Schedule for: 25w5422 - Women in Noncommutative Algebra and Representation Theory 4 (WINART4)

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Meet and Greet at BIRS Lounge (PDC 2nd Floor)

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Subfactor planar algebras first were constructed by Vaughan Jones as a diagrammatic axiomatization of the standard invariant of a subfactor. Planar algebras can be conveniently encoded by diagrams in the plane. These diagrams satisfy some skein relations and have an invariant called an index. The Kuperberg Program asks to find all diagrammatic presentations of subfactor planar algebras. This program has been completed for index less than 4. In this talk, I will introduce subfactor planar algebras and find presentations for subfactor planar algebras of index 4 associated with the affine A Dynkin diagram. Then I will show that categories arising from these planar algebra presentations are also describing categories of representations.

$W_{1+\infty}$ is the central extension of the Lie algebra of differential operators on the circle. It has applications in integrable systems and two-dimensional quantum field theory. There is an interest in the action of this algebra on symmetric functions coming from the studies of such integrable systems as, for example, the KP hierarchy. Our goal is to describe the action on the basis of symmetric functions in terms of generating functions of multiplication operators.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In 2006, BMRRT introduced us to the wonderful world of cluster categories. In 2015, Igusa and Todorov gifted us continuous cluster categories. Until now, these constructions have largely remained separate. Last year, Paquette-R-Yıldırım introduced generalized thread quivers, which introduce non-discreteness to the notion of thread quivers, which come from Berg and van Roosmalen. In this ongoing work, we discuss how to smash the continuous and discrete worlds together and obtain something that sits inbetween. Ongoing joint work with Charles Paquette and Emine Yıldırım.

We begin by recalling the combinatorial definitions by generators and relations of cluster algebras of type A, B and C with principal coefficients in a triangulation of a regular polygon. Then, we present a formula expressing cluster variables of type B and C in terms of cluster variables of type A. This formula allows us to provide the cluster expansion of cluster variables of type B and C in terms of perfect matchings of certain modified snake graphs. Finally, we associate a symmetric quiver Q with any cluster of these cluster algebras. In this framework, cluster variables of type B (resp. C) correspond to orthogonal (resp. symplectic) indecomposable representations of Q.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Grassmannian Gr(k,n) of k-planes in an n-dimensional space is a well-loved algebraic variety and seems to be the keeper of many fascinating combinatorial problems. One way Gr(k,n) can be endowed with a cluster algebra structure is through the combinatorics of plabic graphs. Though its cluster algebra structure is defined combinatorially, generators and bases for these algebras are not well understood for k \geq 3. We will explore how webs seem to be the missing piece of combinatorics, focusing on k=3 and 4, specifically using the new machinery of Gaetz, Pechenick, Pfannerer, Striker, Swanson hourglass webs. In particular, we will discuss web duality, as defined by Fraser, Lam and Le, and show how it can be used to understand Laurent expansions of cluster variables as generating functions of higher dimer covers. This will be based on joint work with two other WINART participants: Esther Banaian and Elise Catania, as well as Christian Gaetz, Miranda Moore, and Gregg Musiker.

Abstract: Given a finite poset, Greene introduced a rational function obtained by summing certain rational functions over the linear extensions of the poset. This function has interesting interpretations, and for certain families of posets, it simplifies surprisingly. In particular, Greene evaluated this rational function for strongly planar posets in his work on the Murnaghan–Nakayama formula. In 2012, Develin, Macauley, and Reiner introduced toric posets, which combinatorially are equivalence classes of posets (or rather acyclic quivers) under the operation of flipping maximum elements into minimum elements and vice versa. In this work, we introduce a toric analogue of Greene's rational function for toric posets, and study its properties. In addition, we use toric posets to show that the Kleiss–Kuijf relations, which appear in scattering amplitudes, are equivalent to a specific instance of Greene's evaluation of his rational function for strongly planar posets. Also in this work, we give an algorithm for finding the set of toric total extensions of a toric poset.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.