Abstracts

Nov. 29 (Saturday)

14:00 -- 14:10  Opening

14:10 -- 15:10  Margarida Camarinha (University of Coimbra)

Dynamic interpolation on Riemannian manifolds

The dynamic interpolation problem arose as an attempt to extend spline-based methods to Riemannian manifolds and led to the study of the so-called Riemannian cubic splines. This optimization problem has been investigated in different fields of research, from calculus of variations to  optimal control, with a wide range of applications in engineering, physics, and medicine, such as rigid body control in robotics, spacecraft control in aeronautics, quantum control in quantum information processing, 3D animation in computer graphics, or regression schemes for computational anatomy in medical imaging.

In this talk, we will survey the main results on dynamic interpolation. The second variational  problem associated with Riemannian cubics can be interpreted as an optimal control problem for an affine connection control system. We study the  optimal control problem in  Lie groups for an affine connection control system defined by a left-invariant distribution.

15:30 -- 16:30  Athanasios Chatzistavrakidis (Ruđer Bošković  Institute)

The Atiyah class of a dg-manifold and gauge theory

Motivated from target space covariant formulations of topological sigma models and from the graded-geometric approach to gauge theory, we discuss dg-manifolds admitting a compatible linear connection. The existence of a compatible connection is generally obstructed by a characteristic class, the Atiyah class of a dg-manifold. In this talk, we discuss the Atiyah class in the case of Lie algebroids and determine the conditions under which it vanishes. We study in detail the module of vector fields on graded manifolds of degree 1 and characterize the geometrical data contained in degree 0 connections on them. We emphasize the role of the basic curvature tensor and its importance in various problems in geometry and physics. Finally, we briefly comment on the extension of these results for Courant algebroids. Based on work in progress with Larisa Jonke and Dmitry Roytenberg. 

16:50 -- 17:50  Larisa Jonke (Ruđer Bošković Institute) 

Noncommutative gravity via classical double copy map

We exploit the idea of the classical double copy prescription connecting perturbative Yang-Mills theory with perturbative double field theory. Within this setting we compute the first nontrivial noncommutative corrections to the perturbative cubic double field theory action. The pure gravitation limit of this action in the transverse-traceless gauge leads to the noncommutative correction to the Einstein-Hilbert Lagrangian. This is joint work with Eric Lescano.

18:30 -- Discussion

Nov. 30 (Sunday)

Nov. 30 (Sunday)

10:00 -- 12:00  Short talk session

10:00 -- 10:25 Kohei Iwamoto (Ritsumeikan University)

Metrics for quandles and their Schreier graphs

A quandle is an algebraic structure that generalizes the conjugation operation in groups, and it has been studied from the viewpoints of knot theory and symmetric space theory. In geometric group theory, a basic object of study is the large-scale geometry of Cayley graphs associated with groups. In this talk, inspired by this point of view, we introduce a graph structure on connected components of quandles satisfying a certain finiteness condition and define metrics arising from this structure. We also present several examples of quandles that are quasi-isometric to typical metric spaces. This talk is based on joint work with Ryoya Kai (Osaka Metropolitan University) and Yuya Kodama (Kagoshima University).

10:30 -- 10:55 Yuya Nishimori (Institute of Science Tokyo)

Operator realization of quantum groups and deformation of module-algebra structures

This study aims to realize concretely the quantum groups $(U_q(g))$ within the Completed Weyl algebra $W_n[[h]]$ using differential operators. As our main results, we first constructed explicit embeddings for quantum groups of types $A, C$, and $D$ into the Weyl algebra $W_m[[h]]$. Notably, the realization for type C necessarily involves second-order q-differential operators. Furthermore, we determined the deformed product structure required to make the function space a module algebra over the Quantum groups. From the viewpoint of deformation quantization, we introduce bi-derivative operators $B$ in $W_m[[h]]^{\otimes 2}$ to provide an explicit formula for $B$ using q-exponentials and q-differential operators.

11:00 -- 11:25 Yohei Ota (Institute of Science Tokyo)

On the quantum groupoid structure of the Heisenberg double of $U_{\hbar}(\mathfrak{sl_2}^{+})$

Let $\mathfrak{g}$ be a Lie bialgebra and let $U_\hbar(\mathfrak{g})$ be a quantization of the universal enveloping algebra $U(\mathfrak{g})$. We denote by $D\mathfrak{g}$ the classical Drinfel'd double of $\mathfrak{g}$. The classical limit of the Drinfel'd double $D(U_\hbar(\mathfrak{g}))$ of $U_\hbar(\mathfrak{g})$ is then $U(D\mathfrak{g})$. In this construction, if we replace $D(U_\hbar(\mathfrak{g}))$ by the Heisenberg double $H(U_\hbar(\mathfrak{g}))$, the same argument no longer applies, since the Heisenberg double is not a Hopf algebra.

In general, for a Lie-Rinehart algebra $(A,L,\rho)$ one can define its universal enveloping algebra $U_A(L)$, which carries the structure of a left bialgebroid. Let $B_{\hbar}$ be a topological left bialgebroid whose limit $B$ as $\hbar \to 0$ is again a left bialgebroid. When $B$ is the universal enveloping algebra of some Lie-Rinehart algebra, we call $B_{\hbar}$ a quantum groupoid. The Heisenberg double $H(U_\hbar(\mathfrak{g}))$ admits a quantum groupoid structure in the following way.

Let $H$ be a Hopf algebra and let $A$ be a quantum commutative Yetter-Drinfel'd $H$-module algebra. By a theorem of Brzezi\'{n}ski-Militaru (B-M) [BM02], the smash product $A \sharp H$ carries a left bialgebroid structure. Let $G$ be a Poisson-Lie group with Lie bialgebra $\mathfrak{g}$. By the quantum duality principle [Gav07], there is a natural Hopf pairing between the quantized enveloping algebra $U_\hbar(\mathfrak{g})$ and the quantized function algebra $\mathcal{F}_\hbar(G)$. The Heisenberg double associated with this pairing is the smash product H(U_\hbar(\mathfrak{g})) = \mathcal{F}_\hbar(G) \sharp U_\hbar(\mathfrak{g}).

By the result of B-M, it therefore acquires a left bialgebroid structure. Moreover, for the transformation Lie-Rinehart algebra $H\mathfrak{g} = \mathcal{F}(G)\sharp \mathfrak{g}$, the universal enveloping algebra $U_{\mathcal{F}(G)}(H\mathfrak{g})$ is a left bialgebroid. In particular, $U_{\mathcal{F}(G)}(H\mathfrak{g})$ is the classical limit of $H(U_\hbar(\mathfrak{g}))$, so that $H(U_\hbar(\mathfrak{g}))$ becomes a quantum groupoid. In this situation, $H\mathfrak{g}$ carries the standard Lie-Rinehart bialgebra structure.

In this talk I will describe explicit computations of the quantum groupoid structure in the case of $H(U_{\hbar}(\mathfrak{sl}_2^{+}))$. This is joint work with Yuya Nishimori (Institute of Science Tokyo).

[BM02] Tomasz Brzezi\'{n}ski and Gigel Militaru, Bialgebroids, $\times_A$-bialgebras and duality, J. Algebra 251 (2002) no.1, 279--294, DOI 10.1006/jabr.2001.9101

[CG15] Sophie Chemla and Fabio Gavarini, Duality functors for quantum groupoids, J. Noncommut. Geom. 9 (2015) no.2, 287--358, DOI 10.4171/JNCG/194

[Gav07] Fabio Gavarini, The global quantum duality principle, J. Reine Angew. Math. 612 (2007) 17--33, DOI 10.1515/CRELLE.2007.082

[Lu94] Jiang-Hua Lu, On the Drinfel'd{} double and the Heisenberg double of a Hopf algebra,  Duke Math. J. 74 (1994) no.3, 763--776,  DOI 10.1215/S0012-7094-94-07428-0

[Lu96] Jiang-Hua Lu, Hopf algebroids and quantum groupoids, Internat. J. Math. 7 (1996) no.1, 47--70, DOI 10.1142/S0129167X96000050

[Sto24] Martina Stoji\'c, Scalar extension Hopf algebroids, J. Algebra Appl. 23 (2024) no.6, Paper No. 2450114, 34, DOI 10.1142/S0219498824501147

11:30 -- 11:55 Wenda Fang (Kyoto University)

Chiralization of classical r-matrices and a generalized AKS integrability scheme via vertex algebras

In this talk, I will outline a generalized AKS scheme of integrability recently introduced by the speaker (IMRN, 2025). Drawing on ideas from graded manifolds, I will show that the classical R-matrix for Lie conformal algebras can be described via the Maurer-Cartan equation of a differential graded Lie algebra.

14:00 -- 15:00  Hitoshi Konno (Tokyo University of Marine Science and Technology)

Elliptic Quantum Groups and Related Geometry

After making some introductory remarks on the elliptic quantum group $U_{q,p}(\widehat{\mathfrak{g}})$,  we discuss its toroidal algebra version  $U_{t_1,t_2,p}({\mathfrak{gl}}_{1,tor})$ and its applications in mathematical physics related to a geometry of the Jordan quiver variety ${\cal M}(n,r)$ i.e. the ADHM instanton moduli space. We especially emphasize a use of the two vertex operators associated with the  two different co-algebra structures defined by  the Drinfeld comultiplication $\Delta^D$ and the standard comultiplication $\Delta$, respectively.  We first show that the vertex operator of $U_{t_1,t_2,p}({\mathfrak{gl}}_{1,tor})$ w.r.t $\Delta^D$ realizes the Jordan quiver $W$-algebras associated with ${\cal M}(n,r)$ ( operator version of Nekrasov's $qq$-characters) introduced by Kimura-Pestun and is used to calculate  instanton partition functions of the $5d$ and $6d$ lifts of the $4d$ ${\cal N}=2^*$ SUSY gauge theories. Secondly, we show that the vertex operator of the same algebra  w.r.t $\Delta$ can be constructed in a consistent way to Okounkov's geometric formulation of the elliptic quantum groups in terms of  the elliptic stable envelopes and is used to calculate the K-theoretic vertex functions for  ${\cal M}(n,r)$, which have many interesting applications such as the 3d mirror symmetry and a formulation of the  quantum K-theory.   

15:20 -- 16:20 Hiroaki Kanno (Nagoya University)

Instanton counting on the blow-up and the super Ding-Iohara-Miki algebra

In the quiver description of the moduli space of instantons on the blow-up of $\mathbb{C}^2$ or $\mathbb{P}^2$, the fixed points of the torus action are labeled by super partitions.  From the equivariant character at a fixed point, we can define the Nekrasov factor for a pair of super partitions. On the other hand, the set of super partitions provides a basis of a representation of the super Ding-Iohara-Miki algebra (the quantum toroidal algebra of type $\mathfrak{gl}_{1|1}$), which simultaneously diagonalizes the Cartan generators. We show the matrix elements of the raising operators of the algebra are related to the variation of the Nekrasov factor, which suggests the existence of a natural action of the super Ding-Iohara-Miki algebra on the K-theory group of the instanton moduli space.