PIMS Geometry, Algebra and Physics Seminar

Organized by Alex Weekes, Steve Rayan and Curtis Wendlandt

Schedule (2021-2022)

Talks are normally held on Thursdays at 4:00-5:30pm CST, and each can be attended virtually via Zoom (including those talks held in-person at the University of Saskatchewan) using the following information:

Meeting ID	Passcode	Link
936 6357 4827	qUSaskGAP	Here

The seminar is intended to be informal and accessible to graduate students studying algebra, geometry, or mathematical physics. Speakers typically prepare an hour-long talk, with the remaining half hour dedicated to discussion and questions.

Date/Location	Speaker	Title and Abstract
February 17th (Zoom Only)	Daniel Douglas	Tropical Fock-Goncharov coordinates for \(\mathrm{SL}_3\)-webs on surfaces For a finite-type surface \(\mathfrak{S}\), we study a preferred basis for the commutative algebra \(\mathbb{C}[\mathcal{R}_{\mathrm{SL}_3}(\mathfrak{S})]\) of regular functions on the \(\mathrm{SL}_3(\mathbb{C})\)-character variety, introduced by Sikora-Westbury. These basis elements come from the trace functions associated to certain tri-valent graphs embedded in the surface \(\mathfrak{S}\). We show that this basis can be naturally indexed by positive integer coordinates, defined by Knutson-Tao rhombus inequalities and modulo 3 congruence conditions. These coordinates are related, by the geometric theory of Fock-Goncharov, to the tropical points at infinity of the dual version of the character variety. This is joint work with Zhe Sun.
March 24th (Zoom Only)	Anne Dranowski	*Computing fusion products of MV cycles using the Mirkovic–Vybornov isomorphism*The fusion of two subvarieties in the affine Grassmannian is a degeneration of their product, defined using the Beilinson-Drinfeld Grassmannian. In this talk, we present a conceptually elementary approach to computing this product in type A when the varieties are MV cycles. As an application, we can compute all cluster exchange relations in the coordinate ring of the upper-triangular subgroup of \(\mathrm{GL}_4\), confirming that all the cluster variables are contained in the basis of MV cycles.
Friday, March 25th (Colloquium) *3:30-4:30pm* (Zoom Only)\(^\star\)	David Hernandez	Categories of representations, spectra of quantum integrable systems and Coulomb branches The structure of the eigenvalues of a quantum system, that is of its spectrum,is crucial to its study. The spectrum of the "ice model" (six vertex model)was computed in the seminal work of Baxter. It has a remarkable form involvingpolynomials and the famous Baxter relations. Later, it was conjectured that thereis an analog form for the spectrum of more general quantum integrable systems.We will discuss how, using the modern mathematical point of view of representation theory,these (Baxter) polynomials occur in a natural way. Besides, Baxter relations can becategorified using relevant categories of representations of quantum groups. This leads toa proof of the general conjecture (joint results with Jimbo and with Frenkel).We will also discuss other recent applications for the representation theory oftruncated quantum groups (quantized Coulomb branches), in the framework ofsymplectic duality (3d mirror symmetry).
Thursday, May 12th (Hybrid) THORV 124	Bryan Kettle	Super Yangians: Where We Are Today Given any finite-dimensional simple Lie algebra \(\mathfrak{a}\) over \(\mathbb{C}\), the Yangian \(\mathbf{Y}(\mathfrak{a})\) is a certain unital associative \(\mathbb{C}\)-algebra. In particular, Yangians form a family of so-called quantum groups. The main property these algebras is the foundational fact that their representations produce rational solutions to the quantum Yang-Baxter equation. The structure and representation theory of Yangians has become a study in and of itself and has expanded to the study of super Yangians based on Lie superalgebras; however, the theory of super Yangians is comparatively less developed than its non-super counterpart. In this talk, we will survey what recent advancements have been made in the study of super Yangians and view what else remains to do.
Thursday, May 26th (Hybrid) THORV 124	Matthew Rupert	Constructing braided categories associated to logarithmic vertex algebras Unrolled quantum groups are a family of quantum algebras closely related to the non-restricted specialization of Drinfeld-Jimbo algebras at roots of unity. These quantum groups have played a pivotal role in the study of the Kazhdan-Lusztig correspondence, that is, the study of equivalences between module categories over quantum groups and logarithmic vertex operator algebras. In this talk I will introduce unrolled quantum groups, describe their role in the Kazhdan-Lusztig correspondence, and state some recent results.
Monday, May 30th (Hybrid) THORV 124	Sasha Tsymbaliuk	BGG-type relations for transfer matrices of rational spin chains and the shifted Yangians In this talk, I will discuss: (1) the new BGG-type resolutions of finite dimensional representations of simple Lie algebras that lead to BGG-relations expressing finite-dimensional transfer matrices via infinite-dimensional ones, (2) the factorization of infinite-dimensional ones into the product of two Q-operators, (3) the construction of a large family of rational Lax matrices from antidominantly shifted Yangians. This talk is based on the joint works with R. Frassek, I. Karpov, and V. Pestun.
Thursday, July 14th (Hybrid) THORV 124	Raphaël Belliard	Pushing around loop equations The term “loop equations” was coined by Migdal in the early 90's in the context of quark confinement to refer to certain sets of equations describing the propagation of test particles in quantum field theory. They have since appeared many times at the boundary between mathematics and physics. From statistical spectroscopy to enumerative geometry via integrable hierarchies, they encode conformal features of various problems in an analytic way. The prototypical situation where loop equations are valuable is that of matrix integrals, with applications ranging from all the aforementioned topics to number theory and even artificial neural networks. A seemingly antipodal approach to matrix integrals is that of orthogonal polynomials, unveiling a subtle relationship between loop equations and the geometry of meromorphic connections. This bridge can be abstracted in terms of W-algebra constraints and we shall cross it back and forth.
Thursday, August 18th (Hybrid) THORV 124	Monica Jinwoo Kang	A universal formula for the density of states with global symmetry It has long been the case that representation theory played an important role in understanding the spectrum of states in physics. I will describe how it can be used to describe some universal aspect of d-dimensional unitary conformal field theories with a global symmetry group, which can be a discrete group or a compact Lie group.

Date/Location

Speaker

Title and Abstract

February 17th

(Zoom Only)

Daniel Douglas

Tropical Fock-Goncharov coordinates for \(\mathrm{SL}_3\)-webs on surfaces

For a finite-type surface \(\mathfrak{S}\), we study a preferred basis for the commutative algebra \(\mathbb{C}[\mathcal{R}_{\mathrm{SL}_3}(\mathfrak{S})]\) of regular functions on the \(\mathrm{SL}_3(\mathbb{C})\)-character variety, introduced by Sikora-Westbury. These basis elements come from the trace functions associated to certain tri-valent graphs embedded in the surface \(\mathfrak{S}\). We show that this basis can be naturally indexed by positive integer coordinates, defined by Knutson-Tao rhombus inequalities and modulo 3 congruence conditions. These coordinates are related, by the geometric theory of Fock-Goncharov, to the tropical points at infinity of the dual version of the character variety. This is joint work with Zhe Sun.

March 24th

(Zoom Only)

Anne Dranowski

Computing fusion products of MV cycles using the Mirkovic–Vybornov isomorphismThe fusion of two subvarieties in the affine Grassmannian is a degeneration of their product, defined using the Beilinson-Drinfeld Grassmannian. In this talk, we present a conceptually elementary approach to computing this product in type A when the varieties are MV cycles. As an application, we can compute all cluster exchange relations in the coordinate ring of the upper-triangular subgroup of \(\mathrm{GL}_4\), confirming that all the cluster variables are contained in the basis of MV cycles.

Friday, March 25th (Colloquium)

3:30-4:30pm

(Zoom Only)\(^\star\)

David Hernandez

Categories of representations, spectra of quantum integrable systems and Coulomb branches

The structure of the eigenvalues of a quantum system, that is of its spectrum,is crucial to its study. The spectrum of the "ice model" (six vertex model)was computed in the seminal work of Baxter. It has a remarkable form involvingpolynomials and the famous Baxter relations. Later, it was conjectured that thereis an analog form for the spectrum of more general quantum integrable systems.We will discuss how, using the modern mathematical point of view of representation theory,these (Baxter) polynomials occur in a natural way. Besides, Baxter relations can becategorified using relevant categories of representations of quantum groups. This leads toa proof of the general conjecture (joint results with Jimbo and with Frenkel).We will also discuss other recent applications for the representation theory oftruncated quantum groups (quantized Coulomb branches), in the framework ofsymplectic duality (3d mirror symmetry).

Thursday, May 12th

(Hybrid)

THORV 124

Bryan Kettle

Super Yangians: Where We Are Today

Given any finite-dimensional simple Lie algebra \(\mathfrak{a}\) over \(\mathbb{C}\), the Yangian \(\mathbf{Y}(\mathfrak{a})\) is a certain unital associative \(\mathbb{C}\)-algebra. In particular, Yangians form a family of so-called quantum groups. The main property these algebras is the foundational fact that their representations produce rational solutions to the quantum Yang-Baxter equation. The structure and representation theory of Yangians has become a study in and of itself and has expanded to the study of super Yangians based on Lie superalgebras; however, the theory of super Yangians is comparatively less developed than its non-super counterpart. In this talk, we will survey what recent advancements have been made in the study of super Yangians and view what else remains to do.

Thursday, May 26th

(Hybrid)

THORV 124

Matthew Rupert

Constructing braided categories associated to logarithmic vertex algebras

Unrolled quantum groups are a family of quantum algebras closely related to the non-restricted specialization of Drinfeld-Jimbo algebras at roots of unity. These quantum groups have played a pivotal role in the study of the Kazhdan-Lusztig correspondence, that is, the study of equivalences between module categories over quantum groups and logarithmic vertex operator algebras. In this talk I will introduce unrolled quantum groups, describe their role in the Kazhdan-Lusztig correspondence, and state some recent results.

Monday, May 30th

(Hybrid)

THORV 124

Sasha Tsymbaliuk

BGG-type relations for transfer matrices of rational spin chains and the shifted Yangians

In this talk, I will discuss: (1) the new BGG-type resolutions of finite dimensional representations of simple Lie algebras that lead to BGG-relations expressing finite-dimensional transfer matrices via infinite-dimensional ones, (2) the factorization of infinite-dimensional ones into the product of two Q-operators, (3) the construction of a large family of rational Lax matrices from antidominantly shifted Yangians. This talk is based on the joint works with R. Frassek, I. Karpov, and V. Pestun.

Thursday, July 14th

(Hybrid)

THORV 124

Raphaël Belliard

Pushing around loop equations

The term “loop equations” was coined by Migdal in the early 90's in the context of quark confinement to refer to certain sets of equations describing the propagation of test particles in quantum field theory.

They have since appeared many times at the boundary between mathematics and physics. From statistical spectroscopy to enumerative geometry via integrable hierarchies, they encode conformal features of various problems in an analytic way.

The prototypical situation where loop equations are valuable is that of matrix integrals, with applications ranging from all the aforementioned topics to number theory and even artificial neural networks.

A seemingly antipodal approach to matrix integrals is that of orthogonal polynomials, unveiling a subtle relationship between loop equations and the geometry of meromorphic connections. This bridge can be abstracted in terms of W-algebra constraints and we shall cross it back and forth.

Thursday, August 18th

(Hybrid)

THORV 124

Monica Jinwoo Kang

A universal formula for the density of states with global symmetry

It has long been the case that representation theory played an important role in understanding the spectrum of states in physics. I will describe how it can be used to describe some universal aspect of d-dimensional unitary conformal field theories with a global symmetry group, which can be a discrete group or a compact Lie group.

\(^\star\) This talk will be held in the Colloquium Zoom meeting room. Please contact the organizers for access information.