |
| 67 | D+ |
| 63 | D |
| 60 | D- |
Fusion categories, quantum groups, and quantum invariants of knots and 3-manifolds form a remarkably deep triangle of ideas at the intersection of algebra, topology, and physics. Classically, the semisimple framework of fusion categories has played a central role in producing powerful invariants of knots and 3-manifolds, such as the Jones polynomial. However, to push beyond existing boundaries, it has become increasingly important to generalize these ideas to non-semisimple settings for compelling reasons.
First, the representation categories of quantum groups at roots of unity are not automatically semisimple. Second, it has been shown that 3-manifold invariants derived from non-semisimple tensor categories often capture more subtle and powerful information than their semisimple counterparts. Finally, in dimension four, quantum invariants constructed from semisimple categories fail to distinguish smooth structures on 4-manifolds.
This course introduces the construction of quantum invariants of knots and manifolds from categories that are not necessarily semisimple, with the semisimple case appearing naturally as a special instance. The first half of the course develops the algebraic foundations, beginning with (locally) finite Abelian categories and adding structures such as tensor products, duality, braiding, and twists, while covering key topics including projective covers, projective generators, modified quantum traces, and chromatic morphisms. The second half shifts to a more topological perspective, exploring skein modules on surfaces and 3-manifolds, and constructing quantum invariants of knots, 3-manifolds, and 4-manifolds from finite tensor categories. More generally, the course will show how such constructions give rise to topological quantum field theories that capture richer information than invariants alone, with classical theories such as Reshetikhin-Turaev and Crane-Yetter appearing along the way; background on surgery and handle decompositions of manifolds will also be included.
The course is designed to be largely self-contained, though familiarity with basic category theory (functors, natural transformations), module theory (representations of groups and algebras), and basic topology (manifolds, knots) will be helpful.
Purdue University strives to make learning experiences accessible to all participants. If you anticipate or experience physical or academic barriers based on disability, you are welcome to let me know so that we can discuss options. You are also encouraged to contact the Disability Resource Center at: drc@purdue.edu or by phone: 765-494-1247.
If you have been certified by the Disability Resource Center (DRC) as eligible for accommodations, you should contact your instructor to discuss your accommodations as soon as possible. Here are instructions for sending your Course Accessibility Letter to your instructor: https://www.purdue.edu/drc/students/course-accessibility-letter.php