Math 264R: Quantum Topology (Spring 2024)

Tue Thu 10:30 AM - 11:45 AM at Science Center 309.

Office hours: by appointment at Science Center 231.

This course is an introduction to quantum topology, a branch of low-dimensional topology informed by Chern-Simons theory and its generalizations.

Tentative topics include: * Undergraduate students who would like to take this course for a grade need to submit a final paper at the end of the semester.

Tentative schedule

Date Topic Date Topic
1/23 Kauffman bracket and Jones polynomial, TQFT origin of skein relations 1/25 2d TQFTs and Frobenius algebras,
Temperley-Lieb algebroid $\mathrm{TL}$
1/30 Jones-Wenzl projectors, Temperley-Lieb-Jones algebroid 2/1 monoidal, braided, and ribbon categories
2/6 pivotality of ribbon categories, dimensions and traces, category of colored ribbon graphs 2/8 Reshetikhin-Turaev (RT) functor
2/13 algebras and coalgebras, convolution algebras, restricted dual of an algebra 2/15 bialgebras and Hopf algebras
2/20 braided and cobraided bialgebras, ribbon Hopf algebras 2/22 twisted products and coproducts, quantum double
2/27 Hopf pairing and generalized quantum double 2/29 quantized universal enveloping algebras $U_q(\mathfrak{sl}_N)$, modular categories
3/5 Witten-Reshetikhin-Turaev (WRT) invariants 3/7 Witten-Reshetikhin-Turaev (WRT) invariants (cont.)
3/12 No class (Spring recess) 3/14 No class (Spring recess)
3/19 Class canceled 3/21 Class canceled
3/26 skein modules and character varieties 3/28 Class canceled
4/2 quantum Teichmuller space, 2d quantum trace map 4/4 stated skein algebras of surfaces, splitting map,
bigons and $O_q(SL_N)$-comodule structures
4/9 Hochschild cohomology, reduced stated skein algebras,
stated skein modules of 3-manifolds
4/11 3d quantum trace map,
modified dimensions and renormalized RT invariants, Akutsu-Deguchi-Otsuki (ADO) invariants
4/16 ideals and modified traces, unrolled quantum group $\overline{U}_q^H(\mathfrak{sl}_2)$, $G$-grading, realization of an additive abelian group $Z$ in a ribbon category 4/18 relative $G$-modular categories, Costantino-Geer-Patureau (CGP) invariants, admissible skein modules
4/23 BPS $q$-series $\widehat{Z}$

Turaev - Quantum Invariants of Knots and 3-Manifolds (1994)
Kassel, Rosso & Turaev - Quantum Groups and Knot Invariants (1997)
Wang - Topological Quantum Computation (2010)
Kashaev - A Course on Hopf Algebras (2023)

...on stated skein modules and quantum trace maps:
Bonahon & Wong - Quantum traces for representations of surface groups in SL_2 (2010)
Costantino & Le - Stated skein algebras of surfaces (2019)
Panitch & Park - 3d quantum trace map (2024)

...on modified traces and non-semisimple TQFTs:
Geer, Patureau & Turaev - Modified quantum dimensions and re-normalized link invariants (2007)
Geer, Kujawa & Patureau - Generalized trace and modified dimension functions on ribbon categories (2010)
Costantino, Geer & Patureau - Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories (2012)
Costantino, Geer & Patureau - Some remarks on the unrolled quantum group of sl(2) (2014)
Costantino, Geer & Patureau - Admissible Skein Modules (2023)
Costantino, Geer, Haioun & Patureau - Skein (3+1)-TQFTs from non-semisimple ribbon categories (2023)

...on BPS $q$-series:
Gukov & Manolescu - A two-variable series for knot complements (2019)
Park - 3-manifolds, $q$-series, and topological strings (2022)

* Lecture notes will be updated and posted below as the course progresses:

* By the way, you might be interested in Complex Chern-Simons Theory Learning Seminar organized by Dan Freed and Constantin Teleman, happening this semester on Wednesdays at 4:30 PM - 6:00 PM ET at SC 310.