# Vertex Models and Special Functions Seminar Fall 2021

** Last Updated:**

The goal of this seminar is to understand some examples in which vertex models give insight on the study of special functions, especially Macdonald polynomials.

### Time and location information

Organisers: Sam DeHority, Zoe Himwich, Davis Lazowski

Time/date: Thursdays 5-6pm EST

Location: Hybrid in Math 528 / on Zoom

Please email himwich@math.columbia.edu if you would like to be on the mailing list.

### Plan/references

First, we will cover necessary background on MacDonald polynomials, vertex models and DAHA, following (for instance)

- Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux, Ram
- Algebraic analysis of solvable lattice models, Jimbo–Miwa
- Double Affine Hecke Algebras, Cherednik

Then we will focus on the box-ball system, following

- Integrable structure of box-ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry, Inoue–Kuniba–Takagi

Afterwards, depending on attendee interest, we will move on to several other related systems, for instance those of

- From Multiline Queues to Macdonald Polynomials via the Exclusion Process, Corteel–Mandelshtam–Williams
- Coloured Stochastic Vertex Models and their Spectral Theory, Borodin–Wheeler
- Nonsymmetric Macdonald Polynomials via Integrable Vertex Models, Borodin–Wheeler

### Virtual attendance

This seminar will be hybrid. If you are interested in attending virtually, please let us know when you ask to be on the mailing list. (If interest from attendees at other universities is especially high, it’s possible that the seminar might move entirely online.)

### List of talks

Date | Talk Info |
---|---|

Thursday Sep. 9 |
Speaker: Zoe Himwich Title: DAHA and Macdonald Polynomials |

Thursday Sep. 16 |
NO TALK |

Thursday Sep. 23 |
Speaker: Zoe Himwich Title: Continued |

Thursday Sep. 30 |
Speaker: Davis Lazowski Title: Crystals and box-ball systems Abstract: I will give a basic introduction to crystals and how they arise from the theory of quantum groups. Then we’ll use that to study associated dynamical systems, notably the box-ball system. |

Thursday Oct. 7 |
Speaker: Sam DeHority Title: The Toda lattice and its ultradiscretization Abstract: The Toda lattice is an example of a classical integrable system. I will describe this system and some of its solutions, then discuss its ultradiscretization, which is a certain tropical limit of the system and is related to box ball systems. Notes: pdf |

Thursday Oct. 14 |
Doubleheader First Sam continues from last week, and then Davis will speak on the following “I will briefly prove that the combinatorial R matrix describes soliton-soliton scattering” |

Thursday Oct. 21 |
Speaker: Zoe Himwich Title: The 6-Vertex Model and ASEP Abstract: I will define vertex models and ASEP, show the correspondence between the Stochastic 6 Vertex Model’s height function and ASEP. If time permits, I will start to introduce https://arxiv.org/abs/1904.06804. |

Thursday Oct. 28 |
Speaker: Zoe Himwich Title: Multiline Queues and MacDonald Polynomials Abstract: I will begin to go through a paper of Corteel, Mandelshtam, and Williams which uses multiline queues to produce a new formula for Macdonald polynomials. |

Thursday Nov. 4 |
Speaker: Zoe Himwich Title: Multiline Queues and MacDonald Polynomials Abstract: I will continue my presentation of a paper of Corteel, Mandelshtam, and Williams which uses multiline queues to produce a new formula for Macdonald polynomials. |

Thursday Nov. 11 |
Speaker: Davis Lazowski Title: A vertex model for MacDonald polynomials Abstract: I will explain how a matrix product ansatz for stationary states of the asymmetric exclusion process motivated a vertex model construction of MacDonald polynomials. |

Thursday Nov. 18 |
Speaker: Sam DeHority Title: Towards R matrices associated to elliptic surfaces Abstract: We will discuss work in progress which produces solutions to the Yang-Baxter equation from moduli spaces associated to elliptic surfaces. The relevant geometric objects will be introduced and then an outline of the R matrix construction will be presented. |

Thursday Dec. 2 |
NO TALK |

Thursday Dec. 9 |
Speaker: Sam DeHority Title: Towards R matrices associated to elliptic surfaces II Abstract: Continuing from last time, we will discuss work in progress which produces solutions to the Yang-Baxter equation from moduli spaces associated to elliptic surfaces. The relevant geometric objects will be introduced and then an outline of the R matrix construction will be presented. |

Thursday Dec. 16 |
Speaker: Davis Lazowski Title: Baxter’s Q-operator and representation theory Abstract: I will explain how Baxter’s Q-operator can be understood in terms of prefundamental modules for the shifted quantum affine algebra. |