## Graduate Quantum Algebra and Topology Seminar at UC Santa Barbara

### Organizers: Quinn Kolt and Melody Molander

The Graduate UCSB Quantum Algebra and Topology Seminar is a weekly seminar for graduate students and postdocs. Typically, every quarter we pick a topic to learn about and have weekly discussions. We also occasionally have speakers, typically on Zoom, to speak on their current research. These talks are for graduate students, postdocs, and faculty and are indicated in the below schedules by a *.

### Fall 2023

#### Reference: "Quantum Theory for Mathematicians" by Hall

## Schedule

- 10/13: 2.1 "Motion in \(\mathbb{R}\)" (Melody Molander)
- 10/20: 2.2-2.3 "Motion in \(\mathbb{R}^n\) & Systems of Particles" (Daniel Epelbaum)
- 10/27: 2.4-2.5 "Angular Momentum & Poisson Brackets and Hamiltonian Mechanics" (Quinn Kolt)
- 11/03: 2.6 "The Kepler Problem and Runge-Lenz Vector" (Alex Sabater)
- 11/17: 3.1-3.2 "Waves, Particles, and Probabilities & A Few Words about Operators and their Adjoints" (Ka Lok Lam)
- 12/08: 3.6 "Axioms of Quantum Mechanics: Operators and Measurements" (Ka Lok Lam)

### Spring 2023

#### Topic: Homological Representations of the Braid Groups

#### Reference: "Braid Groups", by Kassel and Turaev

## Schedule

- 04/05: An Introduction to the Burau Representation (Melody Molander)
- 04/12: The Burau Representation (Melody Molander)
- 04/19: Nonfaithfulness of the Burau representation (Choom Moos)

### Winter 2023

#### Topic: Braid Groups

#### Reference: "Braid Groups", by Kassel and Turaev

## Schedule

- 01/09: 1.1.1-1.2.3 (Melody Molander)
- 01/16: MLK Jr. Day (No Talk)
- 01/23: 1.2.4-1.3.2 (John White)
- 01/30: 1.3.3-1.4.4 (Sam Sehayek)
- 02/06: 1.5.1-1.7.1 (Troy Kling)
- 02/13: 1.7.2-2.1.3 (Choom Moos)
- 02/20: President's Day (No Talk)
- 02/27: Braids, knots, and links (John White)
- 03/06: 2.4-2.6 (John White)
- *03/13: Quinn Kolt (UCSB),
*Monoidal, Rigid, and Fusion Categories*

### Fall 2022

#### Topic: An Overview of Quantum Algebra

#### Reference: "Exercises in Quantum Algebra" by Penneys

## Schedule

- 09/28: The Temperley-Lieb algebras (Melody Molander)
- 10/05: The Kauffman bracket & 3.1.1 Basic facts about \(\text{M}_n(\mathbb{C})\) (Melody Molander)
- 10/12: Exploring Finite-Dimensional *-Algebras (Daniel Halmrast)
- 10/19: Exploring Finite-Dimensional *-Algebras (Part 2) (Daniel Halmrast)
- 10/26: Jones' Basic Construction in Finite Dimensions (Daniel Halmrast)
- *11/02: Melody Molander (UCSB),
*Diagrammatic Presentation of Index 4 Subfactor Planar Algebras* - 11/09: An Exploration of Linear Categories (Jeremy Khoo)
- 11/16: Cauchy Completion and Semisimplicity (Quinn Kolt)

### Spring 2022

## Schedule

- *04/12: Patrick Vecera (UCSB),
*A Mathematical Guide to Classical and Quantum Gauge Theories* - *04/19: Kevin Walker (Microsoft Station Q),
*Higher-dimensional Temperley-Lieb, categorified idempotent completion, and incompressible surfaces in 3-manifolds* - *04/26: Patrick Vecera (UCSB),
*From Chern-Simons to Quantum Topology* - *05/03: Joseph Melby (Michigan State),
*The Turaev-Viro Invariants and their Asymptotics* - *05/10: Benedict Lee (UCSB),
*Kauffman Bracket Skein Algebra and itâ€™s Quantum Trace Map* - *05/31: Anup Poudel (Iowa),
*From Sikora's Webs to CKM Relations*

### Winter 2022

#### Topic: 3-manifold Invariants

#### Reference: "The 3-manifold invariants of Witten & Reshetikhin-Turaev for sl(2,\(\mathbb{C})\)" by Kirby and Melvin

## Schedule

- 01/13: 0. Introduction and 1. The invariants of Reshetikhin and Turaev
- 01/20: 2.1 The Algebra A
- 01/27: 2.2 Representations of A
- *02/03: Patrick Vecera (UCSB),
*A Survey of Physics, Ladder Operators, and the Relation of These Topics to Quantum Topology* - 02/10: 2.3 The R-matrix
- *02/15: Eric Rowell (Texas A&M),
*Charge Conserving Yang-Baxter Operators* - 02/17: 3.1 Tangles and 3.2 Colored Framed Tangle Operators
- 02/24: 3.3 Properties of Tangles and 3.4 Orientations
- 03/03: 3.5 Tangle Operators for A and 4.1 Skein Theory, the Jones Polynomial, & the Kauffman Bracket
- *03/08: Colleen Delaney (Indiana University Bloomington),
*Zesting and Witten-Reshetikhin-Turaev Invariants* - 03/10: 4.2 Cabling and 4.3 The Symmetry Principle

### Fall 2021

#### Topic: Quantum Invariants

#### Reference: "Quantum Invariants of Knots and 3-Manifolds" by Turaev

## Schedule

- 10/06: Ch 1.1 Ribbon categories
- 10/13: Ch 1.2 Operator invariants of ribbon graphs
- 10/20: Ch 2.1 Modular tensor categories and Ch 2.2 Invariants of 3-manifolds
- 10/27: Ch 2.3.1 Kirby calculus and Ch 2.4 Computations in semisimple categories
- 11/03: Ch 2.5 Hermitian and unitary categories and Ch 3.1 Axiomatic definition of TQFT's
- 11/10: Ch 3.2 Fundamental properties and Ch 3.3 Isomorphisms of TQFTs
- *11/17: Jieru Zhu (Okinawa Institute of Science and Technology),
*The \(SL_2\) Tilting Category in Mixed Characteristic* - *12/01: Jin-Cheng Guu (Stony Brook University),
*(Higher) Category-valued Manifold Invariants*